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

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Luc Rey-Bellet works in several directions in statistical mechanics, both in equilibrium and non-equilibrium. (a) The construction and the ergodic properties of stationary states for stochastic partial differential equations arising from a model of a nonlinear Klein-Gordon equation coupled to one or several heat reservoirs. (b) The validity of the fluctuation theorem for entropy production in some class of non-uniformly hyperbolic systems, such as billiards. This involves the development of large deviations techniques for such systems. (c) The validity of the fluctuation theorem for the entropy production in classical and quantum open systems. (d) The development of higher-order coarse-graining numerical schemes for Monte-Carlo methods. Many fundamental problems in non-equilibrium statistical mechanics remain poorly understood, both at the conceptual and mathematical level: for example the characterization of non-equilibrium stationary states of driven open systems. The fluctuation theorem of Gallavotti and Cohen is a new universal property of these states and its study in various systems (deterministic, random and quantum) will be one of the main theme in the work of the investigator. Multiscale numerical methods and the general question of extracting the relevant degrees of freedom out of complex systems is a problem of paramount importance in modern applied mathematics. The investigator proposes to use probabilistic techniques from statistical mechanics (cluster expansion and renormalization group) to develop efficient numerical schemes for the coarse-graining of Monte-Carlo methods. The proposal of the investigator, besides its theoretical aspects, has a number of applications to various concrete physical models. These applications are integrated (via numerical or analytical work) into graduate research projects. The project involves several collaborations with researchers in U.S. institutions and abroad. The project also helps to the dissemination of modern mathematical tools, in particular probabilistic ones, into applied sciences. The field of statistical mechanics is the physical and mathematical theory which attempts to link the microscopic and macroscopic worlds. The microscopic world, the world of atoms and molecules, is described by the laws of Newtonian or Quantum mechanics which involve a huge number of equations. The macroscopic world on the contrary is usually described by a few parameters or equations, such as pressure, temperature, electrical and thermal conductivity, etc... This reduction comes from the fact that a very large number of particles, seen from through macroscopic lenses, behave in a very regular fashion. For example, in a well isolated room, the temperature throughout the room will be nearly constant. Another example is a piece of metal heated at one end and cooled at the other end: there will 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 head will be extremely close to one half and (almost) never exhibit significant deviations. It turns out that to study this typical behavior it is very useful to study and characterize the rare events corresponding to untypical behavior, i.e., "large deviations". Having a detailed understanding of the atypical behavior of a very large number of particles is in fact the clue to a fundamental understanding of what is really typical. Theses ideas, which go back to the founding fathers of physics and probability theory, and their implementations in various physical situations form the core of the proposal.