The Ergodic Hypothesis: A Typicality Statement

TL;DR

This paper examines Boltzmann's ergodic hypothesis, arguing it is a consequence of stationarity and typicality rather than an assumption, thereby grounding Boltzmann's notions of equilibrium and fluctuation estimates.

Contribution

It demonstrates that the ergodic hypothesis naturally follows from stationarity and typicality, clarifying Boltzmann's foundational approach in statistical mechanics.

Findings

Ergodic hypothesis is a consequence, not an assumption, of stationarity and typicality.

Systems with stationary measures behave as if they are ergodic.

Boltzmann's notion of equilibrium is grounded in these concepts.

Abstract

This paper analyzes the ergodic hypothesis in the context of Boltzmann's late work in statistical mechanics, where Boltzmann lays the foundations for what is today known as the typicality account. I argue that, based on the concepts of stationarity (of the measure) and typicality (of the equilibrium state), the ergodic hypothesis, as an idealization, is a consequence rather than an assumption of Boltzmann's approach. More precisely, it can be shown that every system with a stationary measure and an equilibrium state (be it a state of overwhelming phase space or time average) behaves essentially as if it were ergodic. I claim that Boltzmann was aware of this fact as it grounds both his notion of equilibrium, relating it to the thermodynamic notion of equilibrium, and his estimate of the fluctuation rates.