Reversibility, Coarse Graining and the Chaoticity Principle

Abstract

We describe a way of interpreting the chaotic principle of [GC1] more extensively than it was meant in the original works. Mathematically the analysis is based on the dynamical notions of Axiom A and Axiom B and on the notion of Axiom C, that we introduce arguing that it is suggested by the results of an experiment ([BGG]) on chaotic motions. Physically we interpret a breakdown of the Anosov property of a time reversible attractor (replaced, as a control parameter changes, by an Axiom A property) as a spontaneous breakdown of the time reversal symmetry: the relation between time reversal and the symmetry that remains after the breakdown is analogous to the breakdown of T -invariance while T CP still holds.Keywords: Strange attractors, Chaoticity principle, Time reversal, TCP §1: Introduction.In reference [GC2] a general mechanical system in a non equilibrium situation was considered. Calling C the (compact or "finite") phase space of the "observed events", µ 0 the volume measure on it and S the map describing the time evolution (regarded as a discrete invertible mapping of "observed events" into the "next ones", i.e. as a Poincaré's map on some surface in the full phase space) a principle holding when motions have an empirically chaotic nature was introduced:Chaotic hypothesis: A chaotic many particle system in a stationary state can be regarded, for the purpose of computing macroscopic properties, as a smooth dynamical system with a transitive Axiom A globally attracting set. In reversible systems it can be regarded, for the same purposes, as a smooth transitive Anosov system.•Chaotic is an empirical qualitative notion that means that most points of the attracting set have a stable and an unstable manifold with positive dimension. In the applications in [GC1], [GC2], [G1] the use of the hypothesis, which is a natural extension of a principle proposed by Ruelle, was based on reversibility * mp arc@math.utexas.edu, #96-50, and chao-dyn@xyz.lanl.gov,#9512??.

Fluctuation Relation beyond Linear Response Theory

“…The fluctuation theorem predicts X = 1, but other values of X are possible under different hypothesis, see [7,10,29,30]. We define a function χ 2 (a, X) as…”

“…It is expected, [29], that in such a case ζ ∞ (p)− ζ ∞ (−p) is still linear, but the slope is Xσ + , with X given by the ratio of the dimension of the attractive set and of that of the whole phase space. An estimate of such quantity can be given via the number of negative pairs of exponents in the Lyapunov spectrum [29,30]. Unfortunately negative pairs begin to appear in the Lyapunov spectrum only for values of the external force so high that no negative fluctuations are observable anymore.…”

Simulating Rare Events in Dynamical Processes

“…The only way in which the appearance of a Fluctuation Relation for the hydrodynamic modes may be justified, is to invoke the existence of a large effective temperature, related to the macroscopic fluctuations. Bonetto and Gallavotti [52] have conjectured that this could be justified by considering the restricted space in which the macroscopic takes place. These questions are very much open, and in order to make progress it would be useful to simulate the limits beyond which the fluctuation theorem ceases to hold rigorously, because that is where new concepts may arise.…”

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

“…The reason why this property is important will be discussed below. This property can be checked, for instance, by looking at the Lyapunov spectrum, see [44,45] and references therein.…”

“…In particular, while Σ(x) is related to the entropy production rate, it is not obvious to relate σ A (x) to an experimentally accessible quantity. A possible strategy to access σ A (x), that works under some additional hypothesis, has been proposed in [45,44]. Unfortunately for the moment a numerical verification of this proposal is not possible due to prohibitive computational costs [49], so we will not discuss this matter in more detail here.…”