Introduction

The foundations of the establishment of thermal equilibrium constitute an old problem, first investigated by Boltzmann in. Although Boltzmann's kinetic equations assume a probabilistic nature of the system, the starting point was a deterministic and conservative mechanical system consisting of a finite number of elastically colliding balls. This model is called the Boltzmann–Gibbs gas.

In this article the normal distribution of velocities is put on a firm foundation for conservative mechanical systems with a large number of degrees of freedom, without any additional assumptions of a statistical or random kind. At the end, the Boltzmann distribution for density in configuration space is also justified. So, here some of Boltzmann's ideas (see) are proved in a rigorous way.

Summary of the article

In Section 1 it is shown that at most points of an n-sphere, the difference between the joint density of the Cartesian coordinates and the density of a normal distribution vanishes as n tends to infinity. Our approach is elementary but requires some technical lemmas whose proofs are deferred to Section 2.

In Section 3 the whole energy level is considered – the product of a sphere and a compact configuration space. Then, using the individual ergodic theorem and Lemma 3.1, deviations of the distribution of velocities from normal for individual solutions with different initial conditions are investigated.

Finally, it is shown that for systems with sufficiently many degrees of freedom, for most initial conditions the deviation from the normal distribution is small at almost every moment of time.