Lyapunov instability of fluids

Abstract

The key assumptions of classical statistical mechanics, ergodicity and mixing, are facilitated by the instability of a phase-space trajectory with respect to small perturbations of the initial conditions. Such perturbations typically grow or shrink exponentially with time, which is described by a set of rate constants, the Lyapunov exponents. The set of all exponents is referred to as the Lyapunov spectrum. Here, we summarize current ideas and methods for the computation and interpretation of the Lyapunov spectra of simple fluids. Systems in equilibrium and in stationary nonequilibrium states are considered. Emphasis is given to hard-particle systems, for which the equilibrium phase-space perturbations associated with small Lyapunov exponents display periodic patterns in space reminiscent of the modes of fluctuating hydrodynamics, the so-called Lyapunov modes. Stationary nonequilibrium states are generated by forcing the system away from equilibrium and, simultaneously, by removing the irreversibly-produced excessive heat with a thermostat. Dynamical and stochastic thermostats are considered. The phase-space probability function of dynamically thermostated systems is a multifractal distribution with an information dimension smaller than the phase-space dimension. This is related to the irreversible transport in such systems, and to the Second Law of Thermodynamics. A possible extension of these ideas to stochastically-thermostated nonequilibrium flows is also attempted. © 2005 Springer-Verlag Berlin Heidelberg.