Exact Results in Nonequilibrium Statistical Mechanics: Where Do We Stand?

Abstract

Statistical mechanics attempts to account for the observed behavior of macroscopic physical systems on the basis of the microscopic laws which govern the behavior of their “elementary” constituents (particles or molecules). A central fact of the entire endeavor is that the number of particles is very large, so that, of necessity, the concepts of probability theory play an important role. (For large systems a macroscopic statistical description is not only consistent with deterministic ‘laws’ it is indeed the only one feasible.) The success of the problem, at least from the mathematical or rigorous point of view, has so far been largely limited to the realm of equilibrium statistical mechanics, where it explains in particular how even complex systems are susceptible to a complete macroscopic description involving only a small number of parameters (temperature, pressure, energy, entropy, volume, ⋯). The situation with respect to nonequilibrium statistical mechanics is far more tentative. Here one is concerned with why and how systems come to equilibrium: i.e., we would like to account for the experimental fact that starting from a nonequilibrium state the system will evolve, under the action of its time evolution, to the appropriate equilibrium state, and furthermore that this approach to equilibrium satisfies the relevant kinetic (e.g. Boltzmann), hydrodynamic, and transport equations. We shall review briefly the current status of some of these problems and discuss a little more fully some steady state nonequilibrium phenomena, i.e., attempts to rigorously prove the validity of Fourier's law.