Reconceptualising equilibrium in Boltzmannian statistical mechanics and characterising its existence

TL;DR

This paper introduces a new, general conception of equilibrium in Boltzmannian statistical mechanics, proving theorems that characterize the largest macro-region as equilibrium and establishing conditions for the existence of equilibrium states, independent of system specifics.

Contribution

It provides a mathematically rigorous, system-agnostic framework for defining and characterizing equilibrium and its existence in statistical mechanics.

Findings

The largest macro-region is the equilibrium region.

Necessary and sufficient conditions for equilibrium existence are established.

Approach to equilibrium is reframed as conditions for equilibrium state existence.

Abstract

In Boltzmannian statistical mechanics macro-states supervene on micro-states. This leads to a partitioning of the state space of a system into regions of macroscopically indistinguishable micro-states. The largest of these regions is singled out as the equilibrium region of the system. What justifies this association? We review currently available answers to this question and find them wanting both for conceptual and for technical reasons. We propose a new conception of equilibrium and prove a mathematical theorem which establishes in full generality -- i.e. without making any assumptions about the system's dynamics or the nature of the interactions between its components -- that the equilibrium macro-region is the largest macro-region. We then turn to the question of the approach to equilibrium, of which there exists no satisfactory general answer so far. In our account, this question is replaced by the question when an equilibrium state exists. We prove another -- again fully general -- theorem providing necessary and sufficient conditions for the existence of an equilibrium state. This theorem changes the way in which the question of the approach to equilibrium should be discussed: rather than launching a search for a crucial factor (such as ergodicity or typicality), the focus should be on finding triplets of macro-variables, dynamical conditions, and effective state spaces that satisfy the conditions of the theorem.