When Does a Boltzmannian Equilibrium Exist?

TL;DR

This paper investigates the conditions under which a Boltzmannian equilibrium exists in isolated systems, providing a general existence theorem and analyzing various gas models to understand equilibrium behavior.

Contribution

It introduces a new existence theorem for Boltzmannian equilibrium states and applies it to different gas systems, advancing understanding of equilibrium conditions.

Findings

The existence theorem offers general criteria for equilibrium.

Application to various gases illustrates the theorem's practical relevance.

Discussion on ergodic assumptions clarifies their role in equilibrium analysis.

Abstract

The received wisdom in statistical mechanics is that isolated systems, when left to themselves, approach equilibrium. But under what circumstances does an equilibrium state exist and an approach to equilibrium take place? In this paper we address these questions from the vantage point of the long-run fraction of time definition of Boltzmannian equilibrium that we developed in two recent papers (Werndl and Frigg 2015a, 2015b). After a short summary of Boltzmannian statistical mechanics and our definition of equilibrium, we state an existence theorem which provides general criteria for the existence of an equilibrium state. We first illustrate how the theorem works with a toy example, which allows us to illustrate the various elements of the theorem in a simple setting. After commenting on the ergodic programme, we discuss equilibria in a number of different gas systems: the ideal gas, the dilute gas, the Kac gas, the stadium gas, the mushroom gas and the multi-mushroom gas. In the conclusion we briefly summarise the main points and highlight open questions.