Stationary States of Hamiltonian Systems with Noise

Abstract

The ergodic hypothesis is known as one of the most challenging problems of mathematical physics since the fundamental work of Boltzmann and Gibbs. There are several ways to formulate the requirements of ergodicity, roughly speaking we have a mechanical system, like anharmonic vibrations of a lattice L with canonical coordinates ω = (p k ,q k ) k∈L and energy H, thus its time evolution is given by Newton’s equations of motion $$ {{\dot{p}}_{k}} = - \frac{{\partial H}}{{\partial {{q}_{k}}}},{\text{ }}{{\dot{q}}_{k}} = \frac{{\partial H}}{{\partial {{p}_{k}}}},{\text{ }}k \in L.{\text{ }} $$ (1) Under some natural conditions the above system of differential equations defines a flow in an adequately chosen configuration space (χ,F), and we have to describe the stationary measures of this Hamiltonian dynamics. A positive answer to the ergodic hypothesis means that every stationary state is a superposition of Gibbs distributions. This postulate plays a crucial role in the foundations of classical statistical mechanics, its physical relevance is convincingly motivated by numerous applications in equilibrium and non-equilibrium theories including the mathematical description of phase transi tions and the microscopic derivation of the equations of irreversible thermodynamics. On the other hand, essentially the Sinai billiards are the only non-trivial examples for which the ergodic hypothesis has rigorously been proven.