Experiments in Stochastic Thermodynamics: Short History and Perspectives

Experiments in Stochastic Thermodynamics: Short History and Perspectives PHYSICAL REVIEW X 7, 021051 (2017) S. Ciliberto Université de Lyon, CNRS, École Normale Supérieure de Lyon, Laboratoire de Physique (UMR5672), 46 Allée d’Italie 69364 Lyon Cedex 07, France (Received 5 April 2017; published 30 June 2017) We summarize in this article the experiments which have been performed to test the theoretical findings in stochastic thermodynamics such as fluctuation theorem, Jarzynski equality, stochastic entropy, out-of- equilibrium fluctuation dissipation theorem, and the generalized first and second laws. We briefly describe experiments on mechanical oscillators, colloids, biological systems, and electric circuits in which the statistical properties of out-of-equilibrium fluctuations have been measured and characterized using the abovementioned tools. We discuss the main findings and drawbacks. Special emphasis is given to the connection between information and thermodynamics. The perspectives and followup of stochastic thermodynamics in future experiments and in practical applications are also discussed. DOI: 10.1103/PhysRevX.7.021051 Subject Areas: Statistical Physics I. INTRODUCTION We have already mentioned the Brownian particle driven by an external force, but to clarify the kind of questions that When the size of a system is reduced, the role of we want to analyze, let us consider another simple example fluctuations (either quantum or thermal) increases. Thus, of an out-of-equilibrium system, that is, a thermal con- thermodynamic quantities such as internal energy, work, ductor whose extremities are connected to two heat baths at heat, and entropy cannot be characterized only by their different temperatures. The second law of thermodynamics mean values, but also their fluctuations and probability imposes that the mean heat flux flows from the hot to the distributions become relevant and useful to make predic- cold reservoir. However, the second law does not say tions on a small system. Let us consider a simple example, anything about fluctuations, and in principle one can such as the motion of a Brownian particle subjected to a observe for a short time a heat current in the opposite constant external force. Because of thermal fluctuations, direction, which corresponds to an instantaneous negative the work performed on the particle by this force per unit entropy production rate. What is the probability of observ- time, i.e., the injected power, fluctuates, and the smaller ing these rare events? The same problem appears in the the force, the larger is the importance of power fluctuations abovementioned example of the Brownian particle where [1–3]. The goal of stochastic thermodynamics is just that one can ask, what is the probability that the particles move of studying the fluctuations of the abovementioned thermo- in the opposite direction of the force? The answer to these dynamic quantities in systems driven out of equilibrium questions can be found within the framework of stochastic by external forces, temperature differences, and chemical thermodynamics and fluctuation theorems (FTs) [4–12], reactions. For this reason, it has received in the past which uses statistical mechanics to answer questions 20 years an increasing interest for its applications in related to extremes that are well beyond the mean (i.e., microscopic devices and biological systems and for its thermodynamics) and well beyond the standard fluctuation connections with information theory [1–3]. theory normally dominated, away from critical points by In the following we discuss the role of fluctuations in the central limit theorem. We see that the knowledge of out- out-of-equilibrium thermal systems when the energies of-equilibrium fluctuation properties is actually very useful injected or dissipated are smaller than 100k T (k being B B in experiments to extract useful information on equilibrium the Boltzmann constant and T the temperature). This limit and out-of-equilibrium properties of a specific system. is relevant in biological, nano, and micro systems, where Typical examples are the Jarzynski and Crooks equalities fluctuations cannot be neglected. We are interested in [13–15], which estimate equilibrium properties starting knowing the role of these fluctuations on the dynamics from nonequilibrium measurements. The measurement of and how one can gain some information by measuring them. the linear response in out-of-equilibrium systems is another very important aspect. Indeed, the new formulations of the fluctuation dissipation relation (FDR) related to the FT are Published by the American Physical Society under the terms of quite useful for this purpose, because they allow the the Creative Commons Attribution 4.0 International license. estimation of the response starting from the measurement Further distribution of this work must maintain attribution to of fluctuations of different quantities in nonequilibrium the author(s) and the published article’s title, journal citation, steady states (NESSs) [10,16–21]. Within the context of the and DOI. 2160-3308=17=7(2)=021051(26) 021051-1 Published by the American Physical Society S. CILIBERTO PHYS. REV. X 7, 021051 (2017) using different theoretical tools We start in Sec. II with FDR for out-of-equilibrium states, many studies have been done on the slow relaxation toward equilibrium, such as in an analysis of the experimental results on the energy aging glasses after a temperature quench [22–24]. It turns fluctuations in a harmonic oscillator driven out of equilib- rium by an external force. In Sec. III, we describe the out that entropy production plays a unifying role between the FT and the different extended formulations of the FDR properties of FTs, and as illustrative examples we apply it for out-of-equilibrium systems. to (a) a harmonic oscillator (linear case) and (b) a Brownian particle confined in a time-dependent double-well potential Another application of stochastic thermodynamics is the (nonlinear case). The latter is one of the very few examples study of the efficiency of micro or nano devices and the where the FT is applied to a highly nonlinear potential, role of fluctuations in the power production of these because most of the experiments reported in the literature systems. This is of course very useful for understanding are performed for linear potentials. In Sec. IV, we discuss and measuring the efficiency of molecular motors, which the application of the Jarzinsky and Crooks equalities to are isothermal and driven by chemical reactions. the harmonic oscillators and to the measure of the free It is worth mentioning that the study of stochastic energy of a single molecule. In Sec. VI, we introduce the thermodynamics has allowed us to bring more insight to application of stochastic thermodynamics to the study of the connection between information and thermodynamics. the efficiency of micro or nano machines. The contribution Specifically, the study of the energy fluctuations in a small of fluctuations to the power produced by these machines is system has transformed gedanken experiments, such as the described using the results of a proof of principle experi- Maxwell’s demon, in experiments which may actually be ment. In Sec. VII, we briefly present another relevant performed thanks to the new technologies such as optical or aspect of nonequilibrium statistical mechanics: the meas- electrical traps and single electron devices. urement of the linear response of a system in a non- Before explaining how the article is organized, it is equilibrium state. We discuss here only the main relevant mandatory to point out that the tools of stochastic thermo- features without giving any specific example as these FDT dynamics have also been applied to study the properties aspects have already been discussed in other reviews of macroscopic fluctuations in out-of-equilibrium systems. [2,3,34]. In Sec. VIII, the connections between information Indeed, the injected and dissipated energies may also and thermodynamics is analyzed following two comple- fluctuate in macroscopic systems if the dynamics is chaotic. mentary subjects. The first is the energy production by For instance, think of a motor used to stir a fluid strongly. devices controlled by a Maxwell’s demon. The second is the The motor can be driven by imposing a constant velocity. minimum energy needed to process one bit of information. Because of the turbulent motion of the fluid, the power Finally, we conclude in Sec. IX, where we describe other needed to keep the velocity constant fluctuates [25,26]. useful experimental applications of stochastic thermody- This simple example shows that fluctuations of the injected namics. We also discuss the perspectives and the artifacts and dissipated power may be relevant not only in micro- of these applications. scopic but also in macroscopic systems such as hydro- dynamic flows [26], granular media [27–30], mechanical systems [31], and more recently on self-propelling particles II. WORK AND HEAT FLUCTUATIONS IN [32,33]. The main difference is that in macroscopic systems THE HARMONIC OSCILLATOR fluctuations are produced by the dynamics and are sus- The choice of discussing the dynamics of the harmonic tained by a constant energy flux, whereas in small systems oscillator is dictated by the fact that it is relevant for many they are of either thermal or quantum nature. Thus, it is practical applications, such as the measure of the elasticity useful to divide the fluctuation in out-of-equilibrium of nanotubes [35], the dynamics of the tip of an AFM [36], systems into two classes: one where thermal fluctuations MEMS, and the thermal rheometer that we developed play a significant role (thermal systems) and another where several years ago to study the rheology of complex fluids the fluctuations are produced by chaotic flows or fluctuat- [37], whose high sensitivity has actually allowed several ing driving forces (athermal chaotic systems). In this tests of FT. As the results of these experiments have already article, we focus on thermal systems and only a short been discussed in some detail in several reviews [3,38],we discussion on the problems related to the application of describe here only the main results useful to introduce the stochastic thermodynamics to athermal systems is provided experimental activity in stochastic thermodynamics. in Sec. IX G. As the goal of this article is to present several A. Experimental setup general experimental aspects, we follow an experimentalist approach, and the connection with theory is made on the The thermal rheometer is a torsion pendulum whose basis of experimental measurements. Furthermore, we angular displacement θ is measured by a very sensitive organize the sections in terms of the main topics and tools interferometer. The details of the setup can be found in of stochastic thermodynamics. For this reason, the same Refs. [39–43]. A schematic diagram and a picture of the experimental apparatus is analyzed in various sections apparatus are shown in Fig. 1. 021051-2 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) with theoretical predictions often obtained for Markovian processes. (See Ref. [38] for a discussion on this point.) B. Energy balance When the system is driven out of equilibrium by the external deterministic torque M (which is, in general, time dependent), it receives an amount of work, and a fraction of this energy is dissipated into the heat bath. Multiplying Eq. (1) by θ and integrating between t and t þ τ, one i i obtains a formulation of the first law of thermodynamics between the two states at time t and t þ τ [Eq. (2)]. This i i formulation was first proposed in Ref. [44] and widely used in other theoretical and experimental works in the context of stochastic thermodynamics [1,2]. The change in internal energy ΔU of the oscillator over a time τ, starting at a time t , is written as ΔU ¼ Uðt þ τÞ − Uðt Þ¼ W − Q ; ð2Þ τ i i τ τ FIG. 1. (a) The torsion pendulum. (b) The magnetostatic forcing. (c) A picture of the pendulum. (d) The cell where the where W is the work done on the system over a time τ and pendulum is installed. Q is the dissipated heat. The work W is defined in the τ τ classical way, In equilibrium the variance δθ of the thermal fluctua- tions of θ can be obtained from equipartition; i.e., for our t þτ dθðt Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 W ¼ Mðt Þ dt ; ð3Þ pendulum, δθ ¼ k T=C ≃ 2 nrad, where C is the tor- dt sional stiffness of the pendulum and T is the temperature of the surrounding fluid. The measurement noise is 2 orders of and we use a tilde in order to distinguish it from a more magnitude smaller than thermal fluctuations of the pendu- general definition often used in stochastic thermodynamics lum whose resonance frequency f is about 217 Hz. A o [see Eq. (12) and the discussion at the end of Sec. IVA]. magnetostatic forcing [38,40,41] allows the application of The internal energy is the sum of the potential energy and an external torque M, useful to excite the pendulum and to the kinetic energy: drive it out of equilibrium. The typical applied torque is of the order of a few pN m, and the mean power a few k T=s. 2 1 dθðtÞ 1 UðtÞ¼ I þ CθðtÞ : ð4Þ The dynamics of the torsion pendulum can be assimi- eff 2 dt 2 lated to that of a harmonic oscillator damped by the viscosity of the surrounding fluid, whose equation of The heat transfer Q is deduced from Eq. (2). It has two motion reads contributions: d θ dθ t þτ dθ dθ I þ ν þ Cθ ¼ M þ η; ð1Þ eff 0 0 Q ¼ W − ΔU ¼ ν − ηðt Þ dt ; ð5Þ dt dt τ τ τ 0 0 dt dt where I is the effective moment of inertia of the eff where the integrals in Eqs. (3) and (5) are performed using pendulum, which includes the inertia of the surrounding the Stratonovich convention. fluid as discussed in Ref. [40]. The thermal noise η is, The first term in Eq. (5) corresponds to the viscous in this case, delta correlated in time: hηðtÞηðt Þi ¼ dissipation and is always positive, whereas the second term 2k Tνδðt − t Þ. However, if the fluid is viscoelastic, the can be interpreted as the work of the thermal noise, which noise η is correlated and the process is not Markovian, has a fluctuating sign. The second law of thermodynamics whereas in the viscous case the process is Markovian. Thus, imposes hQ i to be positive. Notice that because of the by changing the quality of the fluid surrounding the τ pendulum one can tune the Markovian nature of the fluctuations of θ and θ all the quantities W , ΔU , and Q τ τ τ process. In this review we consider only the experiment fluctuate, too. We are interested in characterizing these in the glycerol-water mixture where the viscoelastic con- fluctuations, which are related by Eq. (2), which is a tribution is visible only at very low frequencies and is formulation of the first law of thermodynamics between the therefore negligible. This allows a more precise comparison two states at time t and t þ τ. Although Eq. (2) has been i i 021051-3 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) FIG. 2. Sinusoidal forcing. (a) Pdf of W . (b) PDF of Q for various n: n ¼ 7 (∘), n ¼ 15 (□), n ¼ 25 (⋄), and n ¼ 50 (×). The τ τ continuous lines are not fits but are analytical predictions obtained from the Langevin dynamics, as discussed in Sec. III C. obtained for the harmonic oscillators, is indeed a general from the measure of W and ΔU . We first make some τ τ statement for the energy fluctuations of any system. comments on the average values. The average of ΔU is obviously vanishing because the time τ is a multiple of the C. Nonequilibrium steady state: Sinusoidal forcing ~ period of the forcing. Therefore, hW i and hQ i are equal, τ τ as it must be. We now consider a periodic forcing of amplitude M and We rescale the work W (the heat Q ) by the average frequency ω , i.e., MðtÞ¼ M sinðω tÞ [41–43], which is a τ τ d o d NESS because all the averages performed on an integer work hW i (the average heat hQ i) and define w ¼ τ τ τ number of the driving periods do not depend on time. This ~ ~ ðW =hW iÞ (q ¼ðQ =hQ iÞ). In the present article, x τ τ τ τ τ τ kind of periodic forcing is very common and it has been (X ) stands for either w or q (W or Q ). τ τ τ τ τ studied in the case of the first-order Langevin equation [45] We compare now the PDF of w and q in Fig. 2. The τ τ and of a two-level system [46] and in a different context for PDFs of heat fluctuations q have exponential tails the second-order Langevin equation [47]. Furthermore, this [Fig. 2(b)]. It is interesting to stress that although the is a very general case, because using Fourier transform any two variables W and Q have the same mean values, they τ τ periodical driving can be decomposed in a sum of sinus- have a very different PDF. The PDFs of w are Gaussian, oidal modes. We explain here the behavior of a single whereas those of q are exponential. On a first approxi- mode. Experiments have been performed at various M and mation, the PDFs of q are the convolution of a Gaussian ω . We present the results for a particular amplitude and (the PDF of W ) and exponential (the PDF of ΔU ) [38,43]. τ τ frequency: M ¼ 0.78 pN m and ω =ð2πÞ¼ 64 Hz. o d In Fig. 2, the continuous lines are analytical predictions obtained from the Langevin dynamics with no adjustable 1. Work fluctuations parameter (see Sec. III C). The work done by MðtÞ is computed from Eq. (3) on a time τ ¼ 2πn=ω , i.e., an integer number n of the driving III. FLUCTUATION THEOREMS period. W fluctuates and its probability density function In the previous section, we see that both W and Q (PDF) is plotted in Fig. 2(a) for various n. This plot has τ τ present negative values; i.e., the second law is verified only interesting features. Specifically, work fluctuations are on average, but the entropy production can have instanta- Gaussian for all values of n, and W takes negative values neously negative values. The probabilities of getting as long as τ is not too large. The probability of having positive and negative entropy production are quantitatively negative values of W decreases when τ is increased. τ n related in nonequilibrium systems by the fluctuation There is a finite probability of having negative values theorems [4–6], of the work; in other words, the system may have an There are two classes of FTs. The stationary state instantaneous negative entropy production rate although fluctuation theorem (SSFT) considers a nonequilibrium the average of the work hW i is, of course, positive (h·i steady state. The transient fluctuation theorem (TFT) stands for ensemble average). In this specific example, describes transient nonequilibrium states where τ measures hW i¼ 0.04nðk TÞ. τ B the time since the system left the equilibrium state. A fluctuation relation examines the symmetry around 0 of 2. Heat fluctuations the probability density function pðx Þ of a quantity x ,as τ τ The dissipated heat Q cannot be directly measured defined in the previous section. It compares the probability because Eq. (5) requires us to compute the work done by of having a positive event (x ¼þx) versus the probability of having a negative event (x ¼ −x). We quantify the FT the thermal noise, which is experimentally unmeasurable since η is unknown. However, Q can be obtained indirectly using a function (symmetry function): 021051-4 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) k T pðx ¼þxÞ Symðx Þ and x . In the region where the symmetry B τ τ τ Symðx Þ¼ ln : ð6Þ function is linear with x , we define the slope Σ ðτÞ, i.e., hX i pðx ¼ −xÞ τ x τ τ Symðx Þ¼ Σ ðτÞx . As a second step, we measure finite- τ x τ time corrections to the value Σ ðτÞ¼ 1, which is the The TFT states that the symmetry function is linear asymptotic value expected from FTs. with x for any values of the time integration τ and the In this article, we focus on the SSFT applied to the proportionality coefficient is equal to 1 for any value of τ: experimental results of Sec. II C and to other examples. The Symðx Þ¼ x ; ∀x ; ∀τ: ð7Þ applications of TFT are not presented in this section, but τ τ τ we discuss them in Sec. IV B, and interested readers may Contrary to TFT, the SSFT holds only in the limit of infinite find more details in Ref. [43]. time (τ): From the PDFs of w and q plotted in Fig. 2,we τ τ compute the symmetry functions defined in Eq. (6). The lim Symðx Þ¼ x : ð8Þ τ τ symmetry functions Symðw Þ are plotted in Fig. 3(a) as a τ→∞ function of w . They are linear in w . The slope Σ ðnÞ is not τ τ w equal to 1 for all n, but there is a correction at finite time. In the following, we assume linearity at finite time τ Nevertheless, Σ ðnÞ tends to 1 for large n. Thus, SSFT is [48,49] and use the following general expression: satisfied for W and for a sinusoidal forcing. The con- Symðx Þ¼ Σ ðτÞx ; ð9Þ τ x τ vergence is very slow and we have to wait a large number of periods of forcing for the slope to be 1 (after 30 periods, where for SSFT Σ ðτÞ takes into account the finite-time the slope is still 0.9). This behavior is independent of the corrections and lim Σ ðτÞ¼ 1, whereas Σ ðτÞ¼ 1, ∀ τ τ→∞ x x amplitude of the forcing M and consequently of the mean for TFT. value of the work hW i, which, as explained in Ref. [38], However, these claims are not universal because they changes only the time needed to observe a negative event. depend on the kind of x that is used. Specifically, we see in The system satisfies the SSFT for all forcing frequencies the next sections that the results are not exactly the same if ω , but finite-time corrections depend on ω [43]. d d X is replaced by any one of W , Q , and the total entropy τ τ τ We now analyze the PDF of q [Fig. 2(b)] and we [11,12]. Furthermore, the definitions given in this section compute the symmetry functions Symðq Þ of q plotted in τ τ are appropriate for stochastic systems, and the differences Fig. 3(b) for different values of n. They are clearly very between stochastic and chaotic systems are not addressed different from those of w plotted in Fig. 3(a). For Symðq Þ n τ in this review. A discussion on this point can be found three different regions appear. in Ref. [38]. (I) For large fluctuations q , Symðq Þ equals 2. When τ n τ tends to infinity, this region spans from q ¼ 3 to infinity. A. FTs for W and Q measured (II) For small fluctuations q , Symðq Þ is a linear τ τ n n in the harmonic oscillator function of q . We then define Σ ðnÞ as the slope of the function Symðq Þ, i.e., Symðq Þ¼ Σ ðnÞq . We have The questions we ask are whether for finite time FTs are n n q n satisfied for either x ¼ w or x ¼ q and what are the measured [43] that Σ ðnÞ¼ Σ ðnÞ for all the values of q w τ τ τ τ finite-time corrections? As a first step, we test the correc- n; i.e., finite-time corrections are the same for heat and tion to the proportionality between the symmetry function work. Thus, Σ ðnÞ tends to 1 when τ is increased and SSFT FIG. 3. Sinusoidal forcing. Symmetry functions for SSFT. (a) Symmetry functions Symðw Þ plotted as a function of w for various n: τ τ n ¼ 7 (circle), n ¼ 15 (square), n ¼ 25 (diamond), and n ¼ 50 (times). For all n, the dependence of Symðw Þ on w is linear, with slope τ τ Σ ðτÞ. (b) Symmetry functions Symðq Þ plotted as a function of q for various n. The dependence of Symðq Þ on q is linear only for τ τ τ τ q < 1. Continuous lines are theoretical predictions. 021051-5 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) holds in this region II, which spans from q ¼ 0 up to discussion on the different quantities for SSFT and TFT q ¼ 1 for large τ. This effect was discussed for the first can be found. time in Refs. [48,49]. Furthermore, in order to avoid the complexity of (III) A smooth connection between the two behaviors. computing ΔS from individual trajectories, another tot These regions define the fluctuation relation from the quantity, which satisfies a SSFT for any τ, has been heat dissipated by the oscillator. The limit for large τ of the proposed in Ref. [54]. This quantity is the joint probability symmetry function Symðq Þ is rather delicate and it is τ PðW ;J Þ of the work and of the energy currents in the τ τ discussed in Ref. [43]. system. Although its measure might be difficult, it is by far The conclusion of this experimental analysis is that easier than the trajectory-dependent entropy. However, this SSFT holds for work for any value of w , whereas for heat it τ method, although very powerful, has never been tested on holds only for q < 1. The finite-time corrections to FTs, τ experimental data, but it will certainly be useful to try. described by 1 − Σ, are not universal. They are the same for both w and q , but they depend on the driving frequency τ τ C. Comparison with theory and on the kind of driving force Ref. [38,43]. The experimental analysis described in Sec. III A allows These kinds of measurements are important because they a very precise comparison with theoretical predictions allow us to test complex theoretical concepts on relatively using the Langevin equation [Eq. (1)] and using two simple systems in order to apply them to more complex experimental observations: (a) the properties of the heat cases. Furthermore, the experimental analysis on model bath are not modified by the driving and (b) the fluctuations systems allows us to check the theoretical hypothesis made of the W are Gaussian (see also Ref. [55], where it is in order to prove the theorems. For example, one of these shown that in Langevin dynamics W has a Gaussian hypotheses is that the properties of the heat bath are not τ distribution for any kind of deterministic driving force if the modified by the forcing. This hypothesis can be precisely properties of the bath are not modified by the driving and checked in the experiments. the potential is harmonic). The observation in point (a) is extremely important because it is always assumed to be true B. Trajectory-dependent entropy in all the theoretical analysis. In Ref. [43], this point has and the total entropy been precisely checked. Using these experimental obser- In the same way of W and Q , the entropy produc- τ τ vations one can compute the PDF of q and the finite-time tion rate can also be defined at the trajectory level. The corrections ΣðτÞ to SSFT (see Ref. [43]). The continuous trajectory-dependent entropy difference δs ðtÞ is defined as lines in Figs. 2 and 3 are not fit but analytical predictions, δs ðtÞ¼ − log½P(r⃗ ðt þ τÞ; λ)=P(r⃗ ðtÞ; λ), where P(r⃗ ðtÞ; λ) with no adjustable parameters, derived from the Langevin is the probability of finding the system in the position dynamics of Eq. (1) (see Ref. [43] for more details). r⃗ ðtÞ of the phase space at a value λ of the control parameter. Thus, the total entropy difference on the time τ is D. Nonlinear case: Stochastic resonance ΔS ðt; τÞ¼ δs ðtÞþ Q ðtÞ=T [11,12], i.e., the sum of tot τ τ In the harmonic oscillator described in the previous the trajectory-dependent entropy and of the entropy change section, the only nonlinearities, which might appear, are in the reservoir due to energy flow. The mean total entropy those related to the elasticity of the torsion wire. However, difference is equal to the entropy production rate; i.e., hΔS ðt; τÞi¼hQ ðtÞ=Ti. Furthermore, ΔS ðt; τÞ fully to reach this nonlinear regime, the system has to be forced tot τ tot to such a high level that thermal fluctuations become characterizes the out-of-equilibrium dynamics as it is negligible. Thus, in order to study the nonlinear effects rigorously zero, both in average and fluctuations. The we change the experiment and we measure the fluctuations fluctuations of this quantity impose several constrains on of a Brownian particle trapped in a nonlinear potential the time reversibility, which is a central result of stochastic produced by two laser beams, as shown in Fig. 4 [56].It is thermodynamics [2,11,15,50] and which has been tested very well known that a particle of small radius R ≃ 2 μm experimentally [51]. The FT for the ΔS in a SSFT implies tot can be trapped by a focused laser beam, which produces a ΣðτÞ¼ 1 for any τ; i.e., the FT does not have an asymptotic harmonic potential, thereby confining the Brownian par- validity but is valid for any τ. This is certainly a useful ticle motion to the potential well. When two laser beams are property in experiment because one does not have to look focused at a distance D ≃ R, as shown in Fig. 4(a), the for very long asymptotic behavior. However, the calcula- particle has two equilibrium positions, i.e., the foci of the tion of S in experiment is not easy and a lot of care must tot two beams. Thermal fluctuations allow the particle to hop be used in order to correctly estimate this quantity [38,52]. from one to the other. The particle feels an equilibrium We do not discuss here the experimental analysis per- 4 2 formed on electric circuits and harmonic oscillators, but an potential U ðxÞ¼ ax − bx − dx, shown in Fig. 4(b), example of the evaluation of ΔS ðt; τÞ is given in Sec. V. where a, b, and d are determined by the laser intensity tot For further information, the interested reader can look at and by the distance of the two focal points. This potential the abovementioned references and Ref. [53], where a has been computed from the measured equilibrium 021051-6 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) (b) (a) FIG. 4. (a) Drawing of the polystyrene particle trapped by two laser beams whose axis distance is about the radius of the bead. (b) Potential felt by the bead trapped by the two laser beams. The barrier height between the two wells is about 2k T. distribution of the particle, PðxÞ ∝ exp½U ðxÞ (see quite similar to that of the harmonic oscillator (Sec. III A) Ref. [56] for more experimental details). To drive the although the PDFs are more complex [56]. The measure- system out of equilibrium we periodically modulate the ment are in full agreement with a realistic model based on intensity of the two beams at low frequency ω. Thus, the Fokker-Planck equations where the measured values of the potential felt by the bead has the following pro- Uðx; tÞ have been inserted [60]. This example shows the file: Uðx; tÞ¼ U ðxÞþ U ðx; tÞ¼ U þ cx sinðωtÞ. 0 p 0 The x position of the particle can be described by an overdamped Langevin equation: dx ∂Uðx; tÞ ν ¼ − þ η; ð10Þ dt ∂x with ν the friction coefficient and η the thermal noise delta correlated in time. When c ≠ 0, the particle can experience a stochastic resonance [57–59], when the forcing frequency is close to the Kramers rate [56]. As already done in the case of the harmonic oscillator, one can compute the work tþτ 0 0 0 W ¼ fðt Þx_ðt Þdt ð11Þ of the external force fðtÞ¼ −c sinð2πftÞ on the time interval ½t; t þ τ, where τ ¼ð2πn=ωÞ is a multiple of the forcing period [56]. We consider the PDF PðW Þ, which is plotted in Fig. 5(a). Notice that for small n the distributions are double peaked and very complex. They tend to a Gaussian for large n [inset of Fig. 5(a)]. In Fig. 5(b) we plot the normalized symmetry function of W . We can see that the curves are close to the line of slope one. For high values of work, the dispersion of the data increases due to the lack of events. The slope tends toward 1 as expected by the SSFT. It is remarkable that straight lines are obtained even for n FIG. 5. (a) Distribution of classical work W for different close to 1, where the distribution presents a very complex numbers of period n ¼ 1, 2, 4, 8, and 12 (f ¼ 0.25 Hz). Inset: and unusual shape [Fig. 5(a)]. The very fast convergence to Same data in lin-log. (b) Normalized symmetry function as the asymptotic value of the SSFT is quite striking in this function of the normalized work for n ¼ 1 (plus), 2 (circle), 4 example. We do not show here SðQ Þ as the behavior is (diamond), 8 (triangle), and 12 (square). 021051-7 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) application of FT in a nonlinear case where the distributions Specifically, in the experiment described in Sec, III D, λ is tþτ are strongly non-Gaussian. Other examples of application the external force and W ¼½fðtÞxðtÞ − W . Thus, in an of FT to nonlinear potentials can be found in Ref. [61]. experiment in order to get the right value of the free energy using Eq. (13), one has to clearly identify the control IV. ESTIMATE THE FREE-ENERGY DIFFERENCE parameter λ, which drives the system, and the definition FROM WORK FLUCTUATIONS of Eq. (3) must be used to estimate W (see Ref. [40] for a discussion on this point). In 1997 [13,14] Jarzynski derived an equality which relates the free-energy difference of a system in contact B. Crooks relation with a heat reservoir to the PDF of the work performed on the system to drive it from A to B along any path γ in the This relation is related to the Jarzynski equality and it system parameter space. gives useful and complementary information on the dis- sipated work. Crooks considers the forward work W to drive the system from A to B and the backward work W to A. Jarzynski equality drive it from B to A [Eq. (3) is used for both W and W ]. If f b Specifically, when a system parameter λ is varied from the work PDFs during the forward and backward processes time t ¼ 0 to t ¼ t , Jarzynski defines for one realization are P ðWÞ and P ðWÞ, one has [15] f b of the “switching process” from A to B the work performed on the system as P ðWÞ Z ¼ exp ðβ½W − ΔFÞ ¼ exp ½βW : ð15Þ diss ∂H ½zðtÞ P ð−WÞ λ b W ¼ λ dt; ð12Þ ∂λ Notice that the TFT defined in Sec. III is a special case of Eq. (15), for which A ≡ B and the backward and forward where z denotes the phase-space point of the system and H protocols are the same. its λ parametrized Hamiltonian. One can consider an A simple calculation leads from Eq. (15) to Eq. (13). ensemble of realizations of this switching process with However, from an experimental point of view, Eq. (15) is initial conditions all starting in the same initial equilibrium extremely useful because one immediately sees that the state. Then W may be computed for each trajectory in the crossing point of the two PDFs, that is, the point where ensemble. The Jarzynski equality states that [13,14] P ðWÞ¼ P ð−WÞ, is precisely ΔF. Thus, one has another f b exp ð−βΔFÞ¼hexp ð−βWÞi; ð13Þ mean to check the computed free energy by looking at the PDFs crossing point W . The draw back is that both states −1 where h·i denotes the ensemble average and β ¼ k T A and B have to be equilibrium states. A very interesting with k the Boltzmann constant and T the temperature. and extended review on the Jarezynski and Crooks relations In other words, hexp ½−βW i ¼ 1, since we can always can be found in Ref. [63]. diss write W ¼ ΔF þ W , where W is the dissipated work. diss diss Thus, it is easy to see that there must exist some paths γ C. Applications of Jarzynski and Crooks equalities such that W ≤ 0. Moreover, the inequality hexp xi ≥ diss The Jarzynski equality has been tested for the first time exphxi allows us to recover the second principle, namely, in a single-molecule experiment [64]. Here, we follow a hW i ≥ 0, i.e., hWi ≥ ΔF. If the probability distribution diss more pedagogical description by discussing first the appli- 2 2 of the work is Gaussian, PðWÞ∝ exp½−ð½W− hWi =2σ Þ, cation to the harmonic oscillator and then to a single- then Eq. (13) leads to molecule experiment in Sec. IV C 2. βσ ΔF ¼hWi − ; ð14Þ 1. Harmonic oscillator As a simple example we apply the Jarzynski equality to i.e., the dissipate energy hW i¼ðβσ =2Þ > 0. the harmonic oscillator described in Sec. II. The oscillator diss From an experimental point of view the Jarzynski equality is driven from a state A (M ¼ 0) and to a state B (where [Eq. (13)] is quite useful because there is no restriction on the M ¼ M ≠ 0) by the external tork MðtÞ¼ M t=t (for- o o s choice of the path γ. Furthermore, the other big advantage is ward transformation). The backward transformation is that only the initial state has to be an equilibrium state. instead MðtÞ¼ M − M t=t . The switching time t is o o s s Indeed even if the system is not in equilibrium at time t , one varied in order to probe either the reversible (or quasistatic) obtains the value of ΔFðλðt ÞÞ, i.e., the value of the free paths (t ≫ τ ) or the irreversible ones (t ≲ τ ), s relax s relax energy that the system would have at equilibrium for the where τ is the harmonic oscillator relaxation time. relax value of the control parameter λ ¼ λðt Þ [62]. Finally, we We apply a torque which is a sequence of linear increasing stress that the definition of work given in Eq. (12) is more or decreasing ramps and plateaus. The latter are necessary general than the classical definition used in Eqs. (3) and (11). to relax the system in equilibrium before starting a new 021051-8 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) (a) (b) FIG. 6. Free-energy measure in the harmonic oscillator. (a) Time evolution of θðtÞ under a periodic torque M which drives the oscillator from A (θ ¼ 0)to B [θðt Þ¼ 4.1 nrad] and vice versa. In this specific case the stiffness is C ≃ 5.5 Nm, the transition time is −1 t ≃ 0.1τ , and M ¼ 22.4 Nm rad . (b) Probability distribution functions of the work for the forward (blue curve) and backward s relax o (red curve) transformation. The crossing point of the two PDFs determines the value of ΔF [see Eq. (15)]. The crossing point is at A;B W ¼ −112k T, which is within experimental errors of the expected theoretical Δ ≃ −110k T (see text). B F B transformation. This periodic driving produces a sequence [64–77]. Let us summarize a typical example extracted of direct A-B paths and reversed ones, B-A,of θðtÞ, whose from Ref. [65], where Eq. (15) has been used to measure time dependence is plotted in Fig. 6(a). The plotted the ΔF between the folded and unfolded states of a DNA dynamics corresponds to quite irreversible transformation hairpin. In Fig. 7(a), a schematic diagram of the experi- as t ≪ τ ; specifically, t =τ ≃ 0.1. The other rel- ment is depicted, and the typical structure of the DNA s relax s relax evant parameters of the experiment are M ¼ 22.4 pN and hairpin is shown in Fig. 7(b). In this experiment the DNA −4 −1 hairpin is attached to two beads via double-stranded the stiffness C ¼ 5.5 × 10 Nm rad . The work W can be DNA handles. As these two handles are much stiffer than computed in each of the reverse or direct paths using the hairpin, they do not deform during the stretching Eq. (12), and the P ðWÞ and P ðWÞ can be constructed. f b cycles that are performed in the following way. The The result is shown in Fig. 6(b), where the probability bottom bead is kept by a micropipette, whereas an optical distribution functions are plotted. We see that P ðWÞ and trap captures the top bead. A piezoelectric actuator P ðWÞ cross at about −112k T, which is the measured b B controls the position of the bottom bead, which, when value of the ΔF computed from Eq. (15). Let us compare moved along the vertical axis, stretches the DNA. The the measured value with the theoretical estimate. In this difference in positions of the bottom and top beads gives specific case of the harmonic oscillator, as the temperature the end-to-end length of the molecule. The optical trap is is the same in states A and B, the free-energy difference of 1 2 B 2 B used to measure the force exerted by the stretched DNA the oscillator alone is ΔF ¼ ΔU ¼½ Cθ  ¼½ðM =2CÞ . 0 o 2 A A on the top bead. By driving the piezoelectric actuator, the However, the Jarzynski [Eq. (13)] and Crooks [Eq. (15)] molecule is repeatedly subjected to unfold-refold cycles. relations compute the free-energy difference of the system Every pulling cycle consists of a stretching process (the harmonic oscillator) plus the driving, i.e., ΔF ¼ ΔF − (hereafter referred to as S) and a releasing (hereafter 2 B ½ðM =CÞ , which in this case gives ΔF ¼ −ΔF . From A 0 referred to as R) process. In the stretching part of the the values of the experimental parameters, one gets cycle the molecule is stretched from a minimum value of ΔF ≃ 110k T, which is very close to the experimental the force (f ≃ 10 pN), so small that the hairpin is min value. This example shows that indeed using Eqs. (13) and always folded, up to a maximum value of the force (15) it is possible to obtain the values of the free-energy (f ≃ 20 pN), so large that the hairpin is always max difference between two states even by doing a very fast unfolded. During the releasing part of the cycle the transformation with t ≪ τ . However, the applications s relax force is decreased from f back to f at the same max min of Eqs. (13) and (15) to the experiments may present rate of the loading cycle, which is the same protocol used several problems, which are discussed in detail in Ref. [40]. for the abovementioned example of the harmonic oscil- lators. As the stretching force and the displacement of 2. Measure of the free-energy difference the bead are independently measured, the work can be of a single molecule computed using Eq. (12) at each cycle S and R. The Jarzynsky and Crooks relations turn out to be a useful corresponding P ðWÞ and P ðWÞ histograms can be S R tool to measure the free energy of a single molecule computed too. The experiment has been repeated for 021051-9 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) (a) (c) FIG. 7. (a) Experimental setup. The DNA hairpin whose sequence is shown in (b) is attached to two beads. The bottom bead is kept by a micropipette and top bead is captured by an optical trap. The drawing is not to scale; the diameter of the beads is around 3000 nm, much greater than the 20-nm length of the DNA. (b) DNA hairpin sequence. Labels 5 and 3 indicate the polarity of the phosphate chain of the hairpin. (c) Typical work distributions for three different loading rates: 1 pN=s (slow, blue), 4.88 pN=s (medium, green), 14.9 pN=s (fast, red). The vertical lines show the range of the estimated experimental errors for the value of ΔF (adapted from Ref. [65]). different pulling-releasing speeds, and the results for shown that fluctuation relations can be used for much more three speeds are shown in Fig. 7(c). We see that when than estimating free-energy differences. They study ligand the speed is increased the difference between the mean binding and use single-molecule force spectroscopy to works in the S and R protocols increases. However, the measure binding energies, selectivity, and allostery of remarkable fact is that the P ðWÞ and P ðWÞ cross, nucleic acids. S R within experimental errors at the same value of W Finally, useful extensions and generalizations of independently of the pulling speed, showing the validity the Jarzynski equality that allow the study of the of Eq. (15). As already explained, the crossing point transition between two nonequilibrium steady states have gives the value of the free-energy difference between the been derived in Ref. [78] and checked experimentally folded and unfolded states. in Ref. [79]. 3. Short discussion on applications of Jarzynski V. TWO HEAT BATHS and Crooks relations In Secs. II, III D, and IV, we discuss systems in contact The examples in Secs. IV C 1 and IV C 2 show the with a single heat bath, which, within the context of power of Jarzynski and Crooks relations which are a very stochastic thermodynamics, are the most studied cases useful tool to estimate the free-energy differences of micro both experimentally and theoretically [2,3]. Conversely, and nano systems where the role of fluctuation is very systems, driven out of equilibrium by a temperature important. gradient, in which the energy exchanges are produced It is worth mentioning that there is a large amount only by the thermal noise, have been analyzed mainly in of work on this topic performed by the biophysics theoretical models [47,80–90]. This problem has been and chemistry communities. The estimation of the studied only in a few very recent experiments [91–95], protein-folding landscapes is an important application, because of the intrinsic difficulties of dealing with large which remains one of the main interests despite many years of investigation; useful examples can be found in temperature differences in small systems. In order to illustrate the main properties of the energy Refs. [73,74]. Furthermore, using extensions to the basic results of fluxes in these systems driven out of equilibrium by a Jarzynski [68], the works in Refs. [72,75,76] collectively temperature gradient, we summarize in this section the main results of Refs. [91,92]. These two articles analyze show that nonequilibrium measurements give the most both experimentally an theoretically the statistical pro- precise reconstructions, to date, of free-energy landscapes perties of the energy exchanged between two conductors for single molecules (DNA hairpins). kept at different temperature and coupled by the electric The reader might also be interested in a recent extension of these relations by Camunas-Solder et al. [77], who have thermal noise. 021051-10 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) C C A. Two electric circuits interacting via 2 R q ¼ −q þðq − q Þ þ η ; ð16Þ 1 1 1 2 1 1 a conservative coupling X X 1. Experimental setup and stochastic variables C C R q_ ¼ −q þðq − q Þ þ η ; ð17Þ The experimental setup is sketched in Fig. 8(a).Itis 2 2 2 1 2 2 X X constituted by two resistances R and R , which are kept 1 2 at different temperatures T and T , respectively. These 1 2 where η is the usual white noise, hη ðtÞη ðt Þi ¼ m i j temperatures are controlled by thermal baths, and T is 0 2δ k T R δðt − t Þ, and where we have introduced the ij B i j fixed at 296 K, whereas T can be set at a value between 88 quantity X ¼ C C þ CðC þ C Þ. Equations (16) and (17) 2 1 1 2 and 296 K using the stratified vapor above a liquid nitrogen are the same as those for the two coupled Brownian bath. In the figure, the two resistances have been drawn particles sketched in Fig. 8(b) when one regards q as with their associated thermal noise generators η and η , 1 2 the displacement of the particle m, i as its velocity, K ¼ m m whose power spectral densities are given by the Nyquist 0 0 C 0=X (m ¼ 2 if m ¼ 1 and m ¼ 1 if m ¼ 2) as the formula jη~ j ¼ 4k R T , with m ¼ 1, 2 [see Eqs. (16) m B m m stiffness of the spring m, K ¼ C=X as the coupling spring, and (17)]. The coupling capacitance C controls the elec- and R the viscosity term. The analogy with the Feymann trical power exchanged between the resistances and, as a ratchet can be made by assuming, as done in Ref. [82], that consequence, the energy exchanged between the two baths. the particle m has an asymmetric shape and on average No other coupling exists between the two resistances which moves faster in one direction than in the other one. are inside two separated screened boxes. The quantities We now rearrange Eqs. (16) and (17) to obtain the C and C are the capacitances of the circuits and the 1 2 Langevin equations for the voltages, which will be useful cables. Two extremely low-noise amplifiers A and A [96] 1 2 in the following discussion. The relationships between the measure the voltage V and V across the resistances R 1 2 1 measured voltages and the charges are and R , respectively. All the relevant quantities considered in this paper can be derived by the measurements of V q ¼ðV − V ÞC þ V C ; ð18Þ 1 1 2 1 1 and V , as we discuss below. q ¼ðV − V ÞC − V C : ð19Þ 2 1 2 2 2 2. Stochastic equations for the voltages We now proceed to derive the equations for the dynami- By plugging Eqs. (18) and (19) into Eqs. (16) and (17), and cal variables V and V . Furthermore, we discuss how our 1 2 rearranging terms, we obtain system can be mapped onto a system with two interacting Brownian particles, in the overdamped regime, coupled to _ _ ðC þ CÞV ¼ CV þ ðη − V Þ; ð20Þ 1 1 2 1 1 two different temperatures; see Fig. 8(b). Let q (m ¼ 1,2) be the charges that have flowed through the resistances R , so that the instantaneous current flowing through them _ _ ðC þ CÞV ¼ CV þ ðη − V Þ: ð21Þ is i ¼ q_ . A circuit analysis shows that the equations for m m 2 2 1 2 2 the charges are FIG. 8. (a) Diagram of the circuit. The resistances R and R are kept at temperature T and T ¼ 296 K, respectively. They are 1 2 1 2 coupled via the capacitance C. The capacitances C and C schematize the capacitances of the cables and of the amplifier inputs. The 1 2 voltages V and V are amplified by the two low-noise amplifiers A and A [96]. The other relevant parameters are q (m ¼ 1, 2), i.e., 1 2 1 2 the charges that have flowed through the resistances R , and the instantaneous current flowing through them, i.e., i ¼ðdq =dtÞ. m m (b) The circuit in (a) is equivalent to two Brownian particles (m and m ) moving inside two different heat baths at T and T . The two 1 2 1 2 particles are trapped by two elastic potentials of stiffness K and K and coupled by a spring of stiffness K [see text and Eqs. (16) and 1 2 (17)]. The analogy is straightforward by considering q the displacement of the particle m, i its velocity, K ¼ C =X (with m ≠ m ) m m m the stiffness of the spring m, and K ¼ C=X the coupling spring. 021051-11 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) 3. Thermodynamics quantities while PðQ Þ ≠ Pð−Q Þ. Indeed, the shape of PðQ Þ is 1;τ 2;τ 1;τ strongly modified by changing T from 296 to 88 K; Two important quantities can be identified in the circuit whereas the shape of Pð−Q Þ is slightly modified by 2;τ depicted in Fig. 8: the energy Q dissipated in each m;τ the large temperature change, only the tails of Pð−Q Þ 2;τ resistor in a time and the work W exerted by one circuit m;τ present a small asymmetry testifying to the presence on the other one in a time τ. These two thermodynamic of a small heat flux. The fact that PðQ Þ ≠ Pð−Q Þ, quantities are related to the internal energy variation in the 1;τ 2;τ whereas PðW Þ¼ Pð−W Þ, can be understood by time τ by the first principle: 1;τ 2;τ noticing that Q ¼ W − ΔU . Indeed, ΔU m;τ m;τ m;τ m;τ [Eq. (22)] depends on the values of C and V .As C ≠ ΔU ¼ W − Q : ð22Þ m m 1 m;τ m;τ m;τ C and σ ≥ σ , this explains the different behavior of Q 2 2 1 1 and Q . On the contrary, W depends only on C and the Furthermore, it has been proved that the measured variance 2 m 2 product V V . _ 1 2 σ of V is related to the mean heat flux hQ i ∝ ∂ hQ i: m m m t m We study whether our data satisfy the fluctuation theorem as given by Eq. (24) in the limit of large τ.It 2 2 2 σ ¼hV i¼ σ þhQ iR ; ð23Þ m m m;eq m m turns out that the symmetry imposed by Eq. (24) is reached for rather small τ for W. On the contrary, it converges very where σ ¼ k T ðC þ C Þ=X is the equilibrium value m;eq B m m slowly for Q. We have only a qualitative argument to of σ . Note that hQ i ∝ ðT 0 − T Þ; thus, in the equilib- explain this difference in the asymptotic behavior: by m m m m rium case T ¼ T 0, and consequently hQ i¼ 0. looking at the data one understands that the slow con- m m m We do not give here the exact expressions of ΔU , vergence is induced by the presence of the exponential tails m;τ of PðQ Þ for small τ. W , Q , and σ , which have been computed and 1;τ m;τ m;τ m To check Eq. (24), we plot in Fig. 9(c) the symmetry measured in Refs. [91,92]. We discuss instead how the function SymðE Þ¼ ln½PðE Þ=Pð−E Þ as a function FT is modified in the case of two heat baths. 1;τ 1;τ 1;τ of E =ðk T Þ measured at different T ,but τ ¼ 0.1s 1;τ B 2 1 for SymðW Þ and τ ¼ 2s for SymðQ Þ. Indeed, 4. Fluctuation theorem for work and heat 1;τ 1;τ SymðQ Þ reaches the asymptotic regime only for 1;τ One expects that the thermodynamic quantities satisfy a τ → 2s. We see that SymðW Þ is a linear function of 1;τ fluctuation theorem of the type [4,6,81,83,88–90] W =ðk T Þ at all T . These straight lines have a slope 1;τ B 2 1 αðT Þ which, according to Eq. (24), should be ðβ k T Þ. 1 12 B 2 PðE Þ m;τ ln ¼ β E ΣðτÞ; ð24Þ In order to check this prediction we fit the slopes of the 12 m;τ Pð−E Þ m;τ straight lines in Fig. 9(c). From the fitted αðT Þ we deduce a temperature T ¼ T =½αðT Þþ 1, which is compared to fit 2 1 where E stands for either W or Q , β ¼ m;τ m;τ m;τ 12 the measured temperature T in Fig. 9(d). In this figure, the ð1=T − 1=T Þ=k , and ΣðτÞ → 1 for τ → ∞. 1 2 B straight line of slope 1 indicates that T ≃ T within a few fit 1 Equation (24) has been proven in Ref. [92]. percent. These experimental results indicate that our data As the system is in a stationary state, we have verify the fluctuation theorem, Eq. (24), for the work and hW i¼hQ i. On the contrary, the comparison of the m;τ τ;m the heat, but that the asymptotic regime is reached for much PDF of W with those of Q , measured at various m;τ τ;m larger time for the latter. temperatures, presents several interesting features. In Fig. 9(a), we plot PðW Þ, Pð−W Þ, PðQ Þ, and 1;τ 2;τ 1;τ 5. Entropy production rate Pð−Q Þ measured in equilibrium at T ¼ T ¼ 296 K 2;τ 1 2 It is now important to analyze the entropy produced by and τ ≃ 0.1 s. We immediately see that the fluctuations the total system, circuit plus heat reservoirs. We consider of the work are almost Gaussian, whereas those of the first the entropy ΔS due to the heat exchanged with the heat present large exponential tails. This well-known r;τ reservoirs, which reads ΔS ¼ Q =T þ Q =T . This difference [48] between PðQ Þ and PðW Þ is induced r;τ 1;τ 1 2;τ 2 m;τ m;τ by the fact that Q depends also on ΔU [Eq. (22)], entropy is a fluctuating quantity as both Q and Q 1 2 m;τ m;τ fluctuate, and its average in a time τ is hΔS i¼ which is the sum of the square of Gaussian distributed r;τ variables, thus inducing exponential tails in PðQ Þ.In hQ ið1=T − 1=T Þ¼ AτðT − T Þ =ðT T Þ. However, m;τ r;τ 1 2 2 1 2 1 Fig. 9(a), we also notice that PðW Þ¼ Pð−W Þ and the reservoir entropy ΔS is not the only component 1;τ 2;τ r;τ PðQ Þ¼ Pð−Q Þ, showing that in equilibrium all fluc- of the total entropy production: one has to take into account 1;τ 2;τ tuations are perfectly symmetric. The same PDFs measured the entropy variation of the system, due to its dynamical in the out-of-equilibrium case at T ¼ 88 K are plotted in evolution. Indeed, the state variables V also fluctuate as an Fig. 9(b). We notice here that in this case the behavior of the effect of the thermal noise, and thus, if one measures their values at regular time interval, one obtains a “trajectory” in PDFs of the heat is different from those of the work. Indeed, although hW i > 0, we observe that PðW Þ¼ Pð−W Þ, the phase space ½V ðtÞ;V ðtÞ. Thus, following Seifert [11], m;τ 1;τ 2;τ 1 2 021051-12 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) 10 0 P(W 1) (a) P(W1) (b) P(−W 2) P(−W 2) T1=T2=296 K −2 P(−Q 2) P(−Q 2) −2 10 T1 = 88 K P(Q 1) P(Q1) τ = 0.1 s τ = 0.1 s −4 −4 −6 −6 −8 −8 −20 −10 0 10 20 −20 −10 0 10 20 E /(k T ) E /(k T ) m,τ B 2 m,τ B 2 (c) (d) T = 90 K Sym(W ) τ=0.1s 250 Sym(Q ) τ=2s T = 150 K T = 293 K 50 100 150 200 250 300 0 0.5 1 1.5 2 E /(k T ) T (K) 1,τ B 2 1 FIG. 9. (a) Equilibrium: PðW Þ and PðQ Þ, measured in equilibrium at T ¼ T ¼ 296 K and τ ¼ 0.1s, are plotted as functions of m;τ m;τ 1 2 E, where E stands for either W or Q. Notice that when the system is in equilibrium, PðW Þ¼ Pð−W Þ and PðQ Þ¼ Pð−Q Þ. 1;τ 2;τ 1;τ 2;τ (b) Out of equilibrium: Same distributions as in (a) but the PDFs are measured at T ¼ 88 K, T ¼ 296 K, and τ ¼ 0.1s. Notice that 1 2 when the system is out of equilibrium, PðW Þ¼ Pð−W Þ but PðQ Þ ≠ Pð−Q Þ. The reason for this difference is explained in the 1;τ 2;τ 1;τ 2;τ text. (c) The symmetry function SymðE Þ, measured at various T , is plotted as a function of E (W or Q ). The theoretical slope of 1;τ 1 1 1 1 these straight lines is T =T − 1. (d) The temperature T estimated from the slopes of the lines in (c) is plotted as a function of the T 2 1 fit 1 measured by the thermometer. The slope of the line is 1, showing that T ≃ T within a few percent. fit 1 who developed this concept for a single heat bath, one dependence on t, as the system is in a steady state, as can introduce a trajectory entropy for the evolving system discussed above. This entropy has several interesting S ðtÞ¼ −k log P½V ðtÞ;V ðtÞ, which extends to none- features. The first one is that hΔS i¼ 0, and as a s B 1 2 s;τ quilibrium systems the standard Gibbs entropy concept. consequence, hΔS i¼hΔS i, which grows with increas- tot r Therefore, when evaluating the total entropy production, ing ΔT. The second and most interesting result is that one has to take into account the contribution over the time independently of ΔT and of τ, the following equality interval τ of always holds: hexpð−ΔS =k Þi ¼ 1; ð26Þ tot B P½V ðt þ τÞ;V ðt þ τÞ 1 2 ΔS ¼ −k log : ð25Þ s;τ B P½V ðtÞ;V ðtÞ 1 2 for which we find experimental evidence, as discussed in the following, and provide a theoretical proof in It is worth noting that the system we consider is in a Refs. [91,92]. Equation (26) represents an extension to nonequilibrium steady state, with a constant external two temperature sources of the result obtained for a system driving ΔT. Therefore, the probability distribution in a single heat bath driven out of equilibrium by a time- PðV ;V Þ does not depend explicitly on the time, and dependent mechanical force [5,11], and our results provide 1 2 ΔS is nonvanishing whenever the final point of the the first experimental verification of the expression in a s;τ trajectory is different from the initial one: ½V ðt þ τÞ; system driven by a temperature difference. Equation (26) V ðt þ τÞ ≠ ½V ðtÞ;V ðtÞ. Thus, the total entropy change implies that hΔS i ≥ 0, as prescribed by the second law. 2 1 2 tot reads ΔS ¼ ΔS þ ΔS , where we omit the explicit From symmetry considerations, it follows immediately tot;τ r;τ s;τ 021051-13 Sym(E ) P(E ) 1,τ m,τ P(E ) T (K) m,τ fit S. CILIBERTO PHYS. REV. X 7, 021051 (2017) T = 88 K 1 (a) T = 88 K −1 1.1 (b) T = 296 K T = 296 K −2 ⟨ exp (− Δ S /k ) ⟩ tot B 0.9 50 100 150 200 250 300 −3 (c) −4 T = 88 K τ = 0.5 s T = 184 K τ = 0.5 s −5 T = 256 K τ = 0.5 s T = 88 K τ = 0.05 s Theory −6 −15 −10 −5 0 5 10 15 0 1 2 3 4 Δ S [k ] x B Δ S [k ] tot B FIG. 10. (a) The probability PðΔS Þ (dashed lines) and PðΔS Þ (continuous lines) measured at T ¼ 296 K (blue line) which r tot 1 corresponds to equilibrium and T ¼ 88 K (green lines) out of equilibrium. Notice that both distributions are centered at zero at equilibrium and shifted towards positive value in the out of equilibrium. (b) hexpð−ΔS Þi as a function of T at two different τ ¼ 0.5s tot 1 and τ ¼ 0.1s. (c) Symmetry function SymðΔS Þ¼ log½PðΔS Þ=Pð−ΔS Þ as a function of ΔS . The black straight line of slope 1 tot tot tot tot corresponds to the theoretical prediction. that, at equilibrium (T ¼ T ), the probability distribution main results. The experimental system is sketched in 1 2 of ΔS is symmetric: P ðΔS Þ¼ P ð−ΔS Þ. Thus, Fig. 11, and it is based on a single-electron box at tot eq tot eq tot low temperature. This is an excellent test benchmark for Eq. (26) implies that the probability density function of thermodynamics in small systems [99,100], and an inter- ΔS is a Dirac δ function when T ¼ T ; i.e., the quantity tot 1 2 esting review of the statistical properties of coupled ΔS is rigorously zero in equilibrium, both in average and tot fluctuations, and so its mean value and variance provide a circuits, both quantum and classical, can be found in measure of the entropy production. The measured proba- Ref. [101]. bilities PðΔS Þ and PðΔS Þ are shown in Fig. 10(a).We In the single-electron box shown in Fig. 11(a) the r tot see that PðΔS Þ and PðΔS Þ are quite different and that the electrons in the normal metal copper island (N) can tunnel r tot latter is close to a Gaussian and reduces to a Dirac δ to the superconducting Al island (S) through the aluminium oxide insulator (I). The integer net number of electrons function in equilibrium, i.e., T ¼ T ¼ 296 K [notice that, 1 2 tunneled from S to N is denoted by n. This number, in Fig. 10(a), the small broadening of the equilibrium monitored by the nearby single-electron transistor (SET) PðΔS Þ is just due to unavoidable experimental noise and tot discretization of the experimental probability density func- shown in Fig. 11(a), is the classical system degree of tions]. The experimental measurements satisfy Eq. (26) as freedom. it is shown in Fig. 10(b). It is worth noting that Eq. (26) Indeed, the device in Fig. 11(a) can be represented with implies that PðΔS Þ should satisfy a fluctuation theorem a classical electric circuit, in which the energy stored in tot of the form log½PðΔS Þ=Pð−ΔS Þ ¼ ΔS =k , ∀ τ; ΔT, the capacitors and in the voltage sources can be exactly tot tot tot B as discussed extensively in Refs. [2,53]. We clearly see in measured [99]. As in the previous section, Sec. VA 1, the Fig. 10(c)that this relation holds for different values of the conductor N and S are not at the same temperature. temperature gradient. Thus, this experiment clearly estab- Furthermore, here the system is driven by a voltage V lishes a relationship between the mean and the variance which oscillates much slower than the relaxation time of the of the entropy production rate in a system driven out of device. Thus, the forward and backward processes from equilibrium by the temperature difference between two the maximum to the minimum of V can be considered. thermal baths coupled by electrical noise. Because of the By the measured values of nðtÞ and V , one can estimate formal analogy with Brownian motion, the results also the heats, Q and Q , exchanged by the two heat baths N S apply to mechanical coupling [95,97,98]. in a time t , which is the period of the driving signal. In this th way the thermal entropy ΔS ¼ Q =T þ Q =T can be N N S S B. Entropy production in a single-electron box computed. Furthermore, the trajectory-dependent entropy can be estimated by measuring Δs ¼ −k log½P(nðt Þ)= Another interesting experiment on the measure of the b f entropy production in a system subjected to a temperature P(nð0Þ), where P(nðtÞ) is the probability that at time t the difference is presented in Ref. [93]. We summarize here the system is in the state nðtÞ for a value of the driving V ðtÞ. 021051-14 P(Δ S ) Sym(Δ S ) tot EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) th The total entropy is, of course, ΔS ¼ ΔS þ ΔS, and tot its probability distribution PðΔS Þ can be measured. tot The results for the forward and back processes are shown in Fig. 12(a), and the corresponding symmetry func- tions SymðΔS Þ¼ log½PðΔS Þ=Pð−ΔS Þ are plotted tot tot tot Fig. 12(b). In spite of the fact that PðΔS Þ are highly non- tot Gaussian, we notice that SymðΔS Þ¼ k ΔS , which tot B tot implies that Eq. (26) is also satisfied by these data. As in the previous section, Sec. VA 5, the main result of this experiment is that stochastic entropy production extracted from the trajectories is related to thermodynamic entropy production from dissipated heat in the respective thermal baths. VI. MOTOR POWER AND EFFICIENCY Historically, one of the main purposes of thermodynam- ics has been the study of the efficiency of thermal machines and power plants. Nowadays there is a wide interest in extending these studies to micro and nano motors which FIG. 11. (a) Sketch of the measured system together with a play a major role in biological mechanisms and small scanning electron micrograph of a typical sample. The colors devices. In Sec. II B, we see that in small systems all of the on the micrograph indicate the correspondingly colored circuit thermodynamics quantities fluctuate. Thus, we are inter- elements in the sketch. (b) Typical trace of the measured detector ested in knowing the influence of these fluctuations on the signal under a sinusoidal protocol for the drive V , plotted in efficiency of small devices where the dissipated energies green. This trace covers three realizations of the forward protocol and the produced work are a few k T. Furthermore, it is (V from −0.1 to 1 mV), and three realizations of the backward useful to know this efficiency at the maximum power and protocol (V from 1 to −0.1 mV). The SET current I , plotted in g det not in the quasistatic regimes, such as the Carnot cycle, black, indicates the charge state of the box. The output of the threshold detection is shown in solid blue, with the threshold level where the produced power is close to zero. These important indicated by the dashed red line (adapted from Ref. [93]). questions have been theoretically studied in several articles [2,102–107] and only in a few proof-of-principle experi- ments [108–111]. The first stochastic Carnot machine was reported in Ref. [108]. In this experiment a Brownian (a) particle trapped by an optical tweezer is subjected to a kind of Carnot cycle, inspired by a theoretical model proposed in Ref. [103]. The cycle, used in a very similar experiment [109], is sketched in Fig. 13(a), which we describe in some detail. The Brownian particle is trapped by a harmonic potential [bottom row in Fig. 13(a)] whose stiffness is changed as a function of time. The increase of the stiffness is equivalent to a compression (the motion of the particle (b) is more confined), the decrease to an expansion. In the experiment the bead is subjected to a random force which plays the role of an effective temperature, which can be easily changed by changing the amplitude of the random forcing. As in the Carnot cycle, the cycle in Fig. 13(a) is composed by an isothermal and an adiabatic compression and by an isothermal and an adiabatic expansion. Notice that the construction of adiabatic processes for a Brownian particle is a real challenge, which has been achieved by changing simultaneously the temperature in such a way FIG. 12. (a) Probability distribution of the total entropy ΔS , tot that the exchanged heat, during the adiabatics, is zero on which has been measured in the circuit shown in Fig. 11 and average (see Refs. [109,112] for details). The work and the described in the text. (b) The symmetry functions SymðΔS Þ¼ tot heat in this experiment are computed as described in log½PðΔS Þ=Pð−ΔS Þ of PðΔS Þ as a function of ΔS .In tot tot tot tot Secs. II B, IV, and V. As these two quantities fluctuate, spite of the highly non-Gaussian nature of PðΔS Þ, we see that tot SymðΔS Þ¼ k ΔS (adapted from Ref. [93]). the contribution of the fluctuations during adiabatics must tot B tot 021051-15 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) Another interesting article [113] presents the theoretical and experimental results on the conversion of one form of work to another. Using a Brownian particle as an isothermal machine driven by two independent periodic forces, the authors of Ref. [113] analytically compute and experimen- tally measure the stochastic thermodynamic properties of this Brownian engine. Specifically, the efficiency of the energy transfer between the two driving forces and the Onsager coefficients of the coupling are evaluated. The results of these experiments show the kind of problems that one encounters in the study of the efficiency of nano devices. For example, the study of the efficiency at maximum power and the behavior of η fluctuations have been subjects of extensive theoretical investigation [2,104–106], to understand their system dependence and eventually their universality. We do not discuss these theoretical results because they are far from the purposes of this review. The interested reader can look at the abovementioned references. However, it is worth mention- ing that several small devices, such as a molecular motor, are driven by chemical reaction, and the efficiency of these devices has been studied theoretically [107,114–116], but to my knowledge no proof-of-principle experiment, as the one presented here, has been performed for this chemically driven system. FIG. 13. (a) Schematic of the Carnot cycle applied to a Brownian particle trapped in a harmonic potential by a laser beam. The bottom row indicates the harmonic potential as a VII. FLUCTUATION DISSIPATION function of time. The stiffness of the potential is controlled by the RELATIONS FOR NESS laser intensity, as indicated by the green line. A random force, As we see in the previous section, current theoretical applied to the particle, plays the role of an effective temperature developments in nonequilibrium statistical mechanics whose value is changed as indicated by the magenta line. have led to significant progress in the study of systems (b) Contour lines of the probability distribution of the cycle efficiency averaged on n number of cycles. The efficiency η is around states far from thermal equilibrium. Systems in normalized to the Carnot efficiency η . The black dashed line c nonequilibrium steady states are the simplest examples indicates the mean value of η=η (adapted from Ref. [109]). because the dynamics of their degrees of freedom x under fixed control parameters λ can be statistically described be taken into account in computing the efficiency, which is by time-independent probability densities ρ ðx; λÞ. NESSs defined as η ¼ W =Q , where W is the work cycle cycle hot cycle naturally occur in mesoscopic systems such as colloidal produced during a cycle and Q is the heat absorbed from hot particles dragged by optical tweezeres, Brownian ratches, the hot sources. Because of W and Q fluctuations, η cycle and molecular motors because of the presence of non- also fluctuates a lot because, as we have seen in previous conservative or time-dependent forces [117].Atthese sections, Q can be zero and even negative. The measured length scales fluctuations are important, so it is essential to probability distribution of η is plotted in Fig. 13(b) as cycle establish a quantitative link between the statistical proper- function of the number of cycles used to average it. The ties of the NESS fluctuations and the response of the efficiency η is normalized to the standard Carnot system to external perturbations. Around thermal equi- cycle efficiency, η ¼ 1 − T =T . Although the mean value librium this link is provided by the fluctuation-dissipation c cold hot hη i is smaller than η , we clearly see that, for a small theorem [34]. cycle c number of cycles, η has big fluctuations which extend from The validity of the FDR in systems out of thermal values much larger than η to negative values. Furthermore, equilibrium has been the subject of intensive study during recent years. We recall that for a system in equilibrium with as the cycle is performed in finite time, the power pro- a thermal bath at temperature T the FDR establishes a duced by the system can be computed as a function of simple relation between the two-time correlation function the cycle duration. This power has a maximum, and the Cðt − sÞ of a given observable and the linear response other interesting result of this experiment is that the mean function Rðt; sÞ of this observable to a weak external efficiency at the maximum power follows the Curzon- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi perturbation, Ahlborn expression hη i¼ 1 − T =T [102]. cycle cold hot 021051-16 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) ∂ Cðt; sÞ¼ k TRðt; sÞ; ð27Þ theoretically and experimentally. This relationship is s B related to the famous paradox of a Maxwell’s demon, where in equilibrium Cðt; sÞ and Rðt; sÞ depend only on the which is an intelligent creature able to monitor individual time difference (t − s). However, Eq. (27) is not necessarily molecules of a gas contained in two neighboring chambers fulfilled out of equilibrium and violations are observed in a [141,142]. Initially, the two chambers are at the same variety of systems, such as glassy materials [24,118–123], temperature, defined by the mean kinetic energy of the granular matter [124], biological systems [125], and res- molecules and proportional to their mean-square velocity. onators [126]. Some of the particles, however, travel faster than others. This motivated a theoretical and experimental work By opening and closing a molecule-sized trapdoor in the devoted to a search of a general framework describing partitioning wall, the demon can collect the faster mole- FD relations; see the review Ref. [34]. The generalization of cules in one chamber and the slower ones in the other. The the fluctuation-dissipation theorem around NESS for sys- two chambers then contain gases with different temper- tems with Markovian dynamics has been achieved from atures, and that temperature difference may be used to different theoretical approaches [17,19,20,127–136]. The power a heat engine and produce mechanical work. By different generalized formulations of FDR link correlation gathering information about the particles positions and functions of the fluctuations of the observable of interest velocities and using that knowledge to sort them, the OðxÞ in the unperturbed NESS with the linear response demon is able to decrease the entropy of the system and function of OðxÞ due to a small external time-dependent convert information into energy. Assuming the trapdoor perturbation around the NESS. The observables involved is frictionless, the demon is able to do all that without in such relations are not unique, but they are equivalent in performing any work himself in an apparent violation of the the sense that they lead to the same values of the linear second law of thermodynamics. This paradox has origi- response function. These theoretical relations may be nated a long debate on the connection between information useful in experiments and simulations to know the linear and thermodynamics. A solution of the problem was response of the system around NESS. Indeed, the response proposed in 1929 by Leo Szilard, who used a simplified can be obtained from measurements entirely done at the one-particle engine to explain it. This gedanken experiment unperturbed NESS of the system of interest without any can nowadays be realized [141]. need to perform the actual perturbation. Nevertheless, the theoretical equivalence of the different observables A. Szilard engine: Work production from information involved in those relations does not translate into equiv- Modern technologies allow us to realize these gedanken alent experimental accessibility: e.g., strongly fluctuating experiments related to the Maxwell’s demon original idea. observables such as instantaneous velocities may lead to large statistical errors in the measurements [137]. 1. Sizlard engine Additionally, NESS quantities themselves, such as local For example, a Szilard engine was realized in 2010 mean velocities, joint stationary densities, and the stochas- [143] by using a single microscopic Brownian particle in a tic entropy, are not in general as easily measurable as fluid and confined to a spiral-staircase-like potential shown dynamical observables directly related to the degrees of in Fig. 14. Driven by thermal fluctuations, the particle freedom [18]. Hence, before implementing the different performs an erratic up and down motion along the staircase. fluctuation-response formulas in real situations, it is However, because of the potential gradient, downward important to test its experimental validity under very well steps will be more frequent than upward steps, and the controlled conditions and to assess the influence of particle will on average fall down. The position of the finite data analysis. The experimental test of some fluc- particle is measured with the help of a CCD camera. tuation-dissipation relations has been recently done in Each time the particle is observed to jump upwards, Refs. [18,61,137,138] for colloidal particles in toroidal this information is used to insert a potential barrier that optical traps and in systems subjected to thermal gradients hinders the particle to move down. By repeating this [139,140]. We do not describe here specific experimental procedure, the average particle motion is now upstairs results which have already been widely discussed in the and work is done against the potential gradient. By lifting abovementioned articles (see also Refs. [2,3,34]). What is the particle, mechanical work has therefore been produced important to recall is that this term is related to the out-of- by gathering information about its position. This is the first equilibrium current of the system, which is proportional to example of a device that converts information into energy the mean total entropy production for a NESS. for a system coupled to a single thermal environment. However, there is not a contradiction with the second law VIII. THERMODYNAMICS, INFORMATION, because Sagawa and Ueda [144] formalized the idea that AND THE MAXWELL DEMONS information gained through microlevel measurements can The relationship between stochastic thermodynamics be used to extract added work from a heat engine. Their and information now has an increasing importance both formula for the maximum extractable work is 021051-17 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) final equilibrium state can be smaller than the free-energy difference between the two states. Equation (29) has been directly tested in a single- electron transistor [146], similar to the one described in Sec. VIII A. 2. Autonomous Maxwell’s demon improves cooling In the previous section, Sec. VIII A, the Maxwell’s demon has been realized using an external feedback. However, working at low temperature and coupling in a suitable way the single-electron devices, already described in Sec. VIII A, one can construct a local feedback which behaves as an autonomous Maxwell’s demon and allows an efficient cooling of the system [147,148]. The device, whose principle is sketched in Fig. 15(a), is composed by a SET formed by a small normal metallic island connected to two normal metallic leads by tunnel junctions, which permit electron transport between the leads and the island. The SET is biased by a FIG. 14. (a) Experimental realization of Szilard’s engine. (a) A colloidal particle in a staircase potential moves downward on potential V and a gate voltage V , applied to the island via average, but energy fluctuations can push it upward from time to a capacitance, controls the current I flowing through the time. (b) When the demon observes such an event, it inserts a wall SET. The island is coupled capacitively with a single- to prevent downward steps. By repeating this procedure, the electron box which acts as a demon which detects the particle can be brought to move upwards, performing work presence of an electron in the island and applies a against the force created by the staircase potential. In the actual feedback. Specifically, when an electron tunnels to the experiment, the staircase potential is implemented by a tilted island, the demon traps it with a positive charge [illus- periodic potential and the insertion of the wall is simply realized trations 1 and 2 in Fig. 15(a)]. Conversely, when an by switching the potential, replacing a minimum (no wall) by a electron leaves the island, the demon applies a negative maximum (wall) (adapted from Ref. [143]). charge to repel further electrons that would enter the island [illustrations 3 and 4 in Fig. 15(a)]. This effect is obtained W ¼ −ΔF þ k ThIi; ð28Þ max B by designing the electrodes of the demon in such a way that when an electron enters the island from a source where ΔF is the free-energy difference between the electrode, an electron tunnels out of the demon island final and initial state and the extra term represents the as a response, exploiting the mutual Coulomb repulsion so-called mutual information I. In the absence of meas- between the two electrons. Similarly, when an electron urement errors this quantity reduces to the Shannon enters to the drain electrode from the system island, an entropy: I ¼ − PðΓ Þ ln½PðΓ Þ, where PðΓ Þ is the k k m k electron tunnels back to the demon island, attracted by the probability of finding the system in the state Γ . Then in overall positive charge. The cycle of these interactions the specific case of the previously described staircase between the two devices realizes the autonomous demon, potential [143]: I ¼ −p ln p − ð1 − pÞ ln p, where p is which allows the cooling of the leads. In the experimental the probability of finding the particle in a specific region. realization presented in Ref. [147], the leads and the In this context the Jarzynski equality discussed in demon were thermally insulated, and the measurements of Sec. IV also contains this extra term and it becomes their temperatures is used to characterize the effect of the demon on the device operation. In Fig. 15(b) we plot the hexpð−βW þ IÞi ¼ expð−βΔFÞ; ð29Þ variation of the lead temperatures as a function of n ∝ V g g when the demon acts on the system. We clearly see that which leads to around n ¼ 1=2 the two leads are both cooled of 1 mK at a mean temperature of 50 mK. This occurs because the hWi ≥ ΔF − k ThIi: ð30Þ tunneling electrons have to take the energy from the thermal energy of the leads, which, being thermally Equations (29) and (30) generalize the second law of isolated, cool down. This increases the rate at which thermodynamics, taking into account the amount of infor- electrons tunnel against Coulomb repulsion, giving rise to mation introduced into the system [142,145]. Indeed increased cooling power. At the same time, the demon Eq. (30) indicates that, thanks to information, the work increases its temperature because it has to dissipate energy performed on the system to drive it between an initial and a in order to process information, as discussed in Ref. [149]. 021051-18 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) B. Energy cost of information erasure The experiments in the previous two sections show that one can extract work from information. In the rest of this section, we discuss the reverse process, i.e., the energy needed to erase information. By applying the second law of thermodynamics, Landauer demonstrated that information erasure is necessarily a dissipative process: the erasure of one bit of information is accom- panied by the production of at least k T lnð2Þ of heat into the environment. This result is known as Landauer’s erasure principle. It emphasizes the fundamental differ- ence between the process of writing and erasing infor- mation. Writing is akin to copying information from one device to another: state left is mapped to left and state right is mapped to right, for example. This one-to-one mapping can be realized in principle without dissipating any heat (in statistical mechanics one would say that it conserves the volume in phase space). By contrast, erasing information is a two-to-one transformation: states left and right are mapped onto one single state, say, right (this process does not conserve the volume in phase space and is thus dissipative). FIG. 15. (a) Principle of the experimental realization of the Landauer’s original thought experiment was realized autonomous Maxwell’s demon. The horizontal top row schema- [150,151] for the first time in a real system in 2011 using tizes a single-electron transistor. Electrons (blue circle) can tunnel a colloidal Brownian particle in a fluid trapped in a double- inside the central island from the left wall and outside from the well potential produced by two strongly focused laser right wall. The demon watches at the state of the island and it beams. This system has two distinct states (particle in the applies a positive charge to attract the electrons when they tunnel right or left well) and may thus be used to store one bit of inside and repels them when they tunnel outside. The systems cools because of the energy released toward the heat bath by information. The erasure principle has been verified by the tunneling events, and the presence of the demon makes the implementing a protocol proposed by Bennett and illus- cooling processes more efficient. The energy variation of the trated in Fig. 16. At the beginning of the erasure process, processes is negative because of the information introduced by the colloidal particle may be either in the left or right well the demon. (b) The measured temperature variations of the left with equal probability of one-half. The erasure protocol is (blue line) and right (green line) leads as a function of the external composed of the following steps: (1) the barrier height is control parameter n when the demon is active and the bath first decreased by varying the laser intensity, (2) the particle temperature is 50 mK. We see that at the optimum value, is then pushed to the right by gently inclining the potential. n ¼ 1=2, both leads are cooled to about 1 mK and the current and (3) the potential is brought back to its initial shape. I flowing through the SET (black line) has a maximum. At the At the end of the process, the particle is in the right well same time, in order to processes information the temperature of the demon (red line) increases a few mK. (c) The same parameters with unit probability, irrespective of its departure position. of (b) are measured when the demon is not active. We see that the As in the previous experiment, the position of the particle is demon temperature does not change, whereas both leads are now recorded with the help of a camera. For a full erasure cycle, heated by the current I. the average heat dissipated into the environment is equal to the average work needed to modulate the form of the Thus, the total (system plus demon) energy production is double-well potential. This quantity was evaluated from the positive. The coupling of the demon with the SET can be measured trajectory and shown to always be larger than controlled by a second gate which acts on the single- the Landauer bound, which is asymptotically approached electron box. In Fig. 15(c) we plot the measured temper- in the limit of long erasure times. However, in order to atures when the demon has been switched off. We clearly reach the bound, the protocol must be accurately chosen see that in such a case the demon temperature does not because, as discussed in Ref. [150] and shown experimen- change and the two electrodes are heating up because of tally [152], there are protocols that are intrinsically irre- the current flow. As far as I know, this is the only example versible no matter how slowly they are performed. The way that shows that under specific conditions an autonomous in which a protocol can be optimized has been theoretically local Maxwell’s demon, which does not use the external solved in Ref. [153], but the optimal protocol is not often feedback, can be realized. easy to apply in an experiment. 021051-19 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) FIG. 16. Experimental verification of Landauer’s erasure principle. A colloidal particle is initially confined in one of two wells of a double-well potential with probability one-half. This configuration stores one bit of information. By modulating the height of the barrier and applying a tilt, the particle can be brought to one of the wells with probability one, irrespective of the initial position. This final configuration corresponds to zero bit of information. In the limit of long erasure cycles, the heat dissipated during the erasure process can approach, but not exceed, the Landauer bound indicated by the dashed line in the right panel (see Ref. [150] for details). C. Other examples on the connection between information can be used to produce motion and, on the information and energy contrary, information processing needs energy. We do not discuss in more detail this important topic, which is By having successfully turned gedanken into real experi- developed in two other articles [163,164]. ments, the above three seminal examples provide a firm empirical foundation to the physics of information and the intimate connection existing between information and IX. USE OF STOCHASTIC THERMODYNAMICS energy. This connection is reenforced by the relationship IN EXPERIMENTS: DISCUSSION between the generalized Jarzinsky equality [154] and the AND PERSPECTIVES Landauer bound, which has been proved and tested on In this review we present various experimental results experimental data in Ref. [151]. A recent article [155] that allow us to introduce several fundamental concepts extends the equivalence of information and thermodynamic of stochastic thermodynamics, such as the FT for heat entropies at thermal equilibrium. and work, the Jarzynski equality, and the trajectory entropy. A number of additional experiments have been performed We have already mentioned several applications of these on this subject [156–160]. For example, in Ref. [157] the concepts to the measure of the system response in NESS, to symmetry breaking, induced in the probability distribution the free-energy estimation, and to the relationship between of the position of a Brownian particle, is studied by com- information and thermodynamics. In the experiments that muting the trapping potential from a single- to a double-well we have described, all of the theoretical predictions are potential. The authors measured the time evolution of the perfectly verified; thus, the question here is to see whether system entropy and showed how to produce work from those findings of stochastic thermodynamics might become information. The experiment in Ref. [160] shows that using a a useful measurement tool that allows us to calibrate and Maxwell’s demon in the information erasure costs less to make predictions in experiments. We discuss here energy than the Landauer bound. It is worth mentioning several examples. experiments where the Landauer bound has been reached in nano devices [156,159]. These experiments open the way to A. Using FT to calibrate an experiment insightful applications for future developments of informa- tion technology. FT can be used to have a precise estimation of either Finally, the connection between thermodynamics an offset or a calibration error in an experiment. The and information plays a very important role in the method can be easily understood by considering the understanding of biological systems [161,162]. Indeed, systems modeled by either one or several Langevin 021051-20 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) equations described in Secs. II, III D, and V. For these D. Nano or micro motor efficiency systems we have seen that the work and heat PDF satisfy The other aspect that we have briefly discussed in a FT. Thus, in an experimental system that can be modeled Sec. VI concerns the efficiency of nano or micro motor. by the Langevin equation, one can check the good In spite of the large number of theoretical results on this calibration of the apparatus by computing the work and subject, there is a lack of experiments in this important checking whether the FT holds for the measured variables. field, especially in what concerns the efficiency of power If it does not hold, it means that some error has been made supplies and of chemically driven motors. A bound on the in the calibration or some small offset exists in the efficiency of these motors has been recently fixed by the measured quantities (see also Refs. [38,165]). recently discovered indetermination relations [114–116]. However, as we have already said, no proof-of-principle B. Role of hidden variables experiment, such as the Carnot cycle discussed in Sec. VI, has been performed for chemically driven motors. Along the same line of Sec. IX A, FT can be used to estimate unknown parameters of a device. For example, E. Role of Maxwell’s demons such a method has been used to measure the power of molecular motors [38,166]. The idea is certainly very The role of Maxwell’s demons in increasing the efficiency smart and merits serious consideration for applications. of small devices is certainly very important. This subject is in However, one has to pay attention to the influence of its infancy and it might have very powerful applications. The hidden variables. Indeed, in Ref. [167] it has been described autonomous Maxwell’s demon is certainly a smart pointed out that in the abovementioned experiment of system. However, it works because of the very small Ref. [166] a hidden variable has been neglected; thus, the operational temperature of the device. A very big challenge estimated value of the motor power could be affected is the realization of such an autonomous demon for a device by a large error. Another interesting analysis on hidden working at room temperature. At the moment, such a device does not exist, and it is not even clear in which physical or variables has been done in Ref. [168] for the experiment chemical system it could be realized. of Ref. [169] on a SET. In this experiment the bias obtained from the measured FT does not completely match with the experimental value of the applied bias. In F. Energy information connection Ref. [168] it has been shown that this discrepancy is due The connection of stochastic thermodynamics with the to the additional bias introduced by the SET that is used energy dissipation in each logic operation is the last issue to measure the electron transfers. More recent measures that we have developed. It is clearly very important in proved that this is the case [170]. Another accurate connection to the autonomous Maxwell’s demon to esti- experimental analysis of the role of a hidden variable mate the amount of work that the demon has to perform in on FT properties has been reported in Ref. [171]. This order to process information. This will allow us to decide role can be easily understood by looking at the experi- whether the application of a demon is really an advantage to ment described in Sec. V, which is modeled by the reduce the energy consumption. Furthermore, deeper coupled Langevin equations (20) and (21). Suppose now knowledge of the connection between thermodynamics that in that experiment instead of having access to both and information is certainly useful not only to understand V and V variables, only one of the two can be 1 2 biological processes but also to develop methods that allow measured. We immediately see that there is no way of us to recover energy during reversible logical operation. extracting the good values from the measurements. Thus, Also, this field is in his infancy and future development will the use of FT to extract experimental parameters from an certainly appear. experiment might be a very good method, but it has to be used with caution because of the possible existence of G. Macroscopic and self-propelling systems hidden variables. The are several aspects that we do not present because this will extend too much the purpose of this review. C. Statistical inferences The first is the application to quantum systems, which The problem of hidden variables mentioned in Sec. IX B presents several problems discussed in another article has been attacked in a different way in Refs. [172,173]. [174]. The other aspects are those related to the application This approach, called statistical inferences by the authors, of stochastic thermodynamics to dissipative chaotic sys- analyzes to what extent the fact that FT and FDR do not tems driven out of equilibrium by external forces such as, hold can give information on hidden variables. In for example, shaken granular media, turbulence fields, and Ref. [173] the authors have been able to extract interesting chaotic nonlinear oscillators. In these systems the main information on a single-molecule measurements. Thus, the source of randomness is not the thermal noise but the approach is certainly interesting and it merits further study chaotic behavior produced by the complex dynamics. As in more detail in the future. summarized in Ref. [3], the main problem is the absence of 021051-21 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) Connection with Response Theory, J. Chem. 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Experiments in Stochastic Thermodynamics: Short History and Perspectives

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PHYSICAL REVIEW X 7, 021051 (2017) S. Ciliberto Université de Lyon, CNRS, École Normale Supérieure de Lyon, Laboratoire de Physique (UMR5672), 46 Allée d’Italie 69364 Lyon Cedex 07, France (Received 5 April 2017; published 30 June 2017) We summarize in this article the experiments which have been performed to test the theoretical findings in stochastic thermodynamics such as fluctuation theorem, Jarzynski equality, stochastic entropy, out-of- equilibrium fluctuation dissipation theorem, and the generalized first and second laws. We briefly describe experiments on mechanical oscillators, colloids, biological systems, and electric circuits in which the statistical properties of out-of-equilibrium fluctuations have been measured and characterized using the abovementioned tools. We discuss the main findings and drawbacks. Special emphasis is given to the connection between information and thermodynamics. The perspectives and followup of stochastic thermodynamics in future experiments and in practical applications are also discussed. DOI: 10.1103/PhysRevX.7.021051 Subject Areas: Statistical Physics I. INTRODUCTION We have already mentioned the Brownian particle driven by an external force, but to clarify the kind of questions that When the size of a system is reduced, the role of we want to analyze, let us consider another simple example fluctuations (either quantum or thermal) increases. Thus, of an out-of-equilibrium system, that is, a thermal con- thermodynamic quantities such as internal energy, work, ductor whose extremities are connected to two heat baths at heat, and entropy cannot be characterized only by their different temperatures. The second law of thermodynamics mean values, but also their fluctuations and probability imposes that the mean heat flux flows from the hot to the distributions become relevant and useful to make predic- cold reservoir. However, the second law does not say tions on a small system. Let us consider a simple example, anything about fluctuations, and in principle one can such as the motion of a Brownian particle subjected to a observe for a short time a heat current in the opposite constant external force. Because of thermal fluctuations, direction, which corresponds to an instantaneous negative the work performed on the particle by this force per unit entropy production rate. What is the probability of observ- time, i.e., the injected power, fluctuates, and the smaller ing these rare events? The same problem appears in the the force, the larger is the importance of power fluctuations abovementioned example of the Brownian particle where [1–3]. The goal of stochastic thermodynamics is just that one can ask, what is the probability that the particles move of studying the fluctuations of the abovementioned thermo- in the opposite direction of the force? The answer to these dynamic quantities in systems driven out of equilibrium questions can be found within the framework of stochastic by external forces, temperature differences, and chemical thermodynamics and fluctuation theorems (FTs) [4–12], reactions. For this reason, it has received in the past which uses statistical mechanics to answer questions 20 years an increasing interest for its applications in related to extremes that are well beyond the mean (i.e., microscopic devices and biological systems and for its thermodynamics) and well beyond the standard fluctuation connections with information theory [1–3]. theory normally dominated, away from critical points by In the following we discuss the role of fluctuations in the central limit theorem. We see that the knowledge of out- out-of-equilibrium thermal systems when the energies of-equilibrium fluctuation properties is actually very useful injected or dissipated are smaller than 100k T (k being B B in experiments to extract useful information on equilibrium the Boltzmann constant and T the temperature). This limit and out-of-equilibrium properties of a specific system. is relevant in biological, nano, and micro systems, where Typical examples are the Jarzynski and Crooks equalities fluctuations cannot be neglected. We are interested in [13–15], which estimate equilibrium properties starting knowing the role of these fluctuations on the dynamics from nonequilibrium measurements. The measurement of and how one can gain some information by measuring them. the linear response in out-of-equilibrium systems is another very important aspect. Indeed, the new formulations of the fluctuation dissipation relation (FDR) related to the FT are Published by the American Physical Society under the terms of quite useful for this purpose, because they allow the the Creative Commons Attribution 4.0 International license. estimation of the response starting from the measurement Further distribution of this work must maintain attribution to of fluctuations of different quantities in nonequilibrium the author(s) and the published article’s title, journal citation, steady states (NESSs) [10,16–21]. Within the context of the and DOI. 2160-3308=17=7(2)=021051(26) 021051-1 Published by the American Physical Society S. CILIBERTO PHYS. REV. X 7, 021051 (2017) using different theoretical tools We start in Sec. II with FDR for out-of-equilibrium states, many studies have been done on the slow relaxation toward equilibrium, such as in an analysis of the experimental results on the energy aging glasses after a temperature quench [22–24]. It turns fluctuations in a harmonic oscillator driven out of equilib- rium by an external force. In Sec. III, we describe the out that entropy production plays a unifying role between the FT and the different extended formulations of the FDR properties of FTs, and as illustrative examples we apply it for out-of-equilibrium systems. to (a) a harmonic oscillator (linear case) and (b) a Brownian particle confined in a time-dependent double-well potential Another application of stochastic thermodynamics is the (nonlinear case). The latter is one of the very few examples study of the efficiency of micro or nano devices and the where the FT is applied to a highly nonlinear potential, role of fluctuations in the power production of these because most of the experiments reported in the literature systems. This is of course very useful for understanding are performed for linear potentials. In Sec. IV, we discuss and measuring the efficiency of molecular motors, which the application of the Jarzinsky and Crooks equalities to are isothermal and driven by chemical reactions. the harmonic oscillators and to the measure of the free It is worth mentioning that the study of stochastic energy of a single molecule. In Sec. VI, we introduce the thermodynamics has allowed us to bring more insight to application of stochastic thermodynamics to the study of the connection between information and thermodynamics. the efficiency of micro or nano machines. The contribution Specifically, the study of the energy fluctuations in a small of fluctuations to the power produced by these machines is system has transformed gedanken experiments, such as the described using the results of a proof of principle experi- Maxwell’s demon, in experiments which may actually be ment. In Sec. VII, we briefly present another relevant performed thanks to the new technologies such as optical or aspect of nonequilibrium statistical mechanics: the meas- electrical traps and single electron devices. urement of the linear response of a system in a non- Before explaining how the article is organized, it is equilibrium state. We discuss here only the main relevant mandatory to point out that the tools of stochastic thermo- features without giving any specific example as these FDT dynamics have also been applied to study the properties aspects have already been discussed in other reviews of macroscopic fluctuations in out-of-equilibrium systems. [2,3,34]. In Sec. VIII, the connections between information Indeed, the injected and dissipated energies may also and thermodynamics is analyzed following two comple- fluctuate in macroscopic systems if the dynamics is chaotic. mentary subjects. The first is the energy production by For instance, think of a motor used to stir a fluid strongly. devices controlled by a Maxwell’s demon. The second is the The motor can be driven by imposing a constant velocity. minimum energy needed to process one bit of information. Because of the turbulent motion of the fluid, the power Finally, we conclude in Sec. IX, where we describe other needed to keep the velocity constant fluctuates [25,26]. useful experimental applications of stochastic thermody- This simple example shows that fluctuations of the injected namics. We also discuss the perspectives and the artifacts and dissipated power may be relevant not only in micro- of these applications. scopic but also in macroscopic systems such as hydro- dynamic flows [26], granular media [27–30], mechanical systems [31], and more recently on self-propelling particles II. WORK AND HEAT FLUCTUATIONS IN [32,33]. The main difference is that in macroscopic systems THE HARMONIC OSCILLATOR fluctuations are produced by the dynamics and are sus- The choice of discussing the dynamics of the harmonic tained by a constant energy flux, whereas in small systems oscillator is dictated by the fact that it is relevant for many they are of either thermal or quantum nature. Thus, it is practical applications, such as the measure of the elasticity useful to divide the fluctuation in out-of-equilibrium of nanotubes [35], the dynamics of the tip of an AFM [36], systems into two classes: one where thermal fluctuations MEMS, and the thermal rheometer that we developed play a significant role (thermal systems) and another where several years ago to study the rheology of complex fluids the fluctuations are produced by chaotic flows or fluctuat- [37], whose high sensitivity has actually allowed several ing driving forces (athermal chaotic systems). In this tests of FT. As the results of these experiments have already article, we focus on thermal systems and only a short been discussed in some detail in several reviews [3,38],we discussion on the problems related to the application of describe here only the main results useful to introduce the stochastic thermodynamics to athermal systems is provided experimental activity in stochastic thermodynamics. in Sec. IX G. As the goal of this article is to present several A. Experimental setup general experimental aspects, we follow an experimentalist approach, and the connection with theory is made on the The thermal rheometer is a torsion pendulum whose basis of experimental measurements. Furthermore, we angular displacement θ is measured by a very sensitive organize the sections in terms of the main topics and tools interferometer. The details of the setup can be found in of stochastic thermodynamics. For this reason, the same Refs. [39–43]. A schematic diagram and a picture of the experimental apparatus is analyzed in various sections apparatus are shown in Fig. 1. 021051-2 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) with theoretical predictions often obtained for Markovian processes. (See Ref. [38] for a discussion on this point.) B. Energy balance When the system is driven out of equilibrium by the external deterministic torque M (which is, in general, time dependent), it receives an amount of work, and a fraction of this energy is dissipated into the heat bath. Multiplying Eq. (1) by θ and integrating between t and t þ τ, one i i obtains a formulation of the first law of thermodynamics between the two states at time t and t þ τ [Eq. (2)]. This i i formulation was first proposed in Ref. [44] and widely used in other theoretical and experimental works in the context of stochastic thermodynamics [1,2]. The change in internal energy ΔU of the oscillator over a time τ, starting at a time t , is written as ΔU ¼ Uðt þ τÞ − Uðt Þ¼ W − Q ; ð2Þ τ i i τ τ FIG. 1. (a) The torsion pendulum. (b) The magnetostatic forcing. (c) A picture of the pendulum. (d) The cell where the where W is the work done on the system over a time τ and pendulum is installed. Q is the dissipated heat. The work W is defined in the τ τ classical way, In equilibrium the variance δθ of the thermal fluctua- tions of θ can be obtained from equipartition; i.e., for our t þτ dθðt Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 W ¼ Mðt Þ dt ; ð3Þ pendulum, δθ ¼ k T=C ≃ 2 nrad, where C is the tor- dt sional stiffness of the pendulum and T is the temperature of the surrounding fluid. The measurement noise is 2 orders of and we use a tilde in order to distinguish it from a more magnitude smaller than thermal fluctuations of the pendu- general definition often used in stochastic thermodynamics lum whose resonance frequency f is about 217 Hz. A o [see Eq. (12) and the discussion at the end of Sec. IVA]. magnetostatic forcing [38,40,41] allows the application of The internal energy is the sum of the potential energy and an external torque M, useful to excite the pendulum and to the kinetic energy: drive it out of equilibrium. The typical applied torque is of the order of a few pN m, and the mean power a few k T=s. 2 1 dθðtÞ 1 UðtÞ¼ I þ CθðtÞ : ð4Þ The dynamics of the torsion pendulum can be assimi- eff 2 dt 2 lated to that of a harmonic oscillator damped by the viscosity of the surrounding fluid, whose equation of The heat transfer Q is deduced from Eq. (2). It has two motion reads contributions: d θ dθ t þτ dθ dθ I þ ν þ Cθ ¼ M þ η; ð1Þ eff 0 0 Q ¼ W − ΔU ¼ ν − ηðt Þ dt ; ð5Þ dt dt τ τ τ 0 0 dt dt where I is the effective moment of inertia of the eff where the integrals in Eqs. (3) and (5) are performed using pendulum, which includes the inertia of the surrounding the Stratonovich convention. fluid as discussed in Ref. [40]. The thermal noise η is, The first term in Eq. (5) corresponds to the viscous in this case, delta correlated in time: hηðtÞηðt Þi ¼ dissipation and is always positive, whereas the second term 2k Tνδðt − t Þ. However, if the fluid is viscoelastic, the can be interpreted as the work of the thermal noise, which noise η is correlated and the process is not Markovian, has a fluctuating sign. The second law of thermodynamics whereas in the viscous case the process is Markovian. Thus, imposes hQ i to be positive. Notice that because of the by changing the quality of the fluid surrounding the τ pendulum one can tune the Markovian nature of the fluctuations of θ and θ all the quantities W , ΔU , and Q τ τ τ process. In this review we consider only the experiment fluctuate, too. We are interested in characterizing these in the glycerol-water mixture where the viscoelastic con- fluctuations, which are related by Eq. (2), which is a tribution is visible only at very low frequencies and is formulation of the first law of thermodynamics between the therefore negligible. This allows a more precise comparison two states at time t and t þ τ. Although Eq. (2) has been i i 021051-3 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) FIG. 2. Sinusoidal forcing. (a) Pdf of W . (b) PDF of Q for various n: n ¼ 7 (∘), n ¼ 15 (□), n ¼ 25 (⋄), and n ¼ 50 (×). The τ τ continuous lines are not fits but are analytical predictions obtained from the Langevin dynamics, as discussed in Sec. III C. obtained for the harmonic oscillators, is indeed a general from the measure of W and ΔU . We first make some τ τ statement for the energy fluctuations of any system. comments on the average values. The average of ΔU is obviously vanishing because the time τ is a multiple of the C. Nonequilibrium steady state: Sinusoidal forcing ~ period of the forcing. Therefore, hW i and hQ i are equal, τ τ as it must be. We now consider a periodic forcing of amplitude M and We rescale the work W (the heat Q ) by the average frequency ω , i.e., MðtÞ¼ M sinðω tÞ [41–43], which is a τ τ d o d NESS because all the averages performed on an integer work hW i (the average heat hQ i) and define w ¼ τ τ τ number of the driving periods do not depend on time. This ~ ~ ðW =hW iÞ (q ¼ðQ =hQ iÞ). In the present article, x τ τ τ τ τ τ kind of periodic forcing is very common and it has been (X ) stands for either w or q (W or Q ). τ τ τ τ τ studied in the case of the first-order Langevin equation [45] We compare now the PDF of w and q in Fig. 2. The τ τ and of a two-level system [46] and in a different context for PDFs of heat fluctuations q have exponential tails the second-order Langevin equation [47]. Furthermore, this [Fig. 2(b)]. It is interesting to stress that although the is a very general case, because using Fourier transform any two variables W and Q have the same mean values, they τ τ periodical driving can be decomposed in a sum of sinus- have a very different PDF. The PDFs of w are Gaussian, oidal modes. We explain here the behavior of a single whereas those of q are exponential. On a first approxi- mode. Experiments have been performed at various M and mation, the PDFs of q are the convolution of a Gaussian ω . We present the results for a particular amplitude and (the PDF of W ) and exponential (the PDF of ΔU ) [38,43]. τ τ frequency: M ¼ 0.78 pN m and ω =ð2πÞ¼ 64 Hz. o d In Fig. 2, the continuous lines are analytical predictions obtained from the Langevin dynamics with no adjustable 1. Work fluctuations parameter (see Sec. III C). The work done by MðtÞ is computed from Eq. (3) on a time τ ¼ 2πn=ω , i.e., an integer number n of the driving III. FLUCTUATION THEOREMS period. W fluctuates and its probability density function In the previous section, we see that both W and Q (PDF) is plotted in Fig. 2(a) for various n. This plot has τ τ present negative values; i.e., the second law is verified only interesting features. Specifically, work fluctuations are on average, but the entropy production can have instanta- Gaussian for all values of n, and W takes negative values neously negative values. The probabilities of getting as long as τ is not too large. The probability of having positive and negative entropy production are quantitatively negative values of W decreases when τ is increased. τ n related in nonequilibrium systems by the fluctuation There is a finite probability of having negative values theorems [4–6], of the work; in other words, the system may have an There are two classes of FTs. The stationary state instantaneous negative entropy production rate although fluctuation theorem (SSFT) considers a nonequilibrium the average of the work hW i is, of course, positive (h·i steady state. The transient fluctuation theorem (TFT) stands for ensemble average). In this specific example, describes transient nonequilibrium states where τ measures hW i¼ 0.04nðk TÞ. τ B the time since the system left the equilibrium state. A fluctuation relation examines the symmetry around 0 of 2. Heat fluctuations the probability density function pðx Þ of a quantity x ,as τ τ The dissipated heat Q cannot be directly measured defined in the previous section. It compares the probability because Eq. (5) requires us to compute the work done by of having a positive event (x ¼þx) versus the probability of having a negative event (x ¼ −x). We quantify the FT the thermal noise, which is experimentally unmeasurable since η is unknown. However, Q can be obtained indirectly using a function (symmetry function): 021051-4 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) k T pðx ¼þxÞ Symðx Þ and x . In the region where the symmetry B τ τ τ Symðx Þ¼ ln : ð6Þ function is linear with x , we define the slope Σ ðτÞ, i.e., hX i pðx ¼ −xÞ τ x τ τ Symðx Þ¼ Σ ðτÞx . As a second step, we measure finite- τ x τ time corrections to the value Σ ðτÞ¼ 1, which is the The TFT states that the symmetry function is linear asymptotic value expected from FTs. with x for any values of the time integration τ and the In this article, we focus on the SSFT applied to the proportionality coefficient is equal to 1 for any value of τ: experimental results of Sec. II C and to other examples. The Symðx Þ¼ x ; ∀x ; ∀τ: ð7Þ applications of TFT are not presented in this section, but τ τ τ we discuss them in Sec. IV B, and interested readers may Contrary to TFT, the SSFT holds only in the limit of infinite find more details in Ref. [43]. time (τ): From the PDFs of w and q plotted in Fig. 2,we τ τ compute the symmetry functions defined in Eq. (6). The lim Symðx Þ¼ x : ð8Þ τ τ symmetry functions Symðw Þ are plotted in Fig. 3(a) as a τ→∞ function of w . They are linear in w . The slope Σ ðnÞ is not τ τ w equal to 1 for all n, but there is a correction at finite time. In the following, we assume linearity at finite time τ Nevertheless, Σ ðnÞ tends to 1 for large n. Thus, SSFT is [48,49] and use the following general expression: satisfied for W and for a sinusoidal forcing. The con- Symðx Þ¼ Σ ðτÞx ; ð9Þ τ x τ vergence is very slow and we have to wait a large number of periods of forcing for the slope to be 1 (after 30 periods, where for SSFT Σ ðτÞ takes into account the finite-time the slope is still 0.9). This behavior is independent of the corrections and lim Σ ðτÞ¼ 1, whereas Σ ðτÞ¼ 1, ∀ τ τ→∞ x x amplitude of the forcing M and consequently of the mean for TFT. value of the work hW i, which, as explained in Ref. [38], However, these claims are not universal because they changes only the time needed to observe a negative event. depend on the kind of x that is used. Specifically, we see in The system satisfies the SSFT for all forcing frequencies the next sections that the results are not exactly the same if ω , but finite-time corrections depend on ω [43]. d d X is replaced by any one of W , Q , and the total entropy τ τ τ We now analyze the PDF of q [Fig. 2(b)] and we [11,12]. Furthermore, the definitions given in this section compute the symmetry functions Symðq Þ of q plotted in τ τ are appropriate for stochastic systems, and the differences Fig. 3(b) for different values of n. They are clearly very between stochastic and chaotic systems are not addressed different from those of w plotted in Fig. 3(a). For Symðq Þ n τ in this review. A discussion on this point can be found three different regions appear. in Ref. [38]. (I) For large fluctuations q , Symðq Þ equals 2. When τ n τ tends to infinity, this region spans from q ¼ 3 to infinity. A. FTs for W and Q measured (II) For small fluctuations q , Symðq Þ is a linear τ τ n n in the harmonic oscillator function of q . We then define Σ ðnÞ as the slope of the function Symðq Þ, i.e., Symðq Þ¼ Σ ðnÞq . We have The questions we ask are whether for finite time FTs are n n q n satisfied for either x ¼ w or x ¼ q and what are the measured [43] that Σ ðnÞ¼ Σ ðnÞ for all the values of q w τ τ τ τ finite-time corrections? As a first step, we test the correc- n; i.e., finite-time corrections are the same for heat and tion to the proportionality between the symmetry function work. Thus, Σ ðnÞ tends to 1 when τ is increased and SSFT FIG. 3. Sinusoidal forcing. Symmetry functions for SSFT. (a) Symmetry functions Symðw Þ plotted as a function of w for various n: τ τ n ¼ 7 (circle), n ¼ 15 (square), n ¼ 25 (diamond), and n ¼ 50 (times). For all n, the dependence of Symðw Þ on w is linear, with slope τ τ Σ ðτÞ. (b) Symmetry functions Symðq Þ plotted as a function of q for various n. The dependence of Symðq Þ on q is linear only for τ τ τ τ q < 1. Continuous lines are theoretical predictions. 021051-5 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) holds in this region II, which spans from q ¼ 0 up to discussion on the different quantities for SSFT and TFT q ¼ 1 for large τ. This effect was discussed for the first can be found. time in Refs. [48,49]. Furthermore, in order to avoid the complexity of (III) A smooth connection between the two behaviors. computing ΔS from individual trajectories, another tot These regions define the fluctuation relation from the quantity, which satisfies a SSFT for any τ, has been heat dissipated by the oscillator. The limit for large τ of the proposed in Ref. [54]. This quantity is the joint probability symmetry function Symðq Þ is rather delicate and it is τ PðW ;J Þ of the work and of the energy currents in the τ τ discussed in Ref. [43]. system. Although its measure might be difficult, it is by far The conclusion of this experimental analysis is that easier than the trajectory-dependent entropy. However, this SSFT holds for work for any value of w , whereas for heat it τ method, although very powerful, has never been tested on holds only for q < 1. The finite-time corrections to FTs, τ experimental data, but it will certainly be useful to try. described by 1 − Σ, are not universal. They are the same for both w and q , but they depend on the driving frequency τ τ C. Comparison with theory and on the kind of driving force Ref. [38,43]. The experimental analysis described in Sec. III A allows These kinds of measurements are important because they a very precise comparison with theoretical predictions allow us to test complex theoretical concepts on relatively using the Langevin equation [Eq. (1)] and using two simple systems in order to apply them to more complex experimental observations: (a) the properties of the heat cases. Furthermore, the experimental analysis on model bath are not modified by the driving and (b) the fluctuations systems allows us to check the theoretical hypothesis made of the W are Gaussian (see also Ref. [55], where it is in order to prove the theorems. For example, one of these shown that in Langevin dynamics W has a Gaussian hypotheses is that the properties of the heat bath are not τ distribution for any kind of deterministic driving force if the modified by the forcing. This hypothesis can be precisely properties of the bath are not modified by the driving and checked in the experiments. the potential is harmonic). The observation in point (a) is extremely important because it is always assumed to be true B. Trajectory-dependent entropy in all the theoretical analysis. In Ref. [43], this point has and the total entropy been precisely checked. Using these experimental obser- In the same way of W and Q , the entropy produc- τ τ vations one can compute the PDF of q and the finite-time tion rate can also be defined at the trajectory level. The corrections ΣðτÞ to SSFT (see Ref. [43]). The continuous trajectory-dependent entropy difference δs ðtÞ is defined as lines in Figs. 2 and 3 are not fit but analytical predictions, δs ðtÞ¼ − log½P(r⃗ ðt þ τÞ; λ)=P(r⃗ ðtÞ; λ), where P(r⃗ ðtÞ; λ) with no adjustable parameters, derived from the Langevin is the probability of finding the system in the position dynamics of Eq. (1) (see Ref. [43] for more details). r⃗ ðtÞ of the phase space at a value λ of the control parameter. Thus, the total entropy difference on the time τ is D. Nonlinear case: Stochastic resonance ΔS ðt; τÞ¼ δs ðtÞþ Q ðtÞ=T [11,12], i.e., the sum of tot τ τ In the harmonic oscillator described in the previous the trajectory-dependent entropy and of the entropy change section, the only nonlinearities, which might appear, are in the reservoir due to energy flow. The mean total entropy those related to the elasticity of the torsion wire. However, difference is equal to the entropy production rate; i.e., hΔS ðt; τÞi¼hQ ðtÞ=Ti. Furthermore, ΔS ðt; τÞ fully to reach this nonlinear regime, the system has to be forced tot τ tot to such a high level that thermal fluctuations become characterizes the out-of-equilibrium dynamics as it is negligible. Thus, in order to study the nonlinear effects rigorously zero, both in average and fluctuations. The we change the experiment and we measure the fluctuations fluctuations of this quantity impose several constrains on of a Brownian particle trapped in a nonlinear potential the time reversibility, which is a central result of stochastic produced by two laser beams, as shown in Fig. 4 [56].It is thermodynamics [2,11,15,50] and which has been tested very well known that a particle of small radius R ≃ 2 μm experimentally [51]. The FT for the ΔS in a SSFT implies tot can be trapped by a focused laser beam, which produces a ΣðτÞ¼ 1 for any τ; i.e., the FT does not have an asymptotic harmonic potential, thereby confining the Brownian par- validity but is valid for any τ. This is certainly a useful ticle motion to the potential well. When two laser beams are property in experiment because one does not have to look focused at a distance D ≃ R, as shown in Fig. 4(a), the for very long asymptotic behavior. However, the calcula- particle has two equilibrium positions, i.e., the foci of the tion of S in experiment is not easy and a lot of care must tot two beams. Thermal fluctuations allow the particle to hop be used in order to correctly estimate this quantity [38,52]. from one to the other. The particle feels an equilibrium We do not discuss here the experimental analysis per- 4 2 formed on electric circuits and harmonic oscillators, but an potential U ðxÞ¼ ax − bx − dx, shown in Fig. 4(b), example of the evaluation of ΔS ðt; τÞ is given in Sec. V. where a, b, and d are determined by the laser intensity tot For further information, the interested reader can look at and by the distance of the two focal points. This potential the abovementioned references and Ref. [53], where a has been computed from the measured equilibrium 021051-6 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) (b) (a) FIG. 4. (a) Drawing of the polystyrene particle trapped by two laser beams whose axis distance is about the radius of the bead. (b) Potential felt by the bead trapped by the two laser beams. The barrier height between the two wells is about 2k T. distribution of the particle, PðxÞ ∝ exp½U ðxÞ (see quite similar to that of the harmonic oscillator (Sec. III A) Ref. [56] for more experimental details). To drive the although the PDFs are more complex [56]. The measure- system out of equilibrium we periodically modulate the ment are in full agreement with a realistic model based on intensity of the two beams at low frequency ω. Thus, the Fokker-Planck equations where the measured values of the potential felt by the bead has the following pro- Uðx; tÞ have been inserted [60]. This example shows the file: Uðx; tÞ¼ U ðxÞþ U ðx; tÞ¼ U þ cx sinðωtÞ. 0 p 0 The x position of the particle can be described by an overdamped Langevin equation: dx ∂Uðx; tÞ ν ¼ − þ η; ð10Þ dt ∂x with ν the friction coefficient and η the thermal noise delta correlated in time. When c ≠ 0, the particle can experience a stochastic resonance [57–59], when the forcing frequency is close to the Kramers rate [56]. As already done in the case of the harmonic oscillator, one can compute the work tþτ 0 0 0 W ¼ fðt Þx_ðt Þdt ð11Þ of the external force fðtÞ¼ −c sinð2πftÞ on the time interval ½t; t þ τ, where τ ¼ð2πn=ωÞ is a multiple of the forcing period [56]. We consider the PDF PðW Þ, which is plotted in Fig. 5(a). Notice that for small n the distributions are double peaked and very complex. They tend to a Gaussian for large n [inset of Fig. 5(a)]. In Fig. 5(b) we plot the normalized symmetry function of W . We can see that the curves are close to the line of slope one. For high values of work, the dispersion of the data increases due to the lack of events. The slope tends toward 1 as expected by the SSFT. It is remarkable that straight lines are obtained even for n FIG. 5. (a) Distribution of classical work W for different close to 1, where the distribution presents a very complex numbers of period n ¼ 1, 2, 4, 8, and 12 (f ¼ 0.25 Hz). Inset: and unusual shape [Fig. 5(a)]. The very fast convergence to Same data in lin-log. (b) Normalized symmetry function as the asymptotic value of the SSFT is quite striking in this function of the normalized work for n ¼ 1 (plus), 2 (circle), 4 example. We do not show here SðQ Þ as the behavior is (diamond), 8 (triangle), and 12 (square). 021051-7 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) application of FT in a nonlinear case where the distributions Specifically, in the experiment described in Sec, III D, λ is tþτ are strongly non-Gaussian. Other examples of application the external force and W ¼½fðtÞxðtÞ − W . Thus, in an of FT to nonlinear potentials can be found in Ref. [61]. experiment in order to get the right value of the free energy using Eq. (13), one has to clearly identify the control IV. ESTIMATE THE FREE-ENERGY DIFFERENCE parameter λ, which drives the system, and the definition FROM WORK FLUCTUATIONS of Eq. (3) must be used to estimate W (see Ref. [40] for a discussion on this point). In 1997 [13,14] Jarzynski derived an equality which relates the free-energy difference of a system in contact B. Crooks relation with a heat reservoir to the PDF of the work performed on the system to drive it from A to B along any path γ in the This relation is related to the Jarzynski equality and it system parameter space. gives useful and complementary information on the dis- sipated work. Crooks considers the forward work W to drive the system from A to B and the backward work W to A. Jarzynski equality drive it from B to A [Eq. (3) is used for both W and W ]. If f b Specifically, when a system parameter λ is varied from the work PDFs during the forward and backward processes time t ¼ 0 to t ¼ t , Jarzynski defines for one realization are P ðWÞ and P ðWÞ, one has [15] f b of the “switching process” from A to B the work performed on the system as P ðWÞ Z ¼ exp ðβ½W − ΔFÞ ¼ exp ½βW : ð15Þ diss ∂H ½zðtÞ P ð−WÞ λ b W ¼ λ dt; ð12Þ ∂λ Notice that the TFT defined in Sec. III is a special case of Eq. (15), for which A ≡ B and the backward and forward where z denotes the phase-space point of the system and H protocols are the same. its λ parametrized Hamiltonian. One can consider an A simple calculation leads from Eq. (15) to Eq. (13). ensemble of realizations of this switching process with However, from an experimental point of view, Eq. (15) is initial conditions all starting in the same initial equilibrium extremely useful because one immediately sees that the state. Then W may be computed for each trajectory in the crossing point of the two PDFs, that is, the point where ensemble. The Jarzynski equality states that [13,14] P ðWÞ¼ P ð−WÞ, is precisely ΔF. Thus, one has another f b exp ð−βΔFÞ¼hexp ð−βWÞi; ð13Þ mean to check the computed free energy by looking at the PDFs crossing point W . The draw back is that both states −1 where h·i denotes the ensemble average and β ¼ k T A and B have to be equilibrium states. A very interesting with k the Boltzmann constant and T the temperature. and extended review on the Jarezynski and Crooks relations In other words, hexp ½−βW i ¼ 1, since we can always can be found in Ref. [63]. diss write W ¼ ΔF þ W , where W is the dissipated work. diss diss Thus, it is easy to see that there must exist some paths γ C. Applications of Jarzynski and Crooks equalities such that W ≤ 0. Moreover, the inequality hexp xi ≥ diss The Jarzynski equality has been tested for the first time exphxi allows us to recover the second principle, namely, in a single-molecule experiment [64]. Here, we follow a hW i ≥ 0, i.e., hWi ≥ ΔF. If the probability distribution diss more pedagogical description by discussing first the appli- 2 2 of the work is Gaussian, PðWÞ∝ exp½−ð½W− hWi =2σ Þ, cation to the harmonic oscillator and then to a single- then Eq. (13) leads to molecule experiment in Sec. IV C 2. βσ ΔF ¼hWi − ; ð14Þ 1. Harmonic oscillator As a simple example we apply the Jarzynski equality to i.e., the dissipate energy hW i¼ðβσ =2Þ > 0. the harmonic oscillator described in Sec. II. The oscillator diss From an experimental point of view the Jarzynski equality is driven from a state A (M ¼ 0) and to a state B (where [Eq. (13)] is quite useful because there is no restriction on the M ¼ M ≠ 0) by the external tork MðtÞ¼ M t=t (for- o o s choice of the path γ. Furthermore, the other big advantage is ward transformation). The backward transformation is that only the initial state has to be an equilibrium state. instead MðtÞ¼ M − M t=t . The switching time t is o o s s Indeed even if the system is not in equilibrium at time t , one varied in order to probe either the reversible (or quasistatic) obtains the value of ΔFðλðt ÞÞ, i.e., the value of the free paths (t ≫ τ ) or the irreversible ones (t ≲ τ ), s relax s relax energy that the system would have at equilibrium for the where τ is the harmonic oscillator relaxation time. relax value of the control parameter λ ¼ λðt Þ [62]. Finally, we We apply a torque which is a sequence of linear increasing stress that the definition of work given in Eq. (12) is more or decreasing ramps and plateaus. The latter are necessary general than the classical definition used in Eqs. (3) and (11). to relax the system in equilibrium before starting a new 021051-8 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) (a) (b) FIG. 6. Free-energy measure in the harmonic oscillator. (a) Time evolution of θðtÞ under a periodic torque M which drives the oscillator from A (θ ¼ 0)to B [θðt Þ¼ 4.1 nrad] and vice versa. In this specific case the stiffness is C ≃ 5.5 Nm, the transition time is −1 t ≃ 0.1τ , and M ¼ 22.4 Nm rad . (b) Probability distribution functions of the work for the forward (blue curve) and backward s relax o (red curve) transformation. The crossing point of the two PDFs determines the value of ΔF [see Eq. (15)]. The crossing point is at A;B W ¼ −112k T, which is within experimental errors of the expected theoretical Δ ≃ −110k T (see text). B F B transformation. This periodic driving produces a sequence [64–77]. Let us summarize a typical example extracted of direct A-B paths and reversed ones, B-A,of θðtÞ, whose from Ref. [65], where Eq. (15) has been used to measure time dependence is plotted in Fig. 6(a). The plotted the ΔF between the folded and unfolded states of a DNA dynamics corresponds to quite irreversible transformation hairpin. In Fig. 7(a), a schematic diagram of the experi- as t ≪ τ ; specifically, t =τ ≃ 0.1. The other rel- ment is depicted, and the typical structure of the DNA s relax s relax evant parameters of the experiment are M ¼ 22.4 pN and hairpin is shown in Fig. 7(b). In this experiment the DNA −4 −1 hairpin is attached to two beads via double-stranded the stiffness C ¼ 5.5 × 10 Nm rad . The work W can be DNA handles. As these two handles are much stiffer than computed in each of the reverse or direct paths using the hairpin, they do not deform during the stretching Eq. (12), and the P ðWÞ and P ðWÞ can be constructed. f b cycles that are performed in the following way. The The result is shown in Fig. 6(b), where the probability bottom bead is kept by a micropipette, whereas an optical distribution functions are plotted. We see that P ðWÞ and trap captures the top bead. A piezoelectric actuator P ðWÞ cross at about −112k T, which is the measured b B controls the position of the bottom bead, which, when value of the ΔF computed from Eq. (15). Let us compare moved along the vertical axis, stretches the DNA. The the measured value with the theoretical estimate. In this difference in positions of the bottom and top beads gives specific case of the harmonic oscillator, as the temperature the end-to-end length of the molecule. The optical trap is is the same in states A and B, the free-energy difference of 1 2 B 2 B used to measure the force exerted by the stretched DNA the oscillator alone is ΔF ¼ ΔU ¼½ Cθ  ¼½ðM =2CÞ . 0 o 2 A A on the top bead. By driving the piezoelectric actuator, the However, the Jarzynski [Eq. (13)] and Crooks [Eq. (15)] molecule is repeatedly subjected to unfold-refold cycles. relations compute the free-energy difference of the system Every pulling cycle consists of a stretching process (the harmonic oscillator) plus the driving, i.e., ΔF ¼ ΔF − (hereafter referred to as S) and a releasing (hereafter 2 B ½ðM =CÞ , which in this case gives ΔF ¼ −ΔF . From A 0 referred to as R) process. In the stretching part of the the values of the experimental parameters, one gets cycle the molecule is stretched from a minimum value of ΔF ≃ 110k T, which is very close to the experimental the force (f ≃ 10 pN), so small that the hairpin is min value. This example shows that indeed using Eqs. (13) and always folded, up to a maximum value of the force (15) it is possible to obtain the values of the free-energy (f ≃ 20 pN), so large that the hairpin is always max difference between two states even by doing a very fast unfolded. During the releasing part of the cycle the transformation with t ≪ τ . However, the applications s relax force is decreased from f back to f at the same max min of Eqs. (13) and (15) to the experiments may present rate of the loading cycle, which is the same protocol used several problems, which are discussed in detail in Ref. [40]. for the abovementioned example of the harmonic oscil- lators. As the stretching force and the displacement of 2. Measure of the free-energy difference the bead are independently measured, the work can be of a single molecule computed using Eq. (12) at each cycle S and R. The Jarzynsky and Crooks relations turn out to be a useful corresponding P ðWÞ and P ðWÞ histograms can be S R tool to measure the free energy of a single molecule computed too. The experiment has been repeated for 021051-9 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) (a) (c) FIG. 7. (a) Experimental setup. The DNA hairpin whose sequence is shown in (b) is attached to two beads. The bottom bead is kept by a micropipette and top bead is captured by an optical trap. The drawing is not to scale; the diameter of the beads is around 3000 nm, much greater than the 20-nm length of the DNA. (b) DNA hairpin sequence. Labels 5 and 3 indicate the polarity of the phosphate chain of the hairpin. (c) Typical work distributions for three different loading rates: 1 pN=s (slow, blue), 4.88 pN=s (medium, green), 14.9 pN=s (fast, red). The vertical lines show the range of the estimated experimental errors for the value of ΔF (adapted from Ref. [65]). different pulling-releasing speeds, and the results for shown that fluctuation relations can be used for much more three speeds are shown in Fig. 7(c). We see that when than estimating free-energy differences. They study ligand the speed is increased the difference between the mean binding and use single-molecule force spectroscopy to works in the S and R protocols increases. However, the measure binding energies, selectivity, and allostery of remarkable fact is that the P ðWÞ and P ðWÞ cross, nucleic acids. S R within experimental errors at the same value of W Finally, useful extensions and generalizations of independently of the pulling speed, showing the validity the Jarzynski equality that allow the study of the of Eq. (15). As already explained, the crossing point transition between two nonequilibrium steady states have gives the value of the free-energy difference between the been derived in Ref. [78] and checked experimentally folded and unfolded states. in Ref. [79]. 3. Short discussion on applications of Jarzynski V. TWO HEAT BATHS and Crooks relations In Secs. II, III D, and IV, we discuss systems in contact The examples in Secs. IV C 1 and IV C 2 show the with a single heat bath, which, within the context of power of Jarzynski and Crooks relations which are a very stochastic thermodynamics, are the most studied cases useful tool to estimate the free-energy differences of micro both experimentally and theoretically [2,3]. Conversely, and nano systems where the role of fluctuation is very systems, driven out of equilibrium by a temperature important. gradient, in which the energy exchanges are produced It is worth mentioning that there is a large amount only by the thermal noise, have been analyzed mainly in of work on this topic performed by the biophysics theoretical models [47,80–90]. This problem has been and chemistry communities. The estimation of the studied only in a few very recent experiments [91–95], protein-folding landscapes is an important application, because of the intrinsic difficulties of dealing with large which remains one of the main interests despite many years of investigation; useful examples can be found in temperature differences in small systems. In order to illustrate the main properties of the energy Refs. [73,74]. Furthermore, using extensions to the basic results of fluxes in these systems driven out of equilibrium by a Jarzynski [68], the works in Refs. [72,75,76] collectively temperature gradient, we summarize in this section the main results of Refs. [91,92]. These two articles analyze show that nonequilibrium measurements give the most both experimentally an theoretically the statistical pro- precise reconstructions, to date, of free-energy landscapes perties of the energy exchanged between two conductors for single molecules (DNA hairpins). kept at different temperature and coupled by the electric The reader might also be interested in a recent extension of these relations by Camunas-Solder et al. [77], who have thermal noise. 021051-10 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) C C A. Two electric circuits interacting via 2 R q ¼ −q þðq − q Þ þ η ; ð16Þ 1 1 1 2 1 1 a conservative coupling X X 1. Experimental setup and stochastic variables C C R q_ ¼ −q þðq − q Þ þ η ; ð17Þ The experimental setup is sketched in Fig. 8(a).Itis 2 2 2 1 2 2 X X constituted by two resistances R and R , which are kept 1 2 at different temperatures T and T , respectively. These 1 2 where η is the usual white noise, hη ðtÞη ðt Þi ¼ m i j temperatures are controlled by thermal baths, and T is 0 2δ k T R δðt − t Þ, and where we have introduced the ij B i j fixed at 296 K, whereas T can be set at a value between 88 quantity X ¼ C C þ CðC þ C Þ. Equations (16) and (17) 2 1 1 2 and 296 K using the stratified vapor above a liquid nitrogen are the same as those for the two coupled Brownian bath. In the figure, the two resistances have been drawn particles sketched in Fig. 8(b) when one regards q as with their associated thermal noise generators η and η , 1 2 the displacement of the particle m, i as its velocity, K ¼ m m whose power spectral densities are given by the Nyquist 0 0 C 0=X (m ¼ 2 if m ¼ 1 and m ¼ 1 if m ¼ 2) as the formula jη~ j ¼ 4k R T , with m ¼ 1, 2 [see Eqs. (16) m B m m stiffness of the spring m, K ¼ C=X as the coupling spring, and (17)]. The coupling capacitance C controls the elec- and R the viscosity term. The analogy with the Feymann trical power exchanged between the resistances and, as a ratchet can be made by assuming, as done in Ref. [82], that consequence, the energy exchanged between the two baths. the particle m has an asymmetric shape and on average No other coupling exists between the two resistances which moves faster in one direction than in the other one. are inside two separated screened boxes. The quantities We now rearrange Eqs. (16) and (17) to obtain the C and C are the capacitances of the circuits and the 1 2 Langevin equations for the voltages, which will be useful cables. Two extremely low-noise amplifiers A and A [96] 1 2 in the following discussion. The relationships between the measure the voltage V and V across the resistances R 1 2 1 measured voltages and the charges are and R , respectively. All the relevant quantities considered in this paper can be derived by the measurements of V q ¼ðV − V ÞC þ V C ; ð18Þ 1 1 2 1 1 and V , as we discuss below. q ¼ðV − V ÞC − V C : ð19Þ 2 1 2 2 2 2. Stochastic equations for the voltages We now proceed to derive the equations for the dynami- By plugging Eqs. (18) and (19) into Eqs. (16) and (17), and cal variables V and V . Furthermore, we discuss how our 1 2 rearranging terms, we obtain system can be mapped onto a system with two interacting Brownian particles, in the overdamped regime, coupled to _ _ ðC þ CÞV ¼ CV þ ðη − V Þ; ð20Þ 1 1 2 1 1 two different temperatures; see Fig. 8(b). Let q (m ¼ 1,2) be the charges that have flowed through the resistances R , so that the instantaneous current flowing through them _ _ ðC þ CÞV ¼ CV þ ðη − V Þ: ð21Þ is i ¼ q_ . A circuit analysis shows that the equations for m m 2 2 1 2 2 the charges are FIG. 8. (a) Diagram of the circuit. The resistances R and R are kept at temperature T and T ¼ 296 K, respectively. They are 1 2 1 2 coupled via the capacitance C. The capacitances C and C schematize the capacitances of the cables and of the amplifier inputs. The 1 2 voltages V and V are amplified by the two low-noise amplifiers A and A [96]. The other relevant parameters are q (m ¼ 1, 2), i.e., 1 2 1 2 the charges that have flowed through the resistances R , and the instantaneous current flowing through them, i.e., i ¼ðdq =dtÞ. m m (b) The circuit in (a) is equivalent to two Brownian particles (m and m ) moving inside two different heat baths at T and T . The two 1 2 1 2 particles are trapped by two elastic potentials of stiffness K and K and coupled by a spring of stiffness K [see text and Eqs. (16) and 1 2 (17)]. The analogy is straightforward by considering q the displacement of the particle m, i its velocity, K ¼ C =X (with m ≠ m ) m m m the stiffness of the spring m, and K ¼ C=X the coupling spring. 021051-11 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) 3. Thermodynamics quantities while PðQ Þ ≠ Pð−Q Þ. Indeed, the shape of PðQ Þ is 1;τ 2;τ 1;τ strongly modified by changing T from 296 to 88 K; Two important quantities can be identified in the circuit whereas the shape of Pð−Q Þ is slightly modified by 2;τ depicted in Fig. 8: the energy Q dissipated in each m;τ the large temperature change, only the tails of Pð−Q Þ 2;τ resistor in a time and the work W exerted by one circuit m;τ present a small asymmetry testifying to the presence on the other one in a time τ. These two thermodynamic of a small heat flux. The fact that PðQ Þ ≠ Pð−Q Þ, quantities are related to the internal energy variation in the 1;τ 2;τ whereas PðW Þ¼ Pð−W Þ, can be understood by time τ by the first principle: 1;τ 2;τ noticing that Q ¼ W − ΔU . Indeed, ΔU m;τ m;τ m;τ m;τ [Eq. (22)] depends on the values of C and V .As C ≠ ΔU ¼ W − Q : ð22Þ m m 1 m;τ m;τ m;τ C and σ ≥ σ , this explains the different behavior of Q 2 2 1 1 and Q . On the contrary, W depends only on C and the Furthermore, it has been proved that the measured variance 2 m 2 product V V . _ 1 2 σ of V is related to the mean heat flux hQ i ∝ ∂ hQ i: m m m t m We study whether our data satisfy the fluctuation theorem as given by Eq. (24) in the limit of large τ.It 2 2 2 σ ¼hV i¼ σ þhQ iR ; ð23Þ m m m;eq m m turns out that the symmetry imposed by Eq. (24) is reached for rather small τ for W. On the contrary, it converges very where σ ¼ k T ðC þ C Þ=X is the equilibrium value m;eq B m m slowly for Q. We have only a qualitative argument to of σ . Note that hQ i ∝ ðT 0 − T Þ; thus, in the equilib- explain this difference in the asymptotic behavior: by m m m m rium case T ¼ T 0, and consequently hQ i¼ 0. looking at the data one understands that the slow con- m m m We do not give here the exact expressions of ΔU , vergence is induced by the presence of the exponential tails m;τ of PðQ Þ for small τ. W , Q , and σ , which have been computed and 1;τ m;τ m;τ m To check Eq. (24), we plot in Fig. 9(c) the symmetry measured in Refs. [91,92]. We discuss instead how the function SymðE Þ¼ ln½PðE Þ=Pð−E Þ as a function FT is modified in the case of two heat baths. 1;τ 1;τ 1;τ of E =ðk T Þ measured at different T ,but τ ¼ 0.1s 1;τ B 2 1 for SymðW Þ and τ ¼ 2s for SymðQ Þ. Indeed, 4. Fluctuation theorem for work and heat 1;τ 1;τ SymðQ Þ reaches the asymptotic regime only for 1;τ One expects that the thermodynamic quantities satisfy a τ → 2s. We see that SymðW Þ is a linear function of 1;τ fluctuation theorem of the type [4,6,81,83,88–90] W =ðk T Þ at all T . These straight lines have a slope 1;τ B 2 1 αðT Þ which, according to Eq. (24), should be ðβ k T Þ. 1 12 B 2 PðE Þ m;τ ln ¼ β E ΣðτÞ; ð24Þ In order to check this prediction we fit the slopes of the 12 m;τ Pð−E Þ m;τ straight lines in Fig. 9(c). From the fitted αðT Þ we deduce a temperature T ¼ T =½αðT Þþ 1, which is compared to fit 2 1 where E stands for either W or Q , β ¼ m;τ m;τ m;τ 12 the measured temperature T in Fig. 9(d). In this figure, the ð1=T − 1=T Þ=k , and ΣðτÞ → 1 for τ → ∞. 1 2 B straight line of slope 1 indicates that T ≃ T within a few fit 1 Equation (24) has been proven in Ref. [92]. percent. These experimental results indicate that our data As the system is in a stationary state, we have verify the fluctuation theorem, Eq. (24), for the work and hW i¼hQ i. On the contrary, the comparison of the m;τ τ;m the heat, but that the asymptotic regime is reached for much PDF of W with those of Q , measured at various m;τ τ;m larger time for the latter. temperatures, presents several interesting features. In Fig. 9(a), we plot PðW Þ, Pð−W Þ, PðQ Þ, and 1;τ 2;τ 1;τ 5. Entropy production rate Pð−Q Þ measured in equilibrium at T ¼ T ¼ 296 K 2;τ 1 2 It is now important to analyze the entropy produced by and τ ≃ 0.1 s. We immediately see that the fluctuations the total system, circuit plus heat reservoirs. We consider of the work are almost Gaussian, whereas those of the first the entropy ΔS due to the heat exchanged with the heat present large exponential tails. This well-known r;τ reservoirs, which reads ΔS ¼ Q =T þ Q =T . This difference [48] between PðQ Þ and PðW Þ is induced r;τ 1;τ 1 2;τ 2 m;τ m;τ by the fact that Q depends also on ΔU [Eq. (22)], entropy is a fluctuating quantity as both Q and Q 1 2 m;τ m;τ fluctuate, and its average in a time τ is hΔS i¼ which is the sum of the square of Gaussian distributed r;τ variables, thus inducing exponential tails in PðQ Þ.In hQ ið1=T − 1=T Þ¼ AτðT − T Þ =ðT T Þ. However, m;τ r;τ 1 2 2 1 2 1 Fig. 9(a), we also notice that PðW Þ¼ Pð−W Þ and the reservoir entropy ΔS is not the only component 1;τ 2;τ r;τ PðQ Þ¼ Pð−Q Þ, showing that in equilibrium all fluc- of the total entropy production: one has to take into account 1;τ 2;τ tuations are perfectly symmetric. The same PDFs measured the entropy variation of the system, due to its dynamical in the out-of-equilibrium case at T ¼ 88 K are plotted in evolution. Indeed, the state variables V also fluctuate as an Fig. 9(b). We notice here that in this case the behavior of the effect of the thermal noise, and thus, if one measures their values at regular time interval, one obtains a “trajectory” in PDFs of the heat is different from those of the work. Indeed, although hW i > 0, we observe that PðW Þ¼ Pð−W Þ, the phase space ½V ðtÞ;V ðtÞ. Thus, following Seifert [11], m;τ 1;τ 2;τ 1 2 021051-12 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) 10 0 P(W 1) (a) P(W1) (b) P(−W 2) P(−W 2) T1=T2=296 K −2 P(−Q 2) P(−Q 2) −2 10 T1 = 88 K P(Q 1) P(Q1) τ = 0.1 s τ = 0.1 s −4 −4 −6 −6 −8 −8 −20 −10 0 10 20 −20 −10 0 10 20 E /(k T ) E /(k T ) m,τ B 2 m,τ B 2 (c) (d) T = 90 K Sym(W ) τ=0.1s 250 Sym(Q ) τ=2s T = 150 K T = 293 K 50 100 150 200 250 300 0 0.5 1 1.5 2 E /(k T ) T (K) 1,τ B 2 1 FIG. 9. (a) Equilibrium: PðW Þ and PðQ Þ, measured in equilibrium at T ¼ T ¼ 296 K and τ ¼ 0.1s, are plotted as functions of m;τ m;τ 1 2 E, where E stands for either W or Q. Notice that when the system is in equilibrium, PðW Þ¼ Pð−W Þ and PðQ Þ¼ Pð−Q Þ. 1;τ 2;τ 1;τ 2;τ (b) Out of equilibrium: Same distributions as in (a) but the PDFs are measured at T ¼ 88 K, T ¼ 296 K, and τ ¼ 0.1s. Notice that 1 2 when the system is out of equilibrium, PðW Þ¼ Pð−W Þ but PðQ Þ ≠ Pð−Q Þ. The reason for this difference is explained in the 1;τ 2;τ 1;τ 2;τ text. (c) The symmetry function SymðE Þ, measured at various T , is plotted as a function of E (W or Q ). The theoretical slope of 1;τ 1 1 1 1 these straight lines is T =T − 1. (d) The temperature T estimated from the slopes of the lines in (c) is plotted as a function of the T 2 1 fit 1 measured by the thermometer. The slope of the line is 1, showing that T ≃ T within a few percent. fit 1 who developed this concept for a single heat bath, one dependence on t, as the system is in a steady state, as can introduce a trajectory entropy for the evolving system discussed above. This entropy has several interesting S ðtÞ¼ −k log P½V ðtÞ;V ðtÞ, which extends to none- features. The first one is that hΔS i¼ 0, and as a s B 1 2 s;τ quilibrium systems the standard Gibbs entropy concept. consequence, hΔS i¼hΔS i, which grows with increas- tot r Therefore, when evaluating the total entropy production, ing ΔT. The second and most interesting result is that one has to take into account the contribution over the time independently of ΔT and of τ, the following equality interval τ of always holds: hexpð−ΔS =k Þi ¼ 1; ð26Þ tot B P½V ðt þ τÞ;V ðt þ τÞ 1 2 ΔS ¼ −k log : ð25Þ s;τ B P½V ðtÞ;V ðtÞ 1 2 for which we find experimental evidence, as discussed in the following, and provide a theoretical proof in It is worth noting that the system we consider is in a Refs. [91,92]. Equation (26) represents an extension to nonequilibrium steady state, with a constant external two temperature sources of the result obtained for a system driving ΔT. Therefore, the probability distribution in a single heat bath driven out of equilibrium by a time- PðV ;V Þ does not depend explicitly on the time, and dependent mechanical force [5,11], and our results provide 1 2 ΔS is nonvanishing whenever the final point of the the first experimental verification of the expression in a s;τ trajectory is different from the initial one: ½V ðt þ τÞ; system driven by a temperature difference. Equation (26) V ðt þ τÞ ≠ ½V ðtÞ;V ðtÞ. Thus, the total entropy change implies that hΔS i ≥ 0, as prescribed by the second law. 2 1 2 tot reads ΔS ¼ ΔS þ ΔS , where we omit the explicit From symmetry considerations, it follows immediately tot;τ r;τ s;τ 021051-13 Sym(E ) P(E ) 1,τ m,τ P(E ) T (K) m,τ fit S. CILIBERTO PHYS. REV. X 7, 021051 (2017) T = 88 K 1 (a) T = 88 K −1 1.1 (b) T = 296 K T = 296 K −2 ⟨ exp (− Δ S /k ) ⟩ tot B 0.9 50 100 150 200 250 300 −3 (c) −4 T = 88 K τ = 0.5 s T = 184 K τ = 0.5 s −5 T = 256 K τ = 0.5 s T = 88 K τ = 0.05 s Theory −6 −15 −10 −5 0 5 10 15 0 1 2 3 4 Δ S [k ] x B Δ S [k ] tot B FIG. 10. (a) The probability PðΔS Þ (dashed lines) and PðΔS Þ (continuous lines) measured at T ¼ 296 K (blue line) which r tot 1 corresponds to equilibrium and T ¼ 88 K (green lines) out of equilibrium. Notice that both distributions are centered at zero at equilibrium and shifted towards positive value in the out of equilibrium. (b) hexpð−ΔS Þi as a function of T at two different τ ¼ 0.5s tot 1 and τ ¼ 0.1s. (c) Symmetry function SymðΔS Þ¼ log½PðΔS Þ=Pð−ΔS Þ as a function of ΔS . The black straight line of slope 1 tot tot tot tot corresponds to the theoretical prediction. that, at equilibrium (T ¼ T ), the probability distribution main results. The experimental system is sketched in 1 2 of ΔS is symmetric: P ðΔS Þ¼ P ð−ΔS Þ. Thus, Fig. 11, and it is based on a single-electron box at tot eq tot eq tot low temperature. This is an excellent test benchmark for Eq. (26) implies that the probability density function of thermodynamics in small systems [99,100], and an inter- ΔS is a Dirac δ function when T ¼ T ; i.e., the quantity tot 1 2 esting review of the statistical properties of coupled ΔS is rigorously zero in equilibrium, both in average and tot fluctuations, and so its mean value and variance provide a circuits, both quantum and classical, can be found in measure of the entropy production. The measured proba- Ref. [101]. bilities PðΔS Þ and PðΔS Þ are shown in Fig. 10(a).We In the single-electron box shown in Fig. 11(a) the r tot see that PðΔS Þ and PðΔS Þ are quite different and that the electrons in the normal metal copper island (N) can tunnel r tot latter is close to a Gaussian and reduces to a Dirac δ to the superconducting Al island (S) through the aluminium oxide insulator (I). The integer net number of electrons function in equilibrium, i.e., T ¼ T ¼ 296 K [notice that, 1 2 tunneled from S to N is denoted by n. This number, in Fig. 10(a), the small broadening of the equilibrium monitored by the nearby single-electron transistor (SET) PðΔS Þ is just due to unavoidable experimental noise and tot discretization of the experimental probability density func- shown in Fig. 11(a), is the classical system degree of tions]. The experimental measurements satisfy Eq. (26) as freedom. it is shown in Fig. 10(b). It is worth noting that Eq. (26) Indeed, the device in Fig. 11(a) can be represented with implies that PðΔS Þ should satisfy a fluctuation theorem a classical electric circuit, in which the energy stored in tot of the form log½PðΔS Þ=Pð−ΔS Þ ¼ ΔS =k , ∀ τ; ΔT, the capacitors and in the voltage sources can be exactly tot tot tot B as discussed extensively in Refs. [2,53]. We clearly see in measured [99]. As in the previous section, Sec. VA 1, the Fig. 10(c)that this relation holds for different values of the conductor N and S are not at the same temperature. temperature gradient. Thus, this experiment clearly estab- Furthermore, here the system is driven by a voltage V lishes a relationship between the mean and the variance which oscillates much slower than the relaxation time of the of the entropy production rate in a system driven out of device. Thus, the forward and backward processes from equilibrium by the temperature difference between two the maximum to the minimum of V can be considered. thermal baths coupled by electrical noise. Because of the By the measured values of nðtÞ and V , one can estimate formal analogy with Brownian motion, the results also the heats, Q and Q , exchanged by the two heat baths N S apply to mechanical coupling [95,97,98]. in a time t , which is the period of the driving signal. In this th way the thermal entropy ΔS ¼ Q =T þ Q =T can be N N S S B. Entropy production in a single-electron box computed. Furthermore, the trajectory-dependent entropy can be estimated by measuring Δs ¼ −k log½P(nðt Þ)= Another interesting experiment on the measure of the b f entropy production in a system subjected to a temperature P(nð0Þ), where P(nðtÞ) is the probability that at time t the difference is presented in Ref. [93]. We summarize here the system is in the state nðtÞ for a value of the driving V ðtÞ. 021051-14 P(Δ S ) Sym(Δ S ) tot EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) th The total entropy is, of course, ΔS ¼ ΔS þ ΔS, and tot its probability distribution PðΔS Þ can be measured. tot The results for the forward and back processes are shown in Fig. 12(a), and the corresponding symmetry func- tions SymðΔS Þ¼ log½PðΔS Þ=Pð−ΔS Þ are plotted tot tot tot Fig. 12(b). In spite of the fact that PðΔS Þ are highly non- tot Gaussian, we notice that SymðΔS Þ¼ k ΔS , which tot B tot implies that Eq. (26) is also satisfied by these data. As in the previous section, Sec. VA 5, the main result of this experiment is that stochastic entropy production extracted from the trajectories is related to thermodynamic entropy production from dissipated heat in the respective thermal baths. VI. MOTOR POWER AND EFFICIENCY Historically, one of the main purposes of thermodynam- ics has been the study of the efficiency of thermal machines and power plants. Nowadays there is a wide interest in extending these studies to micro and nano motors which FIG. 11. (a) Sketch of the measured system together with a play a major role in biological mechanisms and small scanning electron micrograph of a typical sample. The colors devices. In Sec. II B, we see that in small systems all of the on the micrograph indicate the correspondingly colored circuit thermodynamics quantities fluctuate. Thus, we are inter- elements in the sketch. (b) Typical trace of the measured detector ested in knowing the influence of these fluctuations on the signal under a sinusoidal protocol for the drive V , plotted in efficiency of small devices where the dissipated energies green. This trace covers three realizations of the forward protocol and the produced work are a few k T. Furthermore, it is (V from −0.1 to 1 mV), and three realizations of the backward useful to know this efficiency at the maximum power and protocol (V from 1 to −0.1 mV). The SET current I , plotted in g det not in the quasistatic regimes, such as the Carnot cycle, black, indicates the charge state of the box. The output of the threshold detection is shown in solid blue, with the threshold level where the produced power is close to zero. These important indicated by the dashed red line (adapted from Ref. [93]). questions have been theoretically studied in several articles [2,102–107] and only in a few proof-of-principle experi- ments [108–111]. The first stochastic Carnot machine was reported in Ref. [108]. In this experiment a Brownian (a) particle trapped by an optical tweezer is subjected to a kind of Carnot cycle, inspired by a theoretical model proposed in Ref. [103]. The cycle, used in a very similar experiment [109], is sketched in Fig. 13(a), which we describe in some detail. The Brownian particle is trapped by a harmonic potential [bottom row in Fig. 13(a)] whose stiffness is changed as a function of time. The increase of the stiffness is equivalent to a compression (the motion of the particle (b) is more confined), the decrease to an expansion. In the experiment the bead is subjected to a random force which plays the role of an effective temperature, which can be easily changed by changing the amplitude of the random forcing. As in the Carnot cycle, the cycle in Fig. 13(a) is composed by an isothermal and an adiabatic compression and by an isothermal and an adiabatic expansion. Notice that the construction of adiabatic processes for a Brownian particle is a real challenge, which has been achieved by changing simultaneously the temperature in such a way FIG. 12. (a) Probability distribution of the total entropy ΔS , tot that the exchanged heat, during the adiabatics, is zero on which has been measured in the circuit shown in Fig. 11 and average (see Refs. [109,112] for details). The work and the described in the text. (b) The symmetry functions SymðΔS Þ¼ tot heat in this experiment are computed as described in log½PðΔS Þ=Pð−ΔS Þ of PðΔS Þ as a function of ΔS .In tot tot tot tot Secs. II B, IV, and V. As these two quantities fluctuate, spite of the highly non-Gaussian nature of PðΔS Þ, we see that tot SymðΔS Þ¼ k ΔS (adapted from Ref. [93]). the contribution of the fluctuations during adiabatics must tot B tot 021051-15 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) Another interesting article [113] presents the theoretical and experimental results on the conversion of one form of work to another. Using a Brownian particle as an isothermal machine driven by two independent periodic forces, the authors of Ref. [113] analytically compute and experimen- tally measure the stochastic thermodynamic properties of this Brownian engine. Specifically, the efficiency of the energy transfer between the two driving forces and the Onsager coefficients of the coupling are evaluated. The results of these experiments show the kind of problems that one encounters in the study of the efficiency of nano devices. For example, the study of the efficiency at maximum power and the behavior of η fluctuations have been subjects of extensive theoretical investigation [2,104–106], to understand their system dependence and eventually their universality. We do not discuss these theoretical results because they are far from the purposes of this review. The interested reader can look at the abovementioned references. However, it is worth mention- ing that several small devices, such as a molecular motor, are driven by chemical reaction, and the efficiency of these devices has been studied theoretically [107,114–116], but to my knowledge no proof-of-principle experiment, as the one presented here, has been performed for this chemically driven system. FIG. 13. (a) Schematic of the Carnot cycle applied to a Brownian particle trapped in a harmonic potential by a laser beam. The bottom row indicates the harmonic potential as a VII. FLUCTUATION DISSIPATION function of time. The stiffness of the potential is controlled by the RELATIONS FOR NESS laser intensity, as indicated by the green line. A random force, As we see in the previous section, current theoretical applied to the particle, plays the role of an effective temperature developments in nonequilibrium statistical mechanics whose value is changed as indicated by the magenta line. have led to significant progress in the study of systems (b) Contour lines of the probability distribution of the cycle efficiency averaged on n number of cycles. The efficiency η is around states far from thermal equilibrium. Systems in normalized to the Carnot efficiency η . The black dashed line c nonequilibrium steady states are the simplest examples indicates the mean value of η=η (adapted from Ref. [109]). because the dynamics of their degrees of freedom x under fixed control parameters λ can be statistically described be taken into account in computing the efficiency, which is by time-independent probability densities ρ ðx; λÞ. NESSs defined as η ¼ W =Q , where W is the work cycle cycle hot cycle naturally occur in mesoscopic systems such as colloidal produced during a cycle and Q is the heat absorbed from hot particles dragged by optical tweezeres, Brownian ratches, the hot sources. Because of W and Q fluctuations, η cycle and molecular motors because of the presence of non- also fluctuates a lot because, as we have seen in previous conservative or time-dependent forces [117].Atthese sections, Q can be zero and even negative. The measured length scales fluctuations are important, so it is essential to probability distribution of η is plotted in Fig. 13(b) as cycle establish a quantitative link between the statistical proper- function of the number of cycles used to average it. The ties of the NESS fluctuations and the response of the efficiency η is normalized to the standard Carnot system to external perturbations. Around thermal equi- cycle efficiency, η ¼ 1 − T =T . Although the mean value librium this link is provided by the fluctuation-dissipation c cold hot hη i is smaller than η , we clearly see that, for a small theorem [34]. cycle c number of cycles, η has big fluctuations which extend from The validity of the FDR in systems out of thermal values much larger than η to negative values. Furthermore, equilibrium has been the subject of intensive study during recent years. We recall that for a system in equilibrium with as the cycle is performed in finite time, the power pro- a thermal bath at temperature T the FDR establishes a duced by the system can be computed as a function of simple relation between the two-time correlation function the cycle duration. This power has a maximum, and the Cðt − sÞ of a given observable and the linear response other interesting result of this experiment is that the mean function Rðt; sÞ of this observable to a weak external efficiency at the maximum power follows the Curzon- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi perturbation, Ahlborn expression hη i¼ 1 − T =T [102]. cycle cold hot 021051-16 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) ∂ Cðt; sÞ¼ k TRðt; sÞ; ð27Þ theoretically and experimentally. This relationship is s B related to the famous paradox of a Maxwell’s demon, where in equilibrium Cðt; sÞ and Rðt; sÞ depend only on the which is an intelligent creature able to monitor individual time difference (t − s). However, Eq. (27) is not necessarily molecules of a gas contained in two neighboring chambers fulfilled out of equilibrium and violations are observed in a [141,142]. Initially, the two chambers are at the same variety of systems, such as glassy materials [24,118–123], temperature, defined by the mean kinetic energy of the granular matter [124], biological systems [125], and res- molecules and proportional to their mean-square velocity. onators [126]. Some of the particles, however, travel faster than others. This motivated a theoretical and experimental work By opening and closing a molecule-sized trapdoor in the devoted to a search of a general framework describing partitioning wall, the demon can collect the faster mole- FD relations; see the review Ref. [34]. The generalization of cules in one chamber and the slower ones in the other. The the fluctuation-dissipation theorem around NESS for sys- two chambers then contain gases with different temper- tems with Markovian dynamics has been achieved from atures, and that temperature difference may be used to different theoretical approaches [17,19,20,127–136]. The power a heat engine and produce mechanical work. By different generalized formulations of FDR link correlation gathering information about the particles positions and functions of the fluctuations of the observable of interest velocities and using that knowledge to sort them, the OðxÞ in the unperturbed NESS with the linear response demon is able to decrease the entropy of the system and function of OðxÞ due to a small external time-dependent convert information into energy. Assuming the trapdoor perturbation around the NESS. The observables involved is frictionless, the demon is able to do all that without in such relations are not unique, but they are equivalent in performing any work himself in an apparent violation of the the sense that they lead to the same values of the linear second law of thermodynamics. This paradox has origi- response function. These theoretical relations may be nated a long debate on the connection between information useful in experiments and simulations to know the linear and thermodynamics. A solution of the problem was response of the system around NESS. Indeed, the response proposed in 1929 by Leo Szilard, who used a simplified can be obtained from measurements entirely done at the one-particle engine to explain it. This gedanken experiment unperturbed NESS of the system of interest without any can nowadays be realized [141]. need to perform the actual perturbation. Nevertheless, the theoretical equivalence of the different observables A. Szilard engine: Work production from information involved in those relations does not translate into equiv- Modern technologies allow us to realize these gedanken alent experimental accessibility: e.g., strongly fluctuating experiments related to the Maxwell’s demon original idea. observables such as instantaneous velocities may lead to large statistical errors in the measurements [137]. 1. Sizlard engine Additionally, NESS quantities themselves, such as local For example, a Szilard engine was realized in 2010 mean velocities, joint stationary densities, and the stochas- [143] by using a single microscopic Brownian particle in a tic entropy, are not in general as easily measurable as fluid and confined to a spiral-staircase-like potential shown dynamical observables directly related to the degrees of in Fig. 14. Driven by thermal fluctuations, the particle freedom [18]. Hence, before implementing the different performs an erratic up and down motion along the staircase. fluctuation-response formulas in real situations, it is However, because of the potential gradient, downward important to test its experimental validity under very well steps will be more frequent than upward steps, and the controlled conditions and to assess the influence of particle will on average fall down. The position of the finite data analysis. The experimental test of some fluc- particle is measured with the help of a CCD camera. tuation-dissipation relations has been recently done in Each time the particle is observed to jump upwards, Refs. [18,61,137,138] for colloidal particles in toroidal this information is used to insert a potential barrier that optical traps and in systems subjected to thermal gradients hinders the particle to move down. By repeating this [139,140]. We do not describe here specific experimental procedure, the average particle motion is now upstairs results which have already been widely discussed in the and work is done against the potential gradient. By lifting abovementioned articles (see also Refs. [2,3,34]). What is the particle, mechanical work has therefore been produced important to recall is that this term is related to the out-of- by gathering information about its position. This is the first equilibrium current of the system, which is proportional to example of a device that converts information into energy the mean total entropy production for a NESS. for a system coupled to a single thermal environment. However, there is not a contradiction with the second law VIII. THERMODYNAMICS, INFORMATION, because Sagawa and Ueda [144] formalized the idea that AND THE MAXWELL DEMONS information gained through microlevel measurements can The relationship between stochastic thermodynamics be used to extract added work from a heat engine. Their and information now has an increasing importance both formula for the maximum extractable work is 021051-17 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) final equilibrium state can be smaller than the free-energy difference between the two states. Equation (29) has been directly tested in a single- electron transistor [146], similar to the one described in Sec. VIII A. 2. Autonomous Maxwell’s demon improves cooling In the previous section, Sec. VIII A, the Maxwell’s demon has been realized using an external feedback. However, working at low temperature and coupling in a suitable way the single-electron devices, already described in Sec. VIII A, one can construct a local feedback which behaves as an autonomous Maxwell’s demon and allows an efficient cooling of the system [147,148]. The device, whose principle is sketched in Fig. 15(a), is composed by a SET formed by a small normal metallic island connected to two normal metallic leads by tunnel junctions, which permit electron transport between the leads and the island. The SET is biased by a FIG. 14. (a) Experimental realization of Szilard’s engine. (a) A colloidal particle in a staircase potential moves downward on potential V and a gate voltage V , applied to the island via average, but energy fluctuations can push it upward from time to a capacitance, controls the current I flowing through the time. (b) When the demon observes such an event, it inserts a wall SET. The island is coupled capacitively with a single- to prevent downward steps. By repeating this procedure, the electron box which acts as a demon which detects the particle can be brought to move upwards, performing work presence of an electron in the island and applies a against the force created by the staircase potential. In the actual feedback. Specifically, when an electron tunnels to the experiment, the staircase potential is implemented by a tilted island, the demon traps it with a positive charge [illus- periodic potential and the insertion of the wall is simply realized trations 1 and 2 in Fig. 15(a)]. Conversely, when an by switching the potential, replacing a minimum (no wall) by a electron leaves the island, the demon applies a negative maximum (wall) (adapted from Ref. [143]). charge to repel further electrons that would enter the island [illustrations 3 and 4 in Fig. 15(a)]. This effect is obtained W ¼ −ΔF þ k ThIi; ð28Þ max B by designing the electrodes of the demon in such a way that when an electron enters the island from a source where ΔF is the free-energy difference between the electrode, an electron tunnels out of the demon island final and initial state and the extra term represents the as a response, exploiting the mutual Coulomb repulsion so-called mutual information I. In the absence of meas- between the two electrons. Similarly, when an electron urement errors this quantity reduces to the Shannon enters to the drain electrode from the system island, an entropy: I ¼ − PðΓ Þ ln½PðΓ Þ, where PðΓ Þ is the k k m k electron tunnels back to the demon island, attracted by the probability of finding the system in the state Γ . Then in overall positive charge. The cycle of these interactions the specific case of the previously described staircase between the two devices realizes the autonomous demon, potential [143]: I ¼ −p ln p − ð1 − pÞ ln p, where p is which allows the cooling of the leads. In the experimental the probability of finding the particle in a specific region. realization presented in Ref. [147], the leads and the In this context the Jarzynski equality discussed in demon were thermally insulated, and the measurements of Sec. IV also contains this extra term and it becomes their temperatures is used to characterize the effect of the demon on the device operation. In Fig. 15(b) we plot the hexpð−βW þ IÞi ¼ expð−βΔFÞ; ð29Þ variation of the lead temperatures as a function of n ∝ V g g when the demon acts on the system. We clearly see that which leads to around n ¼ 1=2 the two leads are both cooled of 1 mK at a mean temperature of 50 mK. This occurs because the hWi ≥ ΔF − k ThIi: ð30Þ tunneling electrons have to take the energy from the thermal energy of the leads, which, being thermally Equations (29) and (30) generalize the second law of isolated, cool down. This increases the rate at which thermodynamics, taking into account the amount of infor- electrons tunnel against Coulomb repulsion, giving rise to mation introduced into the system [142,145]. Indeed increased cooling power. At the same time, the demon Eq. (30) indicates that, thanks to information, the work increases its temperature because it has to dissipate energy performed on the system to drive it between an initial and a in order to process information, as discussed in Ref. [149]. 021051-18 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) B. Energy cost of information erasure The experiments in the previous two sections show that one can extract work from information. In the rest of this section, we discuss the reverse process, i.e., the energy needed to erase information. By applying the second law of thermodynamics, Landauer demonstrated that information erasure is necessarily a dissipative process: the erasure of one bit of information is accom- panied by the production of at least k T lnð2Þ of heat into the environment. This result is known as Landauer’s erasure principle. It emphasizes the fundamental differ- ence between the process of writing and erasing infor- mation. Writing is akin to copying information from one device to another: state left is mapped to left and state right is mapped to right, for example. This one-to-one mapping can be realized in principle without dissipating any heat (in statistical mechanics one would say that it conserves the volume in phase space). By contrast, erasing information is a two-to-one transformation: states left and right are mapped onto one single state, say, right (this process does not conserve the volume in phase space and is thus dissipative). FIG. 15. (a) Principle of the experimental realization of the Landauer’s original thought experiment was realized autonomous Maxwell’s demon. The horizontal top row schema- [150,151] for the first time in a real system in 2011 using tizes a single-electron transistor. Electrons (blue circle) can tunnel a colloidal Brownian particle in a fluid trapped in a double- inside the central island from the left wall and outside from the well potential produced by two strongly focused laser right wall. The demon watches at the state of the island and it beams. This system has two distinct states (particle in the applies a positive charge to attract the electrons when they tunnel right or left well) and may thus be used to store one bit of inside and repels them when they tunnel outside. The systems cools because of the energy released toward the heat bath by information. The erasure principle has been verified by the tunneling events, and the presence of the demon makes the implementing a protocol proposed by Bennett and illus- cooling processes more efficient. The energy variation of the trated in Fig. 16. At the beginning of the erasure process, processes is negative because of the information introduced by the colloidal particle may be either in the left or right well the demon. (b) The measured temperature variations of the left with equal probability of one-half. The erasure protocol is (blue line) and right (green line) leads as a function of the external composed of the following steps: (1) the barrier height is control parameter n when the demon is active and the bath first decreased by varying the laser intensity, (2) the particle temperature is 50 mK. We see that at the optimum value, is then pushed to the right by gently inclining the potential. n ¼ 1=2, both leads are cooled to about 1 mK and the current and (3) the potential is brought back to its initial shape. I flowing through the SET (black line) has a maximum. At the At the end of the process, the particle is in the right well same time, in order to processes information the temperature of the demon (red line) increases a few mK. (c) The same parameters with unit probability, irrespective of its departure position. of (b) are measured when the demon is not active. We see that the As in the previous experiment, the position of the particle is demon temperature does not change, whereas both leads are now recorded with the help of a camera. For a full erasure cycle, heated by the current I. the average heat dissipated into the environment is equal to the average work needed to modulate the form of the Thus, the total (system plus demon) energy production is double-well potential. This quantity was evaluated from the positive. The coupling of the demon with the SET can be measured trajectory and shown to always be larger than controlled by a second gate which acts on the single- the Landauer bound, which is asymptotically approached electron box. In Fig. 15(c) we plot the measured temper- in the limit of long erasure times. However, in order to atures when the demon has been switched off. We clearly reach the bound, the protocol must be accurately chosen see that in such a case the demon temperature does not because, as discussed in Ref. [150] and shown experimen- change and the two electrodes are heating up because of tally [152], there are protocols that are intrinsically irre- the current flow. As far as I know, this is the only example versible no matter how slowly they are performed. The way that shows that under specific conditions an autonomous in which a protocol can be optimized has been theoretically local Maxwell’s demon, which does not use the external solved in Ref. [153], but the optimal protocol is not often feedback, can be realized. easy to apply in an experiment. 021051-19 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) FIG. 16. Experimental verification of Landauer’s erasure principle. A colloidal particle is initially confined in one of two wells of a double-well potential with probability one-half. This configuration stores one bit of information. By modulating the height of the barrier and applying a tilt, the particle can be brought to one of the wells with probability one, irrespective of the initial position. This final configuration corresponds to zero bit of information. In the limit of long erasure cycles, the heat dissipated during the erasure process can approach, but not exceed, the Landauer bound indicated by the dashed line in the right panel (see Ref. [150] for details). C. Other examples on the connection between information can be used to produce motion and, on the information and energy contrary, information processing needs energy. We do not discuss in more detail this important topic, which is By having successfully turned gedanken into real experi- developed in two other articles [163,164]. ments, the above three seminal examples provide a firm empirical foundation to the physics of information and the intimate connection existing between information and IX. USE OF STOCHASTIC THERMODYNAMICS energy. This connection is reenforced by the relationship IN EXPERIMENTS: DISCUSSION between the generalized Jarzinsky equality [154] and the AND PERSPECTIVES Landauer bound, which has been proved and tested on In this review we present various experimental results experimental data in Ref. [151]. A recent article [155] that allow us to introduce several fundamental concepts extends the equivalence of information and thermodynamic of stochastic thermodynamics, such as the FT for heat entropies at thermal equilibrium. and work, the Jarzynski equality, and the trajectory entropy. A number of additional experiments have been performed We have already mentioned several applications of these on this subject [156–160]. For example, in Ref. [157] the concepts to the measure of the system response in NESS, to symmetry breaking, induced in the probability distribution the free-energy estimation, and to the relationship between of the position of a Brownian particle, is studied by com- information and thermodynamics. In the experiments that muting the trapping potential from a single- to a double-well we have described, all of the theoretical predictions are potential. The authors measured the time evolution of the perfectly verified; thus, the question here is to see whether system entropy and showed how to produce work from those findings of stochastic thermodynamics might become information. The experiment in Ref. [160] shows that using a a useful measurement tool that allows us to calibrate and Maxwell’s demon in the information erasure costs less to make predictions in experiments. We discuss here energy than the Landauer bound. It is worth mentioning several examples. experiments where the Landauer bound has been reached in nano devices [156,159]. These experiments open the way to A. Using FT to calibrate an experiment insightful applications for future developments of informa- tion technology. FT can be used to have a precise estimation of either Finally, the connection between thermodynamics an offset or a calibration error in an experiment. The and information plays a very important role in the method can be easily understood by considering the understanding of biological systems [161,162]. Indeed, systems modeled by either one or several Langevin 021051-20 EXPERIMENTS IN STOCHASTIC THERMODYNAMICS: … PHYS. REV. X 7, 021051 (2017) equations described in Secs. II, III D, and V. For these D. Nano or micro motor efficiency systems we have seen that the work and heat PDF satisfy The other aspect that we have briefly discussed in a FT. Thus, in an experimental system that can be modeled Sec. VI concerns the efficiency of nano or micro motor. by the Langevin equation, one can check the good In spite of the large number of theoretical results on this calibration of the apparatus by computing the work and subject, there is a lack of experiments in this important checking whether the FT holds for the measured variables. field, especially in what concerns the efficiency of power If it does not hold, it means that some error has been made supplies and of chemically driven motors. A bound on the in the calibration or some small offset exists in the efficiency of these motors has been recently fixed by the measured quantities (see also Refs. [38,165]). recently discovered indetermination relations [114–116]. However, as we have already said, no proof-of-principle B. Role of hidden variables experiment, such as the Carnot cycle discussed in Sec. VI, has been performed for chemically driven motors. Along the same line of Sec. IX A, FT can be used to estimate unknown parameters of a device. For example, E. Role of Maxwell’s demons such a method has been used to measure the power of molecular motors [38,166]. The idea is certainly very The role of Maxwell’s demons in increasing the efficiency smart and merits serious consideration for applications. of small devices is certainly very important. This subject is in However, one has to pay attention to the influence of its infancy and it might have very powerful applications. The hidden variables. Indeed, in Ref. [167] it has been described autonomous Maxwell’s demon is certainly a smart pointed out that in the abovementioned experiment of system. However, it works because of the very small Ref. [166] a hidden variable has been neglected; thus, the operational temperature of the device. A very big challenge estimated value of the motor power could be affected is the realization of such an autonomous demon for a device by a large error. Another interesting analysis on hidden working at room temperature. At the moment, such a device does not exist, and it is not even clear in which physical or variables has been done in Ref. [168] for the experiment chemical system it could be realized. of Ref. [169] on a SET. In this experiment the bias obtained from the measured FT does not completely match with the experimental value of the applied bias. In F. Energy information connection Ref. [168] it has been shown that this discrepancy is due The connection of stochastic thermodynamics with the to the additional bias introduced by the SET that is used energy dissipation in each logic operation is the last issue to measure the electron transfers. More recent measures that we have developed. It is clearly very important in proved that this is the case [170]. Another accurate connection to the autonomous Maxwell’s demon to esti- experimental analysis of the role of a hidden variable mate the amount of work that the demon has to perform in on FT properties has been reported in Ref. [171]. This order to process information. This will allow us to decide role can be easily understood by looking at the experi- whether the application of a demon is really an advantage to ment described in Sec. V, which is modeled by the reduce the energy consumption. Furthermore, deeper coupled Langevin equations (20) and (21). Suppose now knowledge of the connection between thermodynamics that in that experiment instead of having access to both and information is certainly useful not only to understand V and V variables, only one of the two can be 1 2 biological processes but also to develop methods that allow measured. We immediately see that there is no way of us to recover energy during reversible logical operation. extracting the good values from the measurements. Thus, Also, this field is in his infancy and future development will the use of FT to extract experimental parameters from an certainly appear. experiment might be a very good method, but it has to be used with caution because of the possible existence of G. Macroscopic and self-propelling systems hidden variables. The are several aspects that we do not present because this will extend too much the purpose of this review. C. Statistical inferences The first is the application to quantum systems, which The problem of hidden variables mentioned in Sec. IX B presents several problems discussed in another article has been attacked in a different way in Refs. [172,173]. [174]. The other aspects are those related to the application This approach, called statistical inferences by the authors, of stochastic thermodynamics to dissipative chaotic sys- analyzes to what extent the fact that FT and FDR do not tems driven out of equilibrium by external forces such as, hold can give information on hidden variables. In for example, shaken granular media, turbulence fields, and Ref. [173] the authors have been able to extract interesting chaotic nonlinear oscillators. In these systems the main information on a single-molecule measurements. Thus, the source of randomness is not the thermal noise but the approach is certainly interesting and it merits further study chaotic behavior produced by the complex dynamics. As in more detail in the future. summarized in Ref. [3], the main problem is the absence of 021051-21 S. CILIBERTO PHYS. REV. X 7, 021051 (2017) Connection with Response Theory, J. Chem. 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