AMC-SS: Mathematical and Computational in Nonequilibrium Statistical Mechanics.

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Luc Rey-Bellet works in several directions in statistical mechanics,both in equilibrium and non-equilibrium. (a) The construction and theergodic properties of stationary states for stochastic partialdifferential equations arising from a model of a nonlinearKlein-Gordon equation coupled to one or several heat reservoirs. (b)The validity of the fluctuation theorem for entropy production in someclass of non-uniformly hyperbolic systems, such as billiards. Thisinvolves the development of large deviations techniques for suchsystems. (c) The validity of the fluctuation theorem for the entropyproduction in classical and quantum open systems. (d) The developmentof higher-order coarse-graining numerical schemes for Monte-Carlomethods. Many fundamental problems in non-equilibrium statisticalmechanics remain poorly understood, both at the conceptual andmathematical level: for example the characterization of non-equilibriumstationary states of driven open systems. The fluctuation theorem ofGallavotti and Cohen is a new universal property of these states andits study in various systems (deterministic, random and quantum) willbe one of the main theme in the work of the investigator. Multiscalenumerical methods and the general question of extracting the relevantdegrees of freedom out of complex systems is a problem of paramountimportance in modern applied mathematics. The investigator proposesto use probabilistic techniques from statistical mechanics (clusterexpansion and renormalization group) to develop efficient numericalschemes for the coarse-graining of Monte-Carlo methods. The proposalof the investigator, besides its theoretical aspects, has a number ofapplications to various concrete physical models. These applicationsare integrated (via numerical or analytical work) into graduateresearch projects. The project involves several collaborations withresearchers in U.S. institutions and abroad. The project also helps tothe dissemination of modern mathematical tools, in particularprobabilistic ones, into applied sciences.The field of statistical mechanics is the physical and mathematicaltheory which attempts to link the microscopic and macroscopic worlds.The microscopic world, the world of atoms and molecules, is describedby the laws of Newtonian or Quantum mechanics which involve a hugenumber of equations. The macroscopic world on the contrary is usuallydescribed by a few parameters or equations, such as pressure,temperature, electrical and thermal conductivity, etc... Thisreduction comes from the fact that a very large number of particles,seen from through macroscopic lenses, behave in a very regularfashion. For example, in a well isolated room, the temperaturethroughout the room will be nearly constant. Another example is apiece of metal heated at one end and cooled at the other end: therewill be a flow of energy from the hot part to the cold part but(almost) never in the opposite direction. This phenomena are similar,both in spirit and in mathematical terms to the following: if one throws an unbiased coin very many times then the proportion of headwill be extremely close to one half and (almost) never exhibitsignificant deviations. It turns out that to study this typicalbehavior it is very useful to study and characterize the rare eventscorresponding to untypical behavior, i.e., "large deviations". Havinga detailed understanding of the atypical behavior of a very largenumber of particles is in fact the clue to a fundamental understandingof what is really typical. Theses ideas, which go back to the foundingfathers of physics and probability theory, and their implementationsin various physical situations form the core of the proposal.