Initial growth of Boltzmann entropy and chaos in a large assembly of weakly interacting systems

Abstract

We introduce a high dimensional symplectic map, modeling a large system consisting of weakly interacting chaotic subsystems, as a toy model to analyze the interplay between single-particle chaotic dynamics and particles interactions in thermodynamic systems. We study the growth with time of the Boltzmann entropy, S B , in this system as a function of the coarse graining resolution. We show that a characteristic scale emerges, and that the behavior of S B vs t, at variance with the Gibbs entropy, does not depend on the coarse graining resolution, as far as it is finer than this scale. The interaction among particles is crucial to achieve this result, while the rate of entropy growth depends essentially on the single-particle chaotic dynamics (for t not too small). It is possible to interpret the basic features of the dynamics in terms of a suitable Markov approximation.

On the entropy of relaxing deterministic systems

“…Our observations concerning the fine-grained Gibbs entropy are completely consistent with the recent results of Falcioni et al 19 In this paper, Falcioni et al make a number of observations: r The single-particle Boltzmann entropy (in μ-space), of course obeys the Boltzmann H-theorem. This entropy can be calculated from the (possibly fractal) N-particle phase space distribution by projecting out all the degrees of freedom except those of a single particle.…”

“…To quote Ref. 19: "the onset of entropy variation in the (coarse-grained) Gibbs case has no intrinsic meaning. "…”

Applicability of optimal protocols and the Jarzynski equality

“…Ref. [30,31,32]. However, the implications of this asymmetry of the distributions on the Jarzynski equality are discussed and are consistent with our § Note that W is the work done by the system in [23], whereas it is work done on the system in our case, so the signs are reversed observations.…”

On the relationship between dissipation and the rate of spontaneous entropy production from linear irreversible thermodynamics

“…However, unknown to Gibbs, this did not provide a solution because the coarsegrained Gibbs entropy so obtained is not an objective material property [3,13] and its time dependence in nonequilibrium systems is determined by the grain size [13].…”