On the asymptotic behavior of solutions to a class of grand canonical master equations

TL;DR

This paper analyzes the long-term behavior of solutions to a class of master equations modeling quantum systems approaching thermodynamic equilibrium, establishing spectral properties and convergence to equilibrium distributions.

Contribution

It proves the spectral resolution of the generator and demonstrates global solutions converge to grand canonical or Gibbs equilibrium distributions.

Findings

Solutions stabilize to equilibrium distributions over time.

Some solutions converge exponentially fast.

The spectral analysis applies to trace-class self-adjoint operators.

Abstract

In this article we investigate the long time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential. We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitable weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature.