Claude-Alain Pillet

Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures

Abstract:We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of Hörmander used in the study of hypoelliptic differential operators.

On a model for quantum friction. III. Ergodic properties of the spin-boson system

We investigate the dynamics of a 2-level atom (or spin 1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared. From a more philosophical point of view our results show that, contrary to the conventional wisdom, quantum dynamics can be simpler at positive than at zero temperature.

Entropy Production in Nonlinear, Thermally Driven Hamiltonian Systems

We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system

ON A MODEL FOR QUANTUM FRICTION. II: FERMI'S GOLDEN RULE AND DYNAMICS AT POSITIVE TEMPERATURE

We investigate the dynamics of anN-level system linearly coupled to a field of mass-less bosons at positive temperature. Using complex deformation techniques, we develop time-dependent perturbation theory and study spectral properties of the total Hamiltonian. We also calculate the lifetime of resonances to second order in the coupling.

Mathematical Theory of Non-Equilibrium Quantum Statistical Mechanics

We review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics recently proposed in refs. 1–7. In the algebraic formalism of quantum statistical mechanics we introduce notions of non-equilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.

Non-Equilibrium Steady States of the XY Chain

AbstractWe study the non-equilibrium statistical mechanics of the two-sided XY chain. We start from an initial state in which the left and right part of the lattice,

Perturbation theory of W*-dynamics, Liouvilleans and KMS states

\mathbb{Z}_{\text{L}} = \{ x \in \mathbb{Z}|x < - M\} ,{\text{ }}\mathbb{Z}_{\text{R}} = \{ x \in \mathbb{Z}|x >M\} ,

Spectral theory of thermal relaxation

are at inverse temperatures βL and βR. Using a simple scattering theoretic analysis, we construct the unique non-equilibrium steady state (NESS). This state depends on βL and βR, but not on the choice of the decoupling parameter M. We prove that in the non-equilibrium case, βL≠βR, this state has strictly positive entropy production.