Chaotic motion in classical fluids with scale relativistic
methods
Marie-Noëlle Célérier
a͒
LUTH, Observatoire de Paris, CNRS, Université Paris Diderot, 5 Place Jules Janssen,
92195 Meudon Cedex, France
͑Received 10 July 2008; accepted 10 November 2009; published online 18 December 2009
͒
In the framework of the scale relativity theory, the chaotic behavior in time only of
a number of macroscopic systems corresponds to the motion in a space with geo-
desics of fractal dimension 2 and leads to its representation by a Schrödinger-type
equation acting in the macroscopic domain. The fluid interpretation of such a
Schrödinger equation yields Euler and Navier–Stokes equations. We therefore
choose to extend this formalism to study the properties of a system exhibiting a
chaotic behavior both in space and time, which amounts to consider them as issued
from the geodesic features of a mathematical object exhibiting all the properties of
a fractal “space-time.” Starting with the simplest Klein–Gordon-type form that can
be given to the geodesic equation in this case, we obtain a motion equation for a
“three fluid” velocity field and three continuity equations, together with parametric
expressions for the three velocity components which allow us to derive relations
between their nonvanishing curls. At the nonrelativistic limit and owing to the
physical properties exhibited by this solution, we suggest that it could represent
some kind of three-dimensional chaotic behavior in a classical fluid, tentatively
turbulent if particular conditions are fulfilled. The appearance of a transition pa-
rameter D in the equations allows us to consider different ways of testing experi-
mentally our proposal. © 2009 American Institute of Physics.
͓doi:10.1063/1.3271040͔
I. INTRODUCTION
The Madelung transformation of the Schrödinger equation has been long known as providing
a tentative fluidlike interpretation of nonrelativistic quantum mechanics.
1,2
When writing the com-
plex wave function
in polar form,
=
ͱ
Pe
i
, then separating the real and imaginary parts of the
Schrödinger equation, one obtains a Hamilton–Jacobi and a continuity equations where P and
V=ٌ
can be interpreted as the density and the velocity field of an irrotational flow in a com-
pressible fluid, respectively. The gradient of the Hamilton–Jacobi equation is an Euler-type equa-
tion exhibiting a new term, the so-called “quantum” potential, which depends only on the density
and has been, in the hydrodynamical picture, regarded as a pressure potential or a mechanical
stress tensor.
3,4
In scale relativity, the effect on motion of the simplest scale laws, namely, self-similar laws
with fractal dimension 2, allows one to recover the various equations of quantum mechanics.
5–8
However, since the mathematical construction of the quantum equations thus obtained does not
depend on the value of the Planck constant ប ͑see Sec. III͒, its results can be generalized to the
classical realm provided some particular conditions are fulfilled by the system under
consideration.
5,7
This has already been done for the Schrödinger equation with applications to the
chaotic behavior in time of a number of macroscopic systems ͑planetary systems,
5,9,10
astrophysi-
cal structures,
11,12
etc.͒ which have been successfully corroborated by observations.
a͒
Electronic mail: marie-noelle.celerier@obspm.fr.
JOURNAL OF MATHEMATICAL PHYSICS 50, 123101 ͑2009͒
50, 123101-10022-2488/2009/50͑12͒/123101/15/$25.00 © 2009 American Institute of Physics