Nonequilibrium in statistical and fluid mechanics. Ensembles and their equivalence. Entropy driven intermittency

Abstract

We present a review of the chaotic hypothesis and discuss its applications to intermittency in statistical mechanics and fluid mechanics proposing a quantitative definition. Entropy creation rate is interpreted in terms of certain intermittency phenomena. An attempt to a theory of the experiment of Ciliberto-Laroche on the fluctuation law is presented. §1. Introduction.A general theory of non equilibrium stationary phenomena extending classical thermodynamics to stationary non equilibria is, perhaps surprisingly, still a major open problem more than a century past the work of Boltzmann (and Maxwell, Gibbs,...) which made the breakthrough towards an understanding of properties of matter based on microscopic Newton's equations and the atomic model. In the last thirty years, or so, some progress appears to have been achieved since the recognition that non equilibrium statistical mechanics and stationary turbulence in fluids are closely related problems and, in a sense, in spite of the apparently very different nature of the equations describing them they are essentially the same. The unifying principle, originally proposed for turbulent motions by Ruelle, [Ru78], in the early 970's, has been extended to statistical mechanics and eventually called the "chaotic hypothesis", [GC95]:Chaotic hypothesis: Asymptotic motions of a chaotic system, be it a multi particle system of microscopic particles or a turbulent macroscopic fluid, can be regarded as a transitive Anosov system for the purposes of computing time averages in stationary states.

Is it possible to experimentally verify the fluctuation relation? A review of theoretical motivations and numerical evidence

“…In the case of the two ensembles defined by (55), (58), the equivalence is supported by the concentration argument discussed above, namely that for N, V → ∞ the fluctuations of α(p, q) in the isoenergetic ensemble vanish, see [71,77] for more detailed discussions.…”

“…[2,3,5,8]. However the interpretation of these experiments is not clear for the moment (see [77] for a tentative interpretation of [2]), especially because the relation between the measured quantities and the local entropy production is not straightforward. The presence of large tails in the measured distributions suggests also that unbounded terms are affecting the measurements; it would be very interesting to try to apply the analysis described at the end of section 3 to these data.…”

“…We will see that the saddle point values of v i and v f are small. Then, if p Q > 1, the second equation (77) has no solution 30 , because ζ ′ W (p Q ) < 0; this means that v f will stick to the boundary of the integration region, v f = 0. Conversely, for p Q < 1, we have v i = 0.…”

“…For this reason no exact results are available, but only conjectures [76,22,23] and arguments supporting them [71,77], see in particular [77] for a detailed discussion.…”

Fluctuation theorem for a deterministic one-particle system

“…These time series are solutions of the undriven Duffing equation. It is shown that N = 1 is sufficient to render the fluctuation theorem [Gallavotti and Cohen, Phys. Rev.…”