Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs

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. Ver-boven 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.

Automated summary

The KMS condition is a fundamental property of statistical mechanics that characterizes the equilibrium of infinite classical mechanical systems. It was proposed as an alternative to the DLR equation by Gallavotti and Ver-boven in the 1970s. In this article, we discuss the relevance of this concept in the framework of nonlinear Hamiltonian PDEs and prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof relies on Malliavin calculus and Gross-Sobolev spaces. The significance of our work lies in the generality of our results, which apply to various contexts such as white noise, abstract Wiener spaces, and Gaussian probability spaces, as well as fundamental examples of PDEs.