On normal distribution in velocities (Q1860535)

The foundation of the establishment of the heat balance is an old problem. It was first investigated by \textit{L. Boltzmann} in [Wien. Ber. 66, 275-370 (1872; JFM 04.0566.01)]. Boltzmann kinetic equations assume the probabilistic nature of the system, though the starting point was a `pure' deterministic conservative mechanical system of a finite number of elastically colliding balls. This model is called Boltzmann-Gibbs gas. In this article the foundation of relaxation to normal distribution in velocities is achieved for conservative mechanical systems with a huge number of degrees of freedom, without any additional assumptions of statistical or random kind. In the first part it is proved that on the majority of points of the \(n\)-sphere the difference between the joint density of Cartesian coordinates of its point and the density of the normal distribution vanishes as \(n\) tends to infinity. In the second part the whole energy level is considered -- cross product of sphere and compact configuration space. Then, using a.e. an individual ergodic theorem, variations on individual solutions with different initial conditions are investigated. Eventually it is shown for systems with sufficiently great number of degrees of freedom that for the majority of initial conditions the variation from normal distribution is small at almost every moment of time.