Asymptotic Analysis for Markovian models in non-equilibrium Statistical Mechanics

Abstract

This thesis is mainly concerned with the problem of exponential convergence to equilibrium for open classical systems. We consider a model of a small Hamiltonian system coupled to a heat reservoir, which is described by the Generalized Langevin Equation (GLE) and we focus on a class of Markovian approximations to the GLE. The generator of these Markovian dynamics is an hypoelliptic non-selfadjoint operator. We look at the problem of exponential convergence to equilibrium by using and comparing three different approaches: classic ergodic theory, hypocoercivity theory and semiclassical analysis (singular space theory). In particular, we describe a technique to easily determine the spectrum of quadratic hypoelliptic operators (which are in general non-selfadjoint) and hence obtain the exact rate of convergence to equilibrium.