A spectral theorem for the semigroup generated by a class of infinitely many master equations

TL;DR

This paper analyzes the spectral properties of the generator of an infinite system of master equations modeling approach to equilibrium, revealing a spectral theorem despite the generator's non-normality and lack of a spectral gap.

Contribution

It establishes a spectral theorem for the generator of an infinite master equation system with detailed balance, addressing challenges posed by non-normality and compactness.

Findings

Spectral theorem for the generator of the infinite master equations

Exponential convergence to Gibbs-type equilibrium

Handling of non-normal, non-dissipative compact operators

Abstract

In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under investigation thus consists of infinitely many first-order differential equations governing the time evolution of probabilities susceptible of describing jumps between the eigenstates of a differential operator with a discrete point spectrum. The transition rates between eigenstates are chosen in such a way that the so-called detailed balance conditions are satisfied, so that for a large class of initial conditions the given system possesses a global solution which converges exponentially rapidly toward a time independent probability of Gibbs type. A particular feature and a challenge of the problem under investigation is that in the natural functional space where the initial-value problem is well-posed, the infinitesimal generator is realized as a non normal and non dissipative compact operator, whose spectrum therefore does not exhibit a spectral gap around the zero eigenvalue in contrast to the finite-dimensional case.