H-Theorems from Autonomous Equations

TL;DR

This paper rigorously analyzes the connection between the H-theorem and macroscopic autonomy in Hamiltonian dynamics, clarifying conditions under which entropy is non-decreasing without assuming Markovian behavior.

Contribution

It provides a rigorous analysis linking the H-theorem to macroscopic autonomy, clarifying the role of Liouville's theorem and the Markov property in entropy increase.

Findings

Entropy is non-decreasing under autonomous macroscopic evolution.

Liouville's theorem applies in the finite particle case with careful interpretation.

Autonomy does not necessarily imply Markovian dynamics at the macroscopic level.

Abstract

The H-theorem is an extension of the Second Law to a time-sequence of states that need not be equilibrium ones. In this paper we review and we rigorously establish the connection with macroscopic autonomy. If for a Hamiltonian dynamics for many particles, at all times the present macrostate determines the future macrostate, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a Markov property for the macroscopic evolution.