Ergodic Hypothesis and the Thermalization Time for Chaotic Hamiltonian Systems with Few Degrees of Freedom

One of the fundamental assumptions of Classical Statistical Mechanics is the ergodic hypothesis. Using this hypothesis amounts to assuming that, in thermal equilibrium, the time average of a phase function will be equal to the corresponding microcanonical average. Even though this assumption plays an important role in the formulation of the microcanonical ensemble, the introduction of the ergodic hypothesis was always criticized. We argue that this criticism is the result of an incorrect understanding about the hypothesis under consideration. In this article, we shall discuss some results that were established by means of computer simulations in order to provide a modern justification for using the ergodic hypothesis. Moreover, we are going to explain how the validity of this assumption can be checked for a given physical system whose numerical study can actually be carried out with the aid of a computer. For such a system, we shall discuss how the ergodic hypothesis allows us to obtain an estimate for the time it takes to reach thermal equilibrium. Our approach is based on the dynamical calculation of temperatures. To illustrate the ideas we consider in this work, we are going to present the statistical-mechanical description of two chaotic Hamiltonian systems with few degrees of freedom. These systems can be obtained as particular cases of certain classical field theories.