Studying Dynamical Systems with Algebraic Tools

Abstract

The proper mathematical formulation of Ludwig Boltzmann’s ergodic hypothesis, which has been the source of so much interest and discussion in the last hundred years, is still not clear. For systems of elastic hard balls on a torus, however, Yakov Sinai, in 1963, [Sin(1963)] gave a stronger, and at the same time mathematically rigorous, version of Boltzmann’s hypothesis: the system of an arbitrarily fixed number N of identical elastic hard balls moving in the v-torus \(T_L^v = {^v}/L \cdot {^v}\left( {v \ge 2} \right)\) is ergodic — of course, on the submanifold of the phase space specified by the trivial conservation laws. Boltzmann used his ergodic hypothesis when laying the foundations of statistical physics, and its various forms are still intensively used in modern statistical physics. The meaning of Sinai’s hypothesis for the theory of dynamical systems, which was partially based on the physical arguments of Krylov’s 1942 thesis (cf. [K(1979)] and its afterword written by Ya. G. Sinai, [Sin(1979)]), is stressed by the fact that the interaction of elastic hard balls defines the only physical system of an arbitrary number of particles in arbitrary dimension whose dynamical behaviour has been so far at least guessed — except for the completely integrable system of harmonic oscillators. (As to the history of Boltzmann’s hypothesis, see the recent work [Sz(1996)].