2021/02/24 by Zied Ammari, Ammari, Zied, Vedran Sohinger +1 · 2 citations
Mathematics · Physics and Astronomy · #28C20 #35Q55 #37D35 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 35L05 #Probability (math.PR) #Probability and Statistical Research #Secondary 60H07 #Statistical Mechanics and Entropy
paper · doi:10.48550/arxiv.2102.12202
openalex publication_date 2021/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.