Infinite ergodic theory meets Boltzmann statistics

TL;DR

This paper explores how infinite ergodic theory can be applied to describe the statistical behavior of particles in unbounded potentials, extending Boltzmann-Gibbs statistics to infinite systems with diverging partition functions.

Contribution

It introduces a framework combining infinite ergodic theory with Boltzmann statistics, deriving a generalized virial theorem and addressing non-recurrent processes in infinite systems.

Findings

Boltzmann-Gibbs statistics applies to integrable observables in infinite systems.

Derived a heuristic and first-principles approach to the infinite density.

Established a generalized virial theorem linking particle spread delay to virial coefficient.

Abstract

We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.