Nonequilibrium Time Reversibility with Maps and Walks

TL;DR

This paper explores the paradoxes of time-reversible nonequilibrium systems through simple models like the Baker Map and random walks, revealing insights into irreversibility, periodicity, and fractal properties.

Contribution

It introduces and analyzes simple, reversible models that exhibit dissipative and periodic behavior, shedding light on fundamental paradoxes in nonequilibrium thermodynamics.

Findings

Baker Map exhibits fractal properties and periodicity despite being dissipative.

Random walk models reveal ambiguities in fractal dimension calculations.

The study connects classical paradoxes with simple dynamical systems.

Abstract

Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt's and Zerm\'elo's paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the {\it irreversible} Second Law of Thermodynamics (Loschmidt's) as well as {\it periodic} in the time (Zerm\'elo's, illustrating Poincar\'e recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals' information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here. We review our investigations presented in Budapest in 1997 and end with presentday questions posed as the Snook Prize Problems in 2020 and 2021.