Microscopic Reversibility and Macroscopic Behavior: Physical Explanations and Mathematical Derivations

TL;DR

This paper explains how macroscopic irreversibility emerges from microscopic time-symmetric laws, emphasizing the importance of typicality in ensemble descriptions and deriving hydrodynamic equations for systems with short-range interactions.

Contribution

It clarifies the connection between microscopic reversibility and macroscopic irreversibility, highlighting the role of typicality and providing mathematical derivations of hydrodynamic equations.

Findings

Macroscopic irreversibility arises from microscopic time-symmetric laws.

Hydrodynamic equations accurately describe macro system evolution.

Ensemble-based approaches require typicality to be meaningful.

Abstract

The observed general time-asymmetric behavior of macroscopic systems -- embodied in the second law of thermodynamics -- arises naturally from time-symmetric microscopic laws due to the great disparity between macro and micro-scales. More specific features of macroscopic evolution depend on the nature of the microscopic dynamics. In particular, short range interactions with good mixing properties lead, for simple systems, to the quantitative description of such evolutions by means of autonomous hydrodynamic equations, e.g., the diffusion equation. These deterministic time-asymmetric equations accurately describe the observed behavior of {\it individual} macro systems. Derivations using ensembles (or probability distributions) must therefore, to be relevant, hold for almost all members of the ensemble, i.e., occur with probability close to one. Equating observed irreversible macroscopic behavior with the time evolution of ensembles describing systems having only a few degrees of freedom, where no such typicality holds, is misguided and misleading.