Vojkan Jakšić

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.

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.

Transport properties of quasi-free fermions

Using the scattering approach to the construction of Non-Equilibrium Steady States proposed by Ruelle we study the transport properties of systems of independent electrons. We show that Landauer-Buttiker and Green-Kubo formulas hold under very general conditions.

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

Given a W*-algebra

Ergodic properties of classical dissipative systems I

{\mathfrak M}

Spectral theory of thermal relaxation

with a W*-dynamics τ, we prove the existence of the perturbed W*-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If τ has a β-KMS state, and the perturbation satisfies some mild assumptions related to the Golden–Thompson inequality, we prove the existence of a β-KMS state for the perturbed W*-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.

Topics in non-equilibrium quantum statistical mechanics

We consider a class of models in which a Hamiltonian system A, with a finite number of degrees of freedom, is brought into contact with an infinite heat reservoir B. We develop the formalism required to describe these models near thermal equilibrium. Using a combination of abstract spectral techniques and harmonic analysis we investigate the singular spectrum of the Liouvillean L of the coupled system A+B. We provide a natural set of conditions which ensure that the spectrum of L is purely absolutely continuous except for a simple eigenvalue at zero. It then follows from the spectral theory of dynamical systems (Koopmanism) that the system A+B is strongly mixing. From a probabilistic point of view, we study a new class of random processes on finite dimensional manifolds: non-Markovian Ornstein-Uhlenbeck processes. The paths of such a process are solutions of a random integro-differential equation with Gaussian noise which is a natural generalization of the well known Langevin equation. In this context we establish that, under appropriate conditions, the OU process is strongly mixing even when the Langevin equation has memory and is driven by non-white noise.

QUANTUM HYPOTHESIS TESTING AND NON-EQUILIBRIUM STATISTICAL MECHANICS

We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system I, with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperature T. The zeroth law of thermodynamics postulates that, independently of its initial configuration, the system I approaches a unique stationary state as t→∞. By definition, this limiting state is the equilibrium state of I at temperature T. Statistical mechanics further identifies this state with the Gibbs canonical ensemble associated with I. For simple models we prove that the above picture is correct, provided the equilibrium state of the environment R is itself given by its canonical ensemble. In the quantum case we also obtain an exact formula for the thermal relaxation time.

CORRUGATED SURFACES AND A.C. SPECTRUM

1 Zentrum Mathematik M5, Technische Universitat Munchen, D-85747 Garching, Germany e-mail: hbar@ma.tum.de 2 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada e-mail: jaksic@math.mcgill.co 3 Laboratoire de Mathematiques, Universite Paris-Sud, 91405 Orsay cedex, France e-mail: yan.pautrat@math.u-psud.fr 4 CPT-CNRS, UMR 6207, Universite du Sud, Toulon-Var, B.P. 20132, 83957 La Garde Cedex, France e-mail: pillet@univ-tln.fr