Méthodes hamiltoniennes et markoviennes en mécanique quantique hors équilibre – HAM-MARK
We wish to develop mathematical tools in order to understand the thermodynamical properties of quantum systems out of equilibrium. Indeed, despite the fact that the mathematical structure of equilibrium thermodynamics is well-understood, the situation is far from satisfactory out of the equilibrium regime. In contrast with equilibrium states which can be characterized in several ways (KMS condition, Gibbs property, DLR equation, variational principles, stability, dissipativity, etc.) there exists no satisfactory way of characterizing states out of equilibrium. There are even more fondamental problems such as the validity of the Fourier law which is certainly one of the most important challenges of today's mathematical-physics. The strategy we shall adopt is the one which emerged in the recent developments of the subject. It is based on two different approaches. ' The interesting states out of equilibrium are those which are naturally selected by the dynamics of the system. The construction of such a state is thus based on the asymptotic study of the evolution of a class of initial states. We speak of Hamiltonian approach when the study of this asymptotic behaviour is made on the complete dynamics of the coupled system S+R. The mathematical framework of this approach makes use of extremely elaborated tools. Indeed, the description of the reservoirs (with infinitely many degrees of freedom) makes use of the algebraic description of quantum mechanics (C*-algebras and von Neumann algebras). In particular, the Tomita-Takesaki modular theory plays a crucial role in the definition of a pertinent generator of the quantum dynamical system: the standard Liouvillian (Jaksic-Pillet 1996) close to equilibrium, the C-liouvillian (Jaksic-Pillet 2002) out of equilibrium. The first rigourous proofs of return to equilibrium property (Jaksic-Pillet, Bach-Fröhlich-Sigal, Derezinski-Jaksic, Fröhlich-Merkli) are all based on a fine spectral analysis of the standard Liouvillian. ' In some cases, by selecting in some appropriate way the time and the space scales, it is possible to obtain a reduced description of the Hamiltonian dynamics. The Markovian character of the effective dynamics of the small system, obtained this way, presents a great simplification and allows to treat models for which the Hamiltonian approach would be technically impossible to study. We speak of a Markovian approach when the non-equilibrium states are constructed from this effective dynamics. More generally, one can obtain a Markovian dynamics on S when giving up the idea to describe the environment R (either it is considered as being too complicated, or inaccessible or simply unknown). One then replaces the effects of the interactions of the small system S with its environment by quantum noises. The evolution equation of the system become a quantum stochastic differential equation whose solution, a quantum Markov process, induces a semigroup of completely positive maps on the observables (or the states) of the small system. Its generator, the Lindbladian, presents, in the same way as the generator of a usual Markov process: a first order differential part (which carries the natural dynamics of the small system) and a second order differential part (which carries the dissipation of the small system in favour of the environment). The tools for studying the convergence of such systems are then of the type of those used for usual Markov processes (recurrence-transience, invariant measures, potential theory, spectral gap, Sobolev inequalities, ...) The relations between these two approaches are for the moment rather poor, we feel that we need to develop them both. But the link between them is clearly natural and surely source of important results. For example, the spectral analysis of the C-Liouvillian requires a study of the spectrum of the Lindbladian obtained in the weak-coupling limit. Conversely, the Markovian dynamics can be associated to a Hamiltonian dynamics by the construction of a unitary dilation of the Lindblad semigroup (Attal-Pautrat, Derezinski-De Roeck). In some cases the two approaches match: a model of repeated quantum interactions (Attal-Pautrat) allows to exhibit a dynamics which is at the same time Hamiltonian (time-dependent) and Markovian. In a certain limit (continuous interaction limit) it converges to a quantum stochastic differential equation (a quantum Langevin equation) (Attal-Pautrat 2003). This model has allowed to validate a definitive form for the Langevin equation associated to the action of a quantum heat bath (Attal-Joye 2006). These repeated quantum interactions models, are physically relevant as they correspond to effective experiments in quantum optics.
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