Quantitative Boltzmann Gibbs principles via orthogonal polynomial duality

TL;DR

This paper develops a systematic orthogonal decomposition of fluctuation fields in particle systems with duality, leading to a quantitative generalization of the Boltzmann Gibbs principle applicable to various interacting systems.

Contribution

It introduces a new orthogonal polynomial duality framework that quantifies fluctuation fields and extends the Boltzmann Gibbs principle to non-stationary and interacting particle systems.

Findings

Orthogonal decomposition of fluctuation fields achieved

Quantitative Boltzmann Gibbs principle established

Results applicable to symmetric exclusion and other dual systems

Abstract

We study fluctuation fields of orthogonal polynomials in the context of particle systems with duality. We thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann Gibbs principle. In the context of independent random walkers, we complete this program, including also fluctuation fields in non-stationary context (local equilibrium). For other interacting particle systems with duality such as the symmetric exclusion process, similar results can be obtained, under precise conditions on the $n$ particle dynamics