Rethinking Boltzmannian Equilibrium

TL;DR

This paper challenges traditional justifications for identifying equilibrium in Boltzmannian statistical mechanics and offers a new, general proof that equilibrium corresponds to the largest macro-region based on time spent.

Contribution

It introduces a novel, assumption-free theorem linking equilibrium to the largest macro-region, redefining the justification for equilibrium in statistical mechanics.

Findings

Equilibrium is characterized as the macrostate where a system spends most of its time.

A new theorem establishes the equivalence of time spent and macro-region size.

The derivation is fully general, independent of system dynamics or interactions.

Abstract

Boltzmannian statistical mechanics partitions the phase space of a system into macro-regions, and the largest of these is identified with equilibrium. What justifies this identification? Common answers focus on Boltzmann's combinatorial argument, the Maxwell-Boltzmann distribution, and maximum entropy considerations. We argue that they fail and present a new answer. We characterise equilibrium as the macrostate in which a system spends most of its time and prove a new theorem establishing that equilibrium thus defined corresponds to the largest macro-region. Our derivation is completely general, and does not rely on assumptions about the dynamics or internal interactions.