Rethinking Boltzmannian Equilibrium

Abstract

Boltzmannian statistical mechanics partitions the phase space of a system into macroregions, 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 characterize 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 macroregion. Our derivation is completely general and does not rely on assumptions about the dynamics or interparticle interactions.