Mathematical Problems in Nonequilibrium Statistical Mechanics

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0306540Rey-Bellet Luc Rey-Bellet works in three directions: 1. Study of nonequilibrium stationary states for systems in classical mechanics coupled to heat reservoirs, - in particular the study of stochastic partial differential equations arising from a model of semilinear wave equations coupled to one or several reservoirs described by linear wave equations at positive temperature. The goal is to prove existence and uniqueness of the stationary state, determine the rate of convergence and study the transport properties of the system. 2. Development of probabilistic methods to study the ergodic properties of open quantum systems coupled to reservoirs of bosons and fermions - in particular investigating and developing tools to study the ergodic properties of quantum Markov semigroups and quantum stochastic differential equations and combining them with spectral methods to prove ergodic properties for open quantum systems. 3. Develop the study of large deviations in quantum statistical mechanics - in particular prove the large deviation principle for the Gibbs-KMS states of quantum systems on a lattice (quantum spin systems, fermionic and bosonic lattice gases). Statistical mechanics is the physical and mathematical theory, which attempts to link the microscopic and macroscopic worlds. At the microscopic level, atoms or elementary particles are described by the time reversible laws of Newtonian or Quantum mechanics, which involve a huge number of equations. At the macroscopic level, materials are, in their normal state, described, by a few parameters or equations, such as pressure, temperature, electrical and thermal conductivity, etc. Some fundamental problems in nonequilibrium statistical mechanics remain poorly understood, both at the conceptual and mathematical level: for example the characterization of nonequilibrium stationary states of open systems and the derivation of phenomenological transport laws from microscopic mechanical models (for example, the Ohm's law of electric and Fourier's law of heat conduction). The proposal addresses some of these questions, which are related for example to material sciences. Also a new field, loosely speaking ``quantum probability'', has emerged in the last few years, driven by applications such as quantum cryptography and quantum computing. The proposal is also devoted to the study of several mathematical tools in this field.