Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs

TL;DR

This paper demonstrates that Gibbs measures serve as the unique KMS equilibrium states for nonlinear Hamiltonian PDEs, extending the concept of equilibrium in statistical mechanics to complex infinite-dimensional systems.

Contribution

It establishes the uniqueness of Gibbs measures as KMS states for nonlinear Hamiltonian PDEs using Malliavin calculus and Gross-Sobolev spaces, applicable to various PDEs and Gaussian frameworks.

Findings

Gibbs measures are the unique KMS equilibrium states for nonlinear Hamiltonian PDEs.

Results apply to white noise, Wiener spaces, and Gaussian probability spaces.

Includes fundamental PDE examples like nonlinear Schrödinger, Hartree, and Klein-Gordon equations.

Abstract

The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The main feature of our work is the applicability of our results to the general context of white noise, abstract Wiener spaces and Gaussian probability spaces, as well as to fundamental examples of PDEs like the nonlinear Schrodinger, Hartree, and wave (Klein-Gordon) equations.