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A. Barato, U. Seifert (2016)
Cost and Precision of Brownian ClocksarXiv: Statistical Mechanics
Timur Koyuk, U. Seifert, Patrick Pietzonka (2018)
A generalization of the thermodynamic uncertainty relation to periodically driven systemsJournal of Physics A: Mathematical and Theoretical, 52
A. Dechant, S. Sasa (2017)
Current fluctuations and transport efficiency for general Langevin systemsJournal of Statistical Mechanics: Theory and Experiment, 2018
Patrick Pietzonka, A. Barato, U. Seifert (2016)
Universal bound on the efficiency of molecular motorsJournal of Statistical Mechanics: Theory and Experiment, 2016
Todd Gingrich, Grant Rotskoff, J. Horowitz (2016)
Inferring dissipation from current fluctuationsJournal of Physics A: Mathematical and Theoretical, 50
I. Martínez, É. Roldán, L. Dinis, R. Rica (2017)
Colloidal heat engines: a review.Soft matter, 13 1
Tim Schmiedl, U. Seifert (2007)
Efficiency at maximum power: An analytically solvable model for stochastic heat enginesEPL (Europhysics Letters), 81
Patrick Pietzonka, A. Barato, U. Seifert (2015)
Universal bounds on current fluctuations.Physical review. E, 93 5
K. Hayashi, S. Lorenzo, M. Manosas, J. Huguet, F. Ritort (2012)
Single-Molecule Stochastic ResonancePhysical Review X, 2
Karel Proesmans, C. Broeck (2017)
Discrete-time thermodynamic uncertainty relationEurophysics Letters, 119
R. Uzdin, Amikam Levy, R. Kosloff (2015)
Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic SignaturesPhysical Review X, 5
S. Restrepo, J. Cerrillo, P. Strasberg, G. Schaller (2017)
From quantum heat engines to laser cooling: Floquet theory beyond the Born–Markov approximationNew Journal of Physics, 20
S. Ciliberto (2017)
Experiments in Stochastic Thermodynamics: Short History and PerspectivesPhysical Review X, 7
M. Polettini, M. Esposito (2016)
Carnot efficiency at divergent power outputEurophysics Letters, 118
Wonseok Hwang, Changbong Hyeon (2018)
Energetic Costs, Precision, and Transport Efficiency of Molecular Motors.The journal of physical chemistry letters, 9 3
J. Rossnagel, S. Dawkins, K. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, K. Singer (2015)
A single-atom heat engineScience, 352
V. Blickle, C. Bechinger (2011)
Realization of a micrometre-sized stochastic heat engineNature Physics, 8
M. Carrega, M. Sassetti, U. Weiss (2019)
Optimal work-to-work conversion of a nonlinear quantum Brownian duetPhysical Review A
T. Shirai, Juzar Thingna, Takashi Mori, S. Denisov, P. Hānggi, S. Miyashita (2015)
Effective Floquet–Gibbs states for dissipative quantum systemsNew Journal of Physics, 18
Michael Nguyen, Suriyanarayanan Vaikuntanathan (2015)
Design principles for nonequilibrium self-assemblyProceedings of the National Academy of Sciences, 113
M. Campisi, R. Fazio (2016)
The power of a critical heat engineNature Communications, 7
Patrick Pietzonka, U. Seifert (2017)
Universal Trade-Off between Power, Efficiency, and Constancy in Steady-State Heat Engines.Physical review letters, 120 19
V. Holubec, A. Ryabov (2018)
Cycling Tames Power Fluctuations near Optimum Efficiency.Physical review letters, 121 12
Reuben Wang, Bo Xing, G. Carlo, D. Poletti (2017)
Period doubling in period-one steady states.Physical review. E, 97 2-1
Sundus Erbas-Cakmak, D. Leigh, Charlie McTernan, Alina Nussbaumer (2015)
Artificial Molecular MachinesChemical Reviews, 115
Naoto Shiraishi, Keiji Saito, H. Tasaki (2016)
Universal Trade-Off Relation between Power and Efficiency for Heat Engines.Physical review letters, 117 19
J. Pekola (2015)
Towards quantum thermodynamics in electronic circuitsNature Physics, 11
O. Abah, J. Rossnagel, G. Jacob, Sebastian Deffner, F. Schmidt-Kaler, K. Singer, Eric Lutz (2012)
Single-ion heat engine at maximum power.Physical review letters, 109 20
S. Pigolotti, I. Neri, É. Roldán, F. Jülicher (2017)
Generic Properties of Stochastic Entropy Production.Physical review letters, 119 14
Patrick Pietzonka, F. Ritort, U. Seifert (2017)
Finite-time generalization of the thermodynamic uncertainty relation.Physical review. E, 96 1-1
A. Barato, Raphael Chetrite, A. Faggionato, Davide Gabrielli (2018)
A unifying picture of generalized thermodynamic uncertainty relationsJournal of Statistical Mechanics: Theory and Experiment, 2019
J. Garrahan (2017)
Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables.Physical review. E, 95 3-1
Keye Zhang, F. Bariani, P. Meystre (2014)
Quantum optomechanical heat engine.Physical review letters, 112 15
Simone Gasparinetti, P. Solinas, A. Braggio, Maura Sassetti (2014)
Heat-exchange statistics in driven open quantum systemsNew Journal of Physics, 16
Lukas Fischer, Patrick Pietzonka, U. Seifert (2017)
Large deviation function for a driven underdamped particle in a periodic potential.Physical review. E, 97 2-1
Todd Gingrich, J. Horowitz, N. Perunov, Jeremy England (2015)
Dissipation Bounds All Steady-State Current Fluctuations.Physical review letters, 116 12
I. Terlizzi, M. Baiesi (2018)
Kinetic uncertainty relationJournal of Physics A: Mathematical and Theoretical, 52
G. Guarnieri, G. Landi, S. Clark, J. Goold (2019)
Thermodynamics of precision in quantum nonequilibrium steady statesPhysical Review Research
R. Kosloff, Y. Rezek (2016)
The Quantum Harmonic Otto CycleEntropy, 19
F. Carollo, R. Jack, J. Garrahan (2018)
Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems.Physical review letters, 122 13
P. Jung (1993)
Periodically driven stochastic systemsPhysics Reports, 234
C. Nardini, H. Touchette (2017)
Process interpretation of current entropic boundsThe European Physical Journal B, 91
B. Agarwalla, D. Segal (2018)
Assessing the validity of the thermodynamic uncertainty relation in quantum systemsPhysical Review B
M. Hartmann, D. Poletti, M. Ivanchenko, S. Denisov, P. Hānggi (2016)
Asymptotic Floquet states of open quantum systems: the role of interactionNew Journal of Physics, 19
F. Gambetta, F. Carollo, M. Marcuzzi, J. Garrahan, I. Lesanovsky (2018)
Discrete Time Crystals in the Absence of Manifest Symmetries or Disorder in Open Quantum Systems.Physical review letters, 122 1
K. Brandner, Taro Hanazato, Keiji Saito (2017)
Thermodynamic Bounds on Precision in Ballistic Multiterminal Transport.Physical review letters, 120 9
] for details of derivations, examples, and further generalizations
G. Verley, M. Esposito, T. Willaert, C. Broeck (2014)
The unlikely Carnot efficiencyNature Communications, 5
Katarzyna Macieszczak, K. Brandner, J. Garrahan (2018)
Unified Thermodynamic Uncertainty Relations in Linear Response.Physical review letters, 121 13
J. Horowitz, Todd Gingrich (2017)
Proof of the finite-time thermodynamic uncertainty relation for steady-state currents.Physical review. E, 96 2-1
A. Dechant, S. Sasa (2018)
Entropic bounds on currents in Langevin systems.Physical review. E, 97 6-1
Todd Gingrich, J. Horowitz (2017)
Fundamental Bounds on First Passage Time Fluctuations for Currents.Physical review letters, 119 17
M. Polettini, A. Lazarescu, M. Esposito (2016)
Tightening the uncertainty principle for stochastic currents.Physical review. E, 94 5-1
M. Campisi, J. Pekola, R. Fazio (2014)
Nonequilibrium fluctuations in quantum heat engines: theory, example, and possible solid state experimentsNew Journal of Physics, 17
Karel Proesmans, J. Horowitz (2019)
Hysteretic thermodynamic uncertainty relation for systems with broken time-reversal symmetryJournal of Statistical Mechanics: Theory and Experiment, 2019
Hyun-Myung Chun, Lukas Fischer, U. Seifert (2019)
Effect of a magnetic field on the thermodynamic uncertainty relation.Physical review. E, 99 4-1
A. Barato, U. Seifert (2015)
Thermodynamic uncertainty relation for biomolecular processes.Physical review letters, 114 15
Krzysztof Ptaszyński (2018)
Coherence-enhanced constancy of a quantum thermoelectric generatorPhysical Review B
U. Seifert (2019)
From Stochastic Thermodynamics to Thermodynamic InferenceAnnual Review of Condensed Matter Physics
K. Brandner, Keiji Saito, U. Seifert (2015)
Thermodynamics of Micro- and Nano-Systems Driven by Periodic Temperature VariationsarXiv: Statistical Mechanics
M. Esposito, R. Kawai, K. Lindenberg, C. Broeck (2010)
Quantum-dot Carnot engine at maximum power.Physical review. E, Statistical, nonlinear, and soft matter physics, 81 4 Pt 1
A. Barato, R. Chetrite, A. Faggionato, D. Gabrielli (2018)
Bounds on current fluctuations in periodically driven systemsNew Journal of Physics, 20
Grant Rotskoff (2016)
Mapping current fluctuations of stochastic pumps to nonequilibrium steady states.Physical review. E, 95 3-1
C. Maes (2017)
Frenetic Bounds on the Entropy Production.Physical review letters, 119 16
R. Uzdin, Amikam Levy, R. Kosloff (2015)
Quantum Equivalence and Quantum Signatures in Heat EnginesarXiv: Quantum Physics
Operationally accessible bounds on fluctuations and entropy production in periodically driven systems Timur Koyuk and Udo Seifert II. Institut fur ¨ Theoretische Physik, Universit¨at Stuttgart, 70550 Stuttgart, Germany (Dated: April 19, 2019) For periodically driven systems, we derive a family of inequalities that relate entropy production with experimentally accessible data for the mean, its dependence on driving frequency, and the variance of a large class of observables. With one of these relations, overall entropy production can be bounded by just observing the time spent in a set of states. Among further consequences, the thermodynamic efficiency both of isothermal cyclic engines like molecular motors under a periodic load and of cyclic heat engines can be bounded using experimental data without requiring knowledge of the specific interactions within the system. We illustrate these results for a driven three-level system and for a colloidal Stirling engine. Periodically driven open systems typically reach a pe- on the typically not directly accessible entropy produc- riodic steady state since the coupling to the environment tion, this relation can also be interpreted as a bound on prevents unlimited heating up. For meso- and nanoscopic the precision of a process in the sense that small fluc- systems, such a steady state can still exhibit significant tuations, i.e., high precision or low uncertainty, requires fluctuations [1]. Beyond the quantum domain, in which a minimum amount of entropy production. As specific such systems have found considerable attention recently, applications, the efficiency of molecular motors can be see, e.g., [2–7] and refs. therein, colloidal systems provide bounded using only experimental data [26, 27] and de- a major paradigm [8]. Likewise, chemical and biophysical sign principles for self-assembly can be derived [28]. For systems on the molecular and cellular scale that are sub- steady-state heat engines, the relation shows that an in- jected to periodic mechanical, optical or chemical stimuli evitable side-effect of reaching Carnot efficiency at finite fall into this wide class, see, e.g., [9, 10]. Heat engines power are diverging power fluctuations [29–33]. Refine- and cooling devices coupled cyclically to baths of differ- ments and generalizations of the TUR have been found ent temperature provide a further paradigm [11–23], for diffusive dynamics [34–36], for data allocated over a finite time [37–40], for ballistic transport [41], for under- One obvious question for any such system is whether damped Langevin dynamics with and without magnetic or not the entropy production associated with the peri- fields [42–44]. Rather than looking at the fluctuations odic driving can be inferred, or, at least, bounded, using of currents, it is also possible to constrain the fluctua- only experimentally accessible observables without hav- tions of time-symmetric quantities like residence times ing detailed knowledge of the interactions or the internal or activity [45–47] and the fluctuations of first passage structure of the system. This question is thus a non- times [48, 49]. While a few quantum systems have been trivial one whenever power input and power output are investigated, a systematic picture in the quantum realm not both directly measurable, which is the case when not is still missing [41, 50–54]. all degrees of freedom that couple to these currents are observable. This situation is inter alia typical for all sys- An early counter-example has shown that a naive ex- tems undergoing chemical reactions. Likewise, it holds tension of the thermodynamic uncertainty relation from for systems driven by electric fields since one cannot ob- steady-state systems to periodically driven ones is not serve the motion of all charges, in particular, in (soft) admissible [55]. Subsequent attempts to find an analog condensed matter systems. for periodically driven systems comprise Proesman and For the arguably simpler class of non-equilibrium van den Broeck’s bound valid for time-symmetric driving steady states in systems under constant, time- that, however, for a small frequency of driving leads to a independent driving, a universal, experimentally acces- rather weak bound [56]. Barato et al replace σ in eq. (1) sible relation has recently been found that achieves this by a modified entropy production rate that requires de- aim for classical systems. The so-called thermodynamic tailed knowledge of dynamical properties of the whole uncertainty relation (TUR) [24, 25] system [57]. In a follow up [58], a whole class of such modified entropy production rates were discussed that, σD /J ≥ 1 (1) from an operational perspective, are arguably not that relates the entropy production rate σ with the mean value useful since they require input that is typically not avail- of any current J and its variance, or dispersion, D , all able in experiments. The same holds for our generaliza- defined more precisely below. Besides stating a bound tion introducing an effective entropy production for fur- arXiv:1904.08807v1 [cond-mat.stat-mech] 18 Apr 2019 2 ther types of currents [59] and for a further scheme [60]. to find the system in state i becomes periodic as well ps Recently, Proesmans and Horowitz introduced a modi- and will be denoted by p (τ ; Ω), where the correspond- fied uncertainty relation for hysteretic currents that over- ing driving frequency Ω is explicitly introduced through comes some of the above mentioned short-comings [61]. the second argument. However, from the perspective of thermodynamic infer- One class of fluctuating time-integrated current de- ence [62], an operationally accessible relation, which al- pends on the number of transitions between states lows to bound entropy production and which becomes the TUR for steady-state systems, is still missing for pe- 1 J ≡ dt n ˙ (t)d (t) (4) ij ij riodically driven systems. ij As a main result of this Letter, for systems driven with a period T ≡ 2π/Ω, we will derive a family of universal with periodic increments d (t) = −d (t) and 0 ≤ t ≤ T . ij ji bounds that relate, inter alia, the entropy production Here, n (t) is the number of transitions from i to j up ij with fluctuations of observables. Applied to current fluc- to time t along a trajectory i(t) of length T . In the long- tuations, we get specifically time limit T → ∞, the current in eq. (4) reaches the mean value 2 0 2 σ(Ω)D (Ω)/J (Ω) ≥ [1− ΩJ (Ω)/J (Ω)] . (2) d d ps J (Ω) ≡ hJ i = dτ j (τ ; Ω)d (τ ), (5) The left-hand side involves the same combination of vari- ij T ij i>j ables as the ordinary TUR does, where we make the de- pendence on Ω explicit. The right-hand side addition- ps ps ps where j (τ ; Ω) ≡ p (τ ; Ω)k (τ )− p (τ ; Ω)k (τ ). One ij ji ally contains the derivative of the current with respect to ij i j example for the mean value of a current in eq. (4) is the the driving frequency, i.e., the response of the current to entropy production a slight change of the period of driving. In the special case of constant driving, this bound becomes the ordinary T ps 1 p (τ ; Ω)k (τ ) ps ij TUR, eq. (1), since then the current is formally indepen- i σ(Ω) = dτ j (τ ; Ω) ln . ps ij dent of the driving frequency. Thus, the entropy pro- T p (τ ; Ω)k (τ ) 0 ji i>j duction in a periodically driven system can be bounded (6) using experimental data for the mean of any current, its A further class of current can be derived from the resi- fluctuations and its response to a slight change of driving dence time in certain states as frequency. One consequence of this relation is that it provides J ≡ dt δ a ˙ (t), (7) i(t),i i a necessary condition for (almost) dissipation-less pre- cision. The current must be proportional to the fre- quency of driving since then the right-hand side van- where δ is 1 if state i is occupied at time t and 0, i(t),i ishes. This insight rationalizes the earlier construction otherwise. Here, the periodic increment can be written of a dissipation-less Brownian clock [55]. as a time-derivative of a state variable, e.g., for power Applied to the efficiency η ≡ P /P of isothermal ˙ out in input a ˙ (t) = E (t), where E (t) is the energy of state i. i i i a a cyclic engines that convert an input source of energy with The mean value of eq. (7) is denoted by J (Ω) ≡ hJ i. mean P to an output power with mean P = P − σ, in out in An arbitrary average current consists of a superposition the relation implies of the two types of currents defined in eqs. (4) and (7), d a i.e., J (Ω) ≡ J (Ω) + J (Ω). 1− η D out P ≤ . (3) A second class of observables are called residence quan- out 0 2 η [1− ΩP (Ω)/P (Ω)] out out tities that are defined as Hence, in general, the power of a cyclic engine vanishes at least linearly as its efficiency approaches the maximum A ≡ dt δ A (t) (8) T i i(t),i value of 1. A finite power in this limit is, in principle, possible only if the current fluctuations diverge or if the with time-dependent periodic state variables A (t) that output power is proportional to the cycling frequency of cannot be written as time-derivatives. Simple examples the engine. While the first option has been previously of such a quantity are the average fraction of time τ been derived for steady-state engines from the TUR [26], spent in state k during one cycle with increment A (t) = the second one is genuine for periodically driven engines. δ or the average energy with A (t) = E (t). For long i,k i i These results and the further ones derived and dis- times, their mean value is given by cussed below hold for systems described by a Markov dy- namics on a set of states. The transition rate k (τ ) from ij Z ps state i to state j is time-periodic with k (τ +T ) = k (τ ) ij ij A(Ω) ≡ hA i = dτ p (τ ; Ω)A (τ ). (9) T i and 0 ≤ τ ≤ T . In the long-time limit, the probability i 3 Fluctuations of residence and current observables can be quantified via the diffusion coefficient D (Ω) ≡ lim Th(X −hX i) i/2 (10) X T T T→∞ d a with X ∈ {J , J , A }. T T T T For the residence variable A in eq. (8), we will show that the mean value A(Ω) and the fluctuations D obey 0 2 σ(Ω)D (Ω) ≥ [ΩA (Ω)] . (11) FIG. 1. Ratio of estimator (15) and total entropy produc- Thus, by measuring how the mean value of this observ- tion (6) as a function of the driving frequency Ω for different able changes with driving frequency, a lower bound on energy amplitudes E for the driven 3-state level (inset). The the entropy production can be obtained without knowing s 0 0 0 0 parameters E = 1.0, E = 8.0, E = 2.3, E = 4.1, k = 1 1 2 3 12 0 0 further details of the system. It is quite remarkable that 0.01, k = 0.2, k = 10.0, and α = 0.1 are kept fixed. 23 13 by observing a variable that is even under time-reversal a bound on entropy production, which is a hallmark of broken time-reversal symmetry, can be inferred. There is where Ω is an arbitrary frequency and no analog of this relation in the case of constant driving, T h i X 2 since then the right-hand side vanishes. 1 ps ps ˜ ˜ ˜ S (Ω, Ω) ≡ dτ j (Ωτ/Ω; Ω) /t (τ ; Ω), ij ij As a first example, we illustrate relation (11) via an i>j isothermal three state model with the following energy (18) levels ˜ ˜ ˜ ˜ G(Ω, Ω) ≡ A(Ω) + (Ω/Ω)J (Ω) (19) c s 0 E (t) = E cos Ωt + E sin Ωt + E , 1 1 1 a d ˜ ˜ ˜ 0 0 with J (Ω) ≡ J (Ω) + J (Ω). Here, we introduced the E (t) = E , and E (t) = E . (12) 2 3 2 3 probability current at frequency Ω The rates k (τ ) ij ps ps ps ˜ ˜ ˜ ˜ ˜ ˜ j (Ωτ/Ω; Ω) ≡ p (Ωτ/Ω; Ω)k (τ )−p (Ωτ/Ω; Ω)k (τ ) ij ji ij i j (20) k (t) = k exp [−α(E (t)− E (t))] , (13) ij i j ij and the activity at link (i, j) k (t) = k exp [(1− α)(E (t)− E (t))] (14) ji i j ij ps ps ps t (τ ; Ω) ≡ p (τ ; Ω)k (τ ) + p (τ ; Ω)k (τ ) (21) ij ji ij i j fulfill the local detailed balance condition for any α. Here, we have set β = 1 and introduced a rate ampli- at the original frequency Ω. tude k for each link (i, j). ij We choose either the increments d = a ˙ = 0 or A = 0 ij i i We vary the driving frequency Ω for different energy such that G(Ω) = J (Ω) for currents and G(Ω) = A(Ω) amplitudes E and fix all other parameters. The total ˜ for residence quantities and take the limit Ω → Ω. This entropy production can be estimated via eq. (11): the leads to the following two inequalities frequency-dependent fraction of residence time τ (Ω) in 2 0 2 state 1 and its diffusion coefficient D (Ω) are sufficient 1 D (Ω)C (Ω)/J (Ω) ≥ (1− ΩJ (Ω)/J (Ω)) , (22) to obtain the estimator 2 0 2 D (Ω)C (Ω)/A(Ω) ≥ (ΩA (Ω)/A(Ω)) . (23) σ ≡ [Ωτ (Ω)] /D (Ω) ≤ σ, (15) est τ 1 1 Here, C (Ω) ∈ {2S (Ω), 2A(Ω), σ(Ω)} can be chosen as one of three cost terms with S (Ω) ≡ S (Ω, Ω = Ω) and plotted in Fig. 1a as a function of the driving frequency Ω T ps c A(Ω) ≡ 1/T dτ t (τ ; Ω) the average dynamical for different amplitudes E . Hence, this simple estimate i>j ij activity. These two inequalities (22) and (23) are our yields already up to 40% of the total entropy production. most general results, from which our main results (2) The derivation of our main results (2) and (11) can be and (11) are obtained with C (Ω) = σ(Ω). sketched as follows [63]. The diffusion coefficient D (Ω) By choosing C (Ω) = A(Ω), the average dynamical ac- for a quantity tivity can be bounded via eq. (22) by current observables a d and their fluctuations. The steady-state analog of this re- G ≡ A + J + J (16) T T T T lation was proven in [34] and extended to more general observables in [49]. Our generalization of the latter one with mean G(Ω) ≡ hG i can be bounded by for periodic driving is derived in the Supplemental Ma- 2 2 ˜ ˜ ˜ 2D (Ω) ≥ (G(Ω)−G(Ω, Ω)) /[(Ω/Ω−1) S (Ω, Ω)], (17) terial [63]. These bounds on activity can be generalized G 4 to residence observables through eq. (23). They have no 1 2 V (x, τ ) V (x, τ ) steady-state analog. ps ps p (x, τ ) p (x, τ ) Finally, we investigate the implications of our results for heat engines operating cyclically between two baths max βc βc β β of inverse temperature β < β with efficiency h h c λ(τ ) η(Ω) ≡ P (Ω)/Q (Ω) ≤ η ≡ 1− β /β , (24) out h C h c min 4 0 3 T /2 T V (x, τ ) V (x, τ ) ˙ ps ps p (x, τ ) p (x, τ ) where P > 0 is the output power and Q the heat out h current flowing into the system from the hot reservoir. The inequality (2) implies the bound β β h h −1 0 2 (P (Ω)− ΩP (Ω)) out out η(Ω) ≤ η ˆ(Ω) ≡ η 1 + . (β D P (Ω)) c out P (Ω) FIG. 2. Particle in a harmonic trap with time-dependent stiff- out (25) ness λ(τ ) operating as a Stirling heat engine between temper- atures β > β . Carnot efficiency at finite power can thus be reached only c h if the power fluctuations diverge or if the power increases 1.00 linearly with the driving frequency. The latter condi- σ η/η a) b) C 1.5 tion is typical for quasi-static driving [33]. For maxi- est η/ηˆ 0.75 C est,TUR mal output power (P (Ω ) = 0), eq. (25) reduces to ηˆ /η max ss C out 1.0 P /P out max 0.50 the established bound for steady-state systems [32] thus showing a universal trade-off between efficiency, power 0.5 0.25 and constancy at maximum power. 0.0 0.00 So far we discussed systems with a discrete set 0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00 of states. We now illustrate the new bound on ef- T T ficiency (25) for a system obeying an overdamped Langevin equation. Specifically, we consider a Stirling FIG. 3. (a) Entropy production σ, its corresponding estima- heat engine model inspired by [11, 13]. The engine con- tor σ (30), estimator for steady-state systems σ and est est,TUR sists of a colloid in a one-dimensional harmonic potential output power P divided by its maximum P and (b) effi- out max ciency η, bound on the efficiency η ˆ (25), and its steady-state with a time-dependent stiffness λ(τ ) analog η ˆ as function of the cycle duration T for μ = 10.0, ss k = 2.0, k = 1.0, β = 2.0 and β = 1.0. max min c h V (x, λ(τ )) ≡ V (x, τ ) ≡ λ(τ )x /2 ≥ 0. (26) The position x follows the Langevin equation Currents of interest are power, entropy production and x ˙ (t) = −μλ(t)x(t) + ζ (t). (27) heat. Their explicit expressions and the diffusion coeffi- cient for the output power are given in [63]. In order to Here, μ denotes the mobility, and ζ (t) is zero-mean Gaus- illustrate eq. (2) and (25), we define the estimator ˜ ˜ sian noise with correlation hζ (t)ζ (t)i = 2μδ(t − t)/β(t) and a periodic temperature β(t) = β(t + T ). The cor- σ ≡ (P (Ω)− ΩP (Ω)) /D (Ω) ≤ σ (30) est out P out out responding Fokker-Planck equation for the probability for entropy production and vary the cycle duration T . distribution p(x, t) reads The output power, the estimator (30) and the bound on −1 ∂ p(x, t) = μ∂ ∂ V (x, t) + β (t)∂ p(x, t). (28) efficiency (25) are shown in Fig. 3. With increasing T , τ x x x the output power increases for small T . According to ps The periodic stationary solution p (x, τ ) of (28) is a eq. (2) the ordinary TUR, eq. (1), is still valid. At maxi- Gaussian with zero mean. mum power (vertical dotted line) eq. (2) is equivalent to As a protocol for the stiffness, we use the one from the the TUR and the estimator σ ≡ P /D inter- est,TUR P out out experiment in [13], which increases and decreases linearly sects σ . After reaching the maximum value, the power est in time according to decreases with increasing T with the TUR violated for T & 0.6. 2Δkτ/T + k , 0 ≤ τ < T /2 min In summary, for periodically driven systems, we have λ(τ ) = −2Δk(τ/T − 1/2) + k , T /2 ≤ τ < T max derived a class of inequalities that relate, inter alia, en- (29) tropy production with the mean of an observable, its with Δk ≡ k − k , see Fig. 2. The coupling to dependence on driving frequency and its variance. Re- max min two different heat baths leads to a time-dependent tem- markably, apart from the more familiar currents, such perature that is piecewise constant, i.e., β(τ ) = β for observables can also be even under time-reversal like res- 0 ≤ τ < T /2 and β(τ ) = β for T /2 ≤ τ < T . idence times are. For cyclic heat engines and cyclically h 5 driven isothermal engines, the thermodynamic efficiency Christian Van den Broeck, “The unlikely Carnot effi- ciency,” Nat. Commun. 5, 4721 (2014). can now be bounded using data only for the input or [17] Michele Campisi, Jukka Pekola, and Rosario Fazio, output current. Since our results have been obtained for “Nonequilibrium fluctuations in quantum heat engines: a Markov dynamics on a discrete set of states, they triv- theory, example, and possible solid state experiments,” ially hold as well for overdamped Langevin dynamics. It New J. Phys. 17, 035012 (2015). remains an open problem whether they can be extended [18] Jukka P. Pekola, “Towards quantum thermodynamics in to underdamped dynamics. Arguably even more exciting electronic circuits,” Nature Phys. 11, 118–123 (2015). will be to explore along similar lines periodically driven [19] R. Uzdin, A. Levy, and R. Kosloff, “Equivalence of quan- tum heat machines, and quantum-thermodynamic signa- open quantum systems for which coherences are relevant. ture,” Phys. Rev. X 5, 031044 (2015). [20] Kay Brandner, Keiji Saito, and Udo Seifert, “Thermo- dynamics of micro- and nano-systems driven by periodic temperature variations,” Phys. Rev. X 5, 031019 (2015). [21] J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, [1] P. Jung, “Periodically driven stochastic-systems,” Phys. E. Lutz, F. Schmidt-Kaler, and K. Singer, “A single- Rep. 234, 175–295 (1993). atom heat engine,” Science 352, 325–329 (2016). [2] S. Gasparinetti, P. Solinas, A. Braggio, and M. Sassetti, [22] R. Kosloff and Y. Rezek, “The quantum harmonic Otto “Heat-exchange statistics in driven open quantum sys- cycle,” Entropy 19, 136 (2017). tems,” New J. Phys. 16, 115001 (2014). [23] Ignacio A. Martinez, Edgar Roldan, Luis Dinis, and [3] Tatsuhiko Shirai, Juzar Thingna, Takashi Mori, Sergey Raul A. Rica, “Colloidal heat engines: a review,” Soft Denisov, Peter H¨ anggi, and Seiji Miyashita, “Effective Matter 13, 22–36 (2017). Floquet-Gibbs states for dissipative quantum systems,” [24] Andre C. Barato and Udo Seifert, “Thermodynamic un- New J. Phys. 18, 053008 (2016). certainty relation for biomolecular processes,” Phys. Rev. [4] M. Hartmann, D. Poletti, M. Ivanchenko, S. Denisov, Lett. 114, 158101 (2015). and P. H¨ anggi, “Asymptotic Floquet states of open quan- [25] Todd R. Gingrich, Jordan M. Horowitz, Nikolay Perunov, tum systems: the role of interaction,” New J. Phys. 19, and Jeremy L. England, “Dissipation bounds all steady- 083011 (2017). state current fluctuations,” Phys. Rev. Lett. 116, 120601 [5] S. Restrepo, J. Cerrillo, P. Strasberg, and G. Schaller, (2016). “From quantum heat engines to laser cooling: Floquet [26] Patrick Pietzonka, Andre C Barato, and Udo Seifert, theory beyond the Born-Markov approximation,” New “Universal bound on the efficiency of molecular motors,” J. Phys. 20, 053063 (2018). J. Stat. Mech.: Theor. Exp. , 124004 (2016). [6] R. R. W. Wang, B. Xing, G. G. Carlo, and D. Poletti, [27] Wonseok Hwang and Changbong Hyeon, “Energetic “Period doubling in period-one steady states,” Phys. costs, precision, and transport efficiency of molecular mo- Rev. E 97, 020202(R) (2018). tors,” J. Phys. Chem. Lett. 9, 513–520 (2018). [7] F. M. Gambetta, F. Carollo, M. Marcuzzi, J. P. Garra- [28] Michael Nguyen and Suriyanarayanan Vaikuntanathan, han, and I. Lesanovsky, “Discrete time crystals in the “Design principles for nonequilibrium self-assembly,” absence of manifest symmetries or disorder in open quan- Proc. Natl. Acad. Sci. U.S.A. 113, 14231–14236 (2016). tum systems,” Phys. Rev. Lett. 122, 015701 (2019). [29] Naoto Shiraishi, Keiji Saito, and Hal Tasaki, “Universal [8] S. Ciliberto, “Experiments in stochastic thermodynam- trade-off relation between power and efficiency for heat ics: Short history and perspectives,” Phys. Rev. X 7, engines,” Phys. Rev. Lett. 117, 190601 (2016). 021051 (2017). [30] Michele Campisi and Rosario Fazio, “The power of a crit- [9] K. Hayashi, S. de Lorenzo, M. Manosas, J. M. Huguet, ical heat engine,” Nat. Commun. 7, 11895 (2016). and F. Ritort, “Single-molecule stochastic resonance,” [31] Matteo Polettini and Massimiliano Esposito, “Carnot ef- Phys. Rev. X 2, 031012 (2012). ficiency at divergent power output,” EPL 118, 40003 [10] S. Erbas-Cakmak, D. A. Leigh, C. T. McTernan, and (2017). A. L. Nussbaumer, “Artificial molecular machines,” [32] Patrick Pietzonka and Udo Seifert, “Universal trade-off Chem. Rev. 115, 10081–10206 (2015). between power, efficiency and constancy in steady-state [11] T. Schmiedl and U. Seifert, “Efficiency at maximum heat engines,” Phys. Rev. Lett. 120, 190602 (2018). power: An analytically solvable model for stochastic heat [33] Viktor Holubec and Artem Ryabov, “Cycling tames engines,” EPL 81, 20003 (2008). power fluctuations near optimum efficiency,” Phys. Rev. [12] M. Esposito, R. Kawai, K. Lindenberg, and C. Van Lett. 121, 120601 (2018). den Broeck, “Quantum-dot Carnot engine at maximum [34] Patrick Pietzonka, Andre C. Barato, and Udo Seifert, power,” Phys. Rev. E 81, 041106 (2010). “Universal bounds on current fluctuations,” Phys. Rev. [13] V. Blickle and C. Bechinger, “Realization of a E 93, 052145 (2016). micrometre-sized stochastic heat engine,” Nature Phys. [35] Todd R Gingrich, Grant M Rotskoff, and Jordan M 8, 143 (2012). Horowitz, “Inferring dissipation from current fluctua- [14] O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt- tions,” J. Phys. A: Math. Theor. 50, 184004 (2017). Kaler, K. Singer, and E. Lutz, “Single-ion heat engine at [36] Matteo Polettini, Alexandre Lazarescu, and Massimil- maximum power,” Phys. Rev. Lett. 109, 203006 (2012). iano Esposito, “Tightening the uncertainty principle for [15] K. Zhang, F. Bariani, and P. Meystre, “Quantum op- stochastic currents,” Phys. Rev. E 94, 052104 (2016). tomechanical heat engine,” Phys. Rev. Lett. 112, 150602 [37] Patrick Pietzonka, Felix Ritort, and Udo Seifert, “Finite- (2014). time generalization of the thermodynamic uncertainty re- [16] Gatien Verley, Massimiliano Esposito, Tim Willaert, and lation,” Phys. Rev. E 96, 012101 (2017). 6 [38] Jordan M. Horowitz and Todd R. Gingrich, “Proof of the thermodynamic uncertainty relation in quantum of the finite-time thermodynamic uncertainty relation systems,” Phys. Rev. B 98, 155438 (2018). for steady-state currents,” Phys. Rev. E 96, 020103(R) [52] M. Carrega, M. Sassetti, and U. Weiss, “Optimal work- (2017). to-work conversion of a nonlinear quantum brownian [39] Andreas Dechant and Shin ichi Sasa, “Current fluctua- duet,” arXiv:1901.10881 (2019). tions and transport efficiency for general Langevin sys- [53] K. Ptaszynski, ´ “Coherence-enhanced constancy of a tems,” J. Stat. Mech. Theor. Exp. , 063209 (2018). quantum thermoelectric generator,” Phys. Rev. B 98, 085425 (2018). [40] Simone Pigolotti, Izaak Neri, Edgar Rold´ an, and Frank [54] G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold, Julic ¨ her, “Generic properties of stochastic entropy pro- “Thermodynamics of precision in quantum non equilib- duction,” Phys. Rev. Lett. 119, 140604 (2017). rium steady states,” arXiv:1901.10428 (2019). [41] Kay Brandner, Taro Hanazato, and Keiji Saito, “Ther- [55] Andre C. Barato and Udo Seifert, “Cost and precision of modynamic bounds on precision in ballistic multitermi- Brownian clocks,” Phys. Rev. X 6, 041053 (2016). nal transport,” Phys. Rev. Lett. 120, 090601 (2018). [56] Karel Proesmans and Christian Van den Broeck, [42] Lukas P. Fischer, Patrick Pietzonka, and Udo Seifert, “Discrete-time thermodynamic uncertainty relation,” “Large deviation function for a driven underdamped par- EPL 119, 20001 (2017). ticle in a periodic potential,” Phys. Rev. E 97, 022143 [57] Andre C. Barato, Raphael Chetrite, Alessandra Faggion- (2018). ato, and Davide Gabrielli, “Bounds on current fluctua- [43] A. Dechant and S. I. Sasa, “Entropic bounds on currents tions in periodically driven systems,” New J. Phys. 20, in Langevin systems,” Phys. Rev. E 97, 062101 (2018). 103023 (2018). [44] H.-M. Chun, L. P. Fischer, and U. Seifert, “Effect of a [58] A. C. Barato, R. Chetrite, A. Faggionato, and magnetic field on the thermodynamic uncertainty rela- D. Gabrielli, “A unifying picture of generalized ther- tion,” arXiv:1903.06480 (2019). modynamic uncertainty relations,” arXiv:1810.11894 [45] Cesare Nardini and Hugo Touchette, “Process interpre- (2018). tation of current entropic bounds,” Eur. Phys. J. B 91, 16 (2018). [59] T. Koyuk, U. Seifert, and P. Pietzonka, “A generaliza- [46] Christian Maes, “Frenetic bounds on the entropy produc- tion of the thermodynamic uncertainty relation to peri- tion,” Phys. Rev. Lett. 119, 160601 (2017). odically driven systems,” J. Phys. A Math. Theor. 52, [47] I. Terlizzi and M. Baiesi, “Kinetic uncertainty relation,” 02LT02 (2019). J. Phys. A 52, 02LT03 (2018). [60] Grant M. Rotskoff, “Mapping current fluctuations of [48] Todd R. Gingrich and Jordan M. Horowitz, “Fundamen- stochastic pumps to nonequilibrium steady states,” Phys. tal bounds on first passage time fluctuations for cur- Rev. E 95, 030101(R) (2017). rents,” Phys. Rev. Lett. 119, 170601 (2017). [61] K. Proesmans and J. M. Horowitz, “Hysteretic thermo- [49] Juan P. Garrahan, “Simple bounds on fluctuations and dynamic uncertainty relation for systems with broken uncertainty relations for first-passage times of counting time-reversal symmetry,” arXiv:1902.07008 (2019). observables,” Phys. Rev. E 95, 032134 (2017). [62] Udo Seifert, “From stochastic thermodynamics to ther- [50] Katarzyna Macieszczak, Kay Brandner, and Juan P. modynamic inference,” Ann. Rev. Cond. Mat. Phys. 10, Garrahan, “Unified thermodynamic uncertainty relations 171–192 (2019). in linear response,” Phys. Rev. Lett. 121, 130601 (2018). [63] See Supplemental Material at [...] for details of deriva- [51] B. K. Agarwalla and D. Segal, “Assessing the validity tions, examples, and further generalizations. Supplemental Material for ”Operationally accessible bounds on fluctuations and entropy production in periodically driven systems” Timur Koyuk and Udo Seifert II. Institut fur ¨ Theoretische Physik, Universit¨at Stuttgart, 70550 Stuttgart, Germany (Dated: April 19, 2019) I. DERIVATION OF THE MAIN RESULTS Here, X is an arbitrary observable defined along a tra- jectory i(τ ) of length T . The bound on the generating function is expressed in terms of an auxiliary dynamics. We derive our main results by using a lower bound on ˜ This dynamics is described by rates k(τ ) ≡ {k (τ )} lead- ij the scaled cumulant generating function (SCGF) [1, 2] ing to densities p˜(τ ) ≡ {p ˜ (τ )}. The average with respect aux to the auxiliary dynamics is denoted by h·i . A general bound on the SCGF at time T = nT (see, e.g., [2]) is zTX λ (z) ≡ lnhe i. (1) given by aux ps λ (z) ≥ zhX i − F [p ˜(τ ), k(τ )]− D (p˜(0)||p (0; Ω)) , (2) nT T nT with ! ! X ˜ 1 k (τ ) ij ˜ ˜ ˜ F [p˜(τ ), k(τ )] ≡ dτ p ˜ (τ )k (τ ) ln − p ˜ (τ )[k (τ )− k (τ )] (3) i ij i ij ij T k (τ ) ij i,j and the Kullback-Leibler divergence ps ps D (p ˜(0)||p (0; Ω)) ≡ p ˜ (0) ln[p ˜ (0)/p (0; Ω)]. (4) i i Here, k (τ ) are rates of the original dynamics with fre- the auxiliary dynamics (see, e.g., [2]), this bound is ex- ij quency Ω whose periodic stationary state is denoted by pressed in terms of densities p ˜(τ ) ≡ {p ˜ (τ )} and currents ps ˜ ˜ p (τ ; Ω). j(τ ) ≡ {j (τ )} by choosing optimal rates k (τ ). For ij i ij After an optimization with respect to the activity of choosing X = G the bound is given by T T 1 1 aux ps λ (z) ≥ zhG i − dτL p ˜(τ ), j(τ ) − D (p ˜(0)||p (0; Ω)) , (5) nT T T nT where the integrand L p˜(τ ), j(τ ) is defined as " ! !# p˜ q q j (τ ) j (τ ) ij ij p˜ p˜ p˜ ˜ ˜ ˜ 2 2 2 2 L p˜(τ ), j(τ ) ≡ j (τ ) arsinh − arsinh − [j (τ )] + [a (τ )] − [j (τ )] + [a (τ )] , ij ij ij ij ij p˜ p˜ a (τ ) a (τ ) i>j ij ij (6) with p˜ p˜ j (τ ) ≡ p ˜ (τ )k (τ )− p ˜ (τ )k (τ ), a (τ ) ≡ 4p ˜ (τ )p ˜ (τ )k (τ )k (τ ). (7) i ij j ji i j ij ji ij ij The above introduced auxiliary dynamics has to de- scribe a physical process and hence, it has to obey the arXiv:1904.08807v1 [cond-mat.stat-mech] 18 Apr 2019 2 following conditions as quoted in the main text. The nominator in (14) follows from the first term in (5), i.e., the average of observable p ˜ (τ ) = 1, 0 < p ˜ (τ ) < 1, (8) i i G in the auxiliary dynamics. Here, we used the sub- stitution τ ˜ = Ωτ/Ω and the following properties of the ˙ ˜ increments p ˜ (0) = p ˜ (T ) , p ˜ (τ ) = − j (τ ), (9) i i i ij ˜ ˜ d (τ ; Ω) = d (Ωτ ˜/Ω; Ω) = d (τ ˜; Ω), ij ij ij with probability current ˜ ˜ ˜ ˜ a ˙ (τ ; Ω) = (Ω/Ω)a ˙ (Ωτ ˜/Ω; Ω) = (Ω/Ω)a ˙ (τ ˜; Ω), i i i ˜ ˜ j (τ ) ≡ p ˜ (τ )k (τ )− p ˜ (τ )k (τ ) (10) ˜ ˜ ij i ij j ji A (τ ; Ω) = A (Ωτ ˜/Ω; Ω) = A (τ ˜; Ω), i i i in the auxiliary dynamics. The first condition in (9) guar- where we introduced the second argument to denote the antees that the chosen density p ˜ (τ ) has the same cycle corresponding frequency. The denominator in (14), i.e., duration T as the original periodic stationary state. The the cost term, follows from the second term in eq. (5). latter one can be written as After taking the long-time limit, the third term in eq. (5), ps i.e., the Kullback-Leibler divergence, vanishes. Moreover, p (τ ; Ω) ≡ a (Ω) + a (Ω) cos(nΩτ ) 0 n we note that for Ω → ∞ and A (τ ; Ω) = 0 in eq. (14) the n=1 GTUR in [2] is recovered. + b (Ω) sin(nΩτ ) . (11) Next, we choose Ω = αΩ and take the limit α → 1. Both, nominator and denominator on the right-hand side For every frequency Ω there exists a corresponding set in eq. (14) approach zero in this limit. Hence, we expand of Fourier coefficients {a (Ω), b (Ω)} defining the unique n n both around α = 1 leading to ps periodic stationary state p (τ ; Ω). We choose the following ansatz for the auxiliary dy- 2 0 0 x (J (Ω)− ΩJ (Ω)− ΩA (Ω) +O (x)) D (Ω) ≥ (15) namics G 2 3 (x +O (x )) (S (Ω) +O (x)) ps ps ps ˜ ˜ p ˜ (τ ) = p (τ ; Ω) + p (τ ; Ω)− p (Ωτ/Ω; Ω) , i i i with x ≡ α− 1 and ps ps ps ˜ ˜ ˜ ˜ j (τ ) = j (τ ; Ω) + j (τ ; Ω)− Ω/Ωj (Ωτ/Ω; Ω) ij 2 ij ij ij Z ps j (τ ; Ω) ij ˜ (12) S (Ω) ≡ S (Ω, Ω = Ω) = dτ . ps T t (τ ; Ω) ij i>j with small = O (z), where the density (16) X Here, denotes the derivative with respect to frequency ps ˜ ˜ ˜ ˜ p (Ωτ/Ω; Ω) ≡ a (Ω) + a (Ω) cos(nΩτ ) 0 n i Ω. Moreover, by setting either d (τ ; Ω) = a ˙ (τ ; Ω) = 0 or ij i n=1 A (τ ; Ω) = 0 we get the following inequalities for currents J (Ω) and residence quantities A(Ω) + b (Ω) sin(nΩτ ) (13) 2 0 2 2D (Ω)S (Ω)/J (Ω) ≥ (1− ΩJ (Ω)/J (Ω)) , (17) has the same cycle duration T = 2π/Ω as the original ˜ ˜ 2 0 2 process. The Fourier coefficients {a (Ω), b (Ω)} corre- n n 2D (Ω)S (Ω)/A(Ω) ≥ (ΩA (Ω)/A(Ω)) , (18) spond to a periodic stationary process with frequency Ω. Inserting ansatz (12) into (5), taking the long time as quoted in the main text. ps ps ps 2 2 limit n → ∞, choosing small = O (z), and optimizing Finally, using 2j (τ ) ≤ t (τ )σ (τ ) and j (τ ) ≤ ij ij ij ij ps with respect to this parameter leads to a local quadratic t (τ ) implies our main results ij bound on the generating function, which implies the in- 2 0 2 equality D (Ω)C (Ω)/J (Ω) ≥ (1− ΩJ (Ω)/J (Ω)) , (19) 2 0 2 ˜ D (Ω)C (Ω)/A(Ω) ≥ (ΩA (Ω)/A(Ω)) , (20) (G(Ω)− G(Ω, Ω)) 2D (Ω) ≥ (14) ˜ 2 ˜ (Ω/Ω− 1) S (Ω, Ω) where C (Ω) ∈ {2S (Ω), 2A(Ω), σ(Ω)} are cost terms and ps with A(Ω) ≡ 1/T dτ t (τ ) is the average dynamical i>j ij activity. ps Z ˜ ˜ j (Ωτ/Ω; Ω) ij S (Ω, Ω) ≡ dτ , ps T t (τ ; Ω) ij i>j II. FINITE-TIME GENERALIZATION ˜ ˜ ˜ ˜ G(Ω, Ω) ≡ Ω/ΩJ (Ω) + A(Ω), For a finite-time generalization of relation (14) and G(Ω) ≡ G(Ω, Ω = Ω) = J (Ω) + A(Ω) hence, of our main results (19) and (20), an additional 3 cost term arises due to the non-vanishing Kullback- along a trajectory i(t) of length T , where g (t) are arbi- ij Leibler divergence, i.e., trary time-periodic increments. Note, that we do not re- strict g (t) to be antisymmetric or symmetric. For exam- ij ple, X can be the average number of jumps during one 2 0 2 D (nT ; Ω)[C (Ω) + C (Ω)]/J (Ω) ≥ (1− ΩJ (Ω)/J (Ω)) , period from state 1 → 2 with increments g (t) = δ δ . ij 1,i 2,j (21) The average of (25) is denoted by X (Ω) ≡ hX i. 2 0 2 D (nT ; Ω)[C (Ω) + C (Ω)]/A(Ω) ≥ (ΩA (Ω)/A(Ω)) , A To derive the bound on the activity, we use the lower (22) bound (2) on the generating function λ (z) for the nT quantity X = X . We choose the ansatz T T where the additional term ps ˜ ˜ p ˜ (τ ) = p (Ωτ/Ω; Ω), ˜ ˜ ˜ k (τ ) = (Ω/Ω)k (τ )− (Ω/Ω− 1)k (τ )α (τ )δ, (26) 1 ij ij ij ij 2 ps 0 2 ps C (Ω) ≡ Ω [p (0; Ω) ] /p (0; Ω) (23) i i nT with ps is the contribution of the non-vanishing Kullback-Leibler u ˜ ˜ p (Ωτ/Ω; Ω)k (τ ) ji α (τ ) ≡ (27) divergence and ij ps ˜ ˜ p (Ωτ/Ω; Ω)k (τ ) ij D (nT ; Ω) ≡ nT h(X −hX i) i/2 = λ (0)/2 and δ will be chosen as 0 or 1. For δ = 0, the first term X nT nT nT (24) in (2) is given by is the finite-time generalization of the diffusion coefficient 00 ˜ X for an observable X after n periods. Here, λ (0) (Ω/Ω) ps nT aux nT ˜ ˜ ˆ hX i = dτ p (Ωτ/Ω; Ω)k (τ )d (τ ) T ij ij denotes the second derivative of the SCGF with respect ij to z at z = 0. (28) Using the substitution τ ˜ = Ωτ/Ω and the property of the increments ˜ ˜ g (τ ; Ω) = g (Ωτ ˜/Ω; Ω) = g (τ ˜; Ω), (29) ij ij ij III. BOUNDS ON ACTIVITY where the upper index denotes the corresponding fre- We now generalize the bounds on activity A(Ω) in quency, leads to the main text to arbitrary observables depending on the aux ˜ ˜ number of transitions hX i = (Ω/Ω)X (Ω). (30) Inserting (26) into eq. (3), choosing Ω = (1 + )Ω with X ≡ dt n ˙ (t)g (t) (25) small parameter = O (z) leads to T ij ij ij q q 2 T 2 ps ps F [p˜(τ ), k(τ )] = dτ p (τ ; Ω)k (τ )− δ p (τ ; Ω)k (τ ) +O . (31) ij ji i j T 2 ij Choosing δ = 0, expanding (30) for small and optimiz- given by ing with respect to this parameter leads to a quadratic 2 0 2 2D (nT ; Ω)[A(Ω) +C (Ω)]/X (Ω) ≥ (1−X (Ω)/X (Ω)) , bound on the generating function. Taking the long-time X (33) limit n → ∞, this quadratic bound implies where C (Ω) is defined in eq. (23). 2 0 2 2D (Ω)A(Ω)/X (Ω) ≥ (1−X (Ω)/X (Ω)) , (32) IV. STIRLING HEAT ENGINE which is a generalization of the bound on activity in [3] to periodically driven systems. We note, that for δ = 1, the main results for currents J and residence quantities T In this section, we calculate currents of interest and the A can be recovered. T diffusion coefficient for the output power for the Stirling Finally, the finite-time generalization of eq. (32) is engine discussed in the main text. The periodic station- 4 ary solution of the Fokker-Planck equation in the main as well as the hot heat flux flowing into the system text is a Gaussian with zero mean 1 μλ(τ ) 1 x 2 ps Q = − dτ β λ(τ )hx (τ )i− 1 . (42) h h p (x, τ ) = p exp − , (34) 2 T 2β 2hx (τ )i T /2 2πhx (τ )i where the variance can be calculated according to These currents of interest can be calculated by solv- ing (35) for the initial condition leading to the periodic 2 −1 2 ∂ hx (τ )i = 2μ β (τ )− λ(τ )hx (τ )i , (35) stationary state. The diffusion coefficient of the output power (40) is which follows from the Fokker-Planck equation [4]. Here, given by the variance has to be periodic in time, i.e. 2 2 Z Z hx (0)i = hx (T )i. T T 1 T 0 2 2 0 0 2 ˙ ˙ D = dτ dτ hx (τ )x (τ )iλ(τ )λ(τ )/4− P . out out 2T 2 The output power is defined by 0 0 (43) Z Z 2 2 0 T ∞ The the correlation function hx (τ )x (τ )i can be written ps P ≡ − dτ dxp (x, τ )V (x, τ ). (36) out as 0 −∞ Z Z ∞ ∞ The entropy production associated with heat dissipation 2 2 0 0 2 02 0 0 hx (τ )x (τ )i = dxdx x x p(x, x , τ, τ ), in the medium given by −∞ −∞ (44) Z Z T ∞ 0 0 0 0 ps where x ≡ x(τ ), x ≡ x (τ ) and p(x, x , t, t ) is the joint σ ≡ dτ dxβ(τ )j (x, τ )F (x, τ ), (37) 0 −∞ probability density. Here, x and x are Gaussian dis- tributed according to (34) . Due to the linearity of the where Langevin equation, every linear combination of x and x is also Gaussian distributed. Hence, the joint probabil- ps −1 ps j (x, τ ) ≡ −μ ∂ V (x, τ ) + β (τ )∂ p (x, τ ) (38) x x 0 0 ity density p(x, x , τ, τ ) is a bivariate normal distribution and the expectation value in (44) is consequently given is the probability current. If the system has reached the by periodic stationary state, the entropy production σ of the medium averaged over one cycle duration T is identi- 2 2 0 2 2 0 0 2 cal to the average total entropy production of the system. hx (τ )x (τ )i = hx (τ )ihx (τ )i + 2hx(τ )x(τ )i . (45) Hence, we use the notation σ ≡ σ . The heat current flowing into or out of the system from or into the heat By solving the Langevin equation, we can evaluate the bath is defined by correlation function (45) and obtain the following expres- Z Z sion for the diffusion coefficient ps Q ≡ dτ dxj (x, τ )F (x, τ ). (39) −∞ Z Z T τ 1 1 0 0 2(I(τ )−I(τ)) ˙ ˙ D = dτ dτ λ(τ )λ(τ )e (46) out Note that the integration over τ is performed during the T 2 0 0 time interval at which the heat bath is connected to the system. with Inserting (34) into (36), (37) and (39) yields the fol- lowing expressions for power 0 0 I (τ ) ≡ μ dt λ(t ). (47) 1 λ(τ ) 0 P = − dτ hx (τ )i, (40) out T 2 entropy production 1 μλ(τ ) σ = dτ β(τ )λ(τ )hx (τ )i− 1 , T 2 [1] A. Dechant and S. ichi Sasa, J. Stat. Mech. Theor. Exp. , 063209 (2018). and the cold heat flux flowing out of the system [2] T. Koyuk, U. Seifert, and P. Pietzonka, J. Phys. A Math. Theor. 52, 02LT02 (2019). T /2 1 μλ(τ ) [3] J. P. Garrahan, Phys. Rev. E 95, 032134 (2017). Q = dτ β λ(τ )hx (τ )i− 1 , (41) c c T 2β 0 [4] T. Schmiedl and U. Seifert, EPL 81, 20003 (2008).
Condensed Matter – arXiv (Cornell University)
Published: Apr 18, 2019
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