A remark on the canonical distribution

V. A. Zorich

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2014), pp. 23-33

Résumé

It is а well-known fact that if we take the $n$-dimensional ball of unit volume and project it orthogonally onto a straight line, observing how the measure is distributed on the projection, then in the limit as $n \to \infty$ we get the normal distribution on the line. It is also well known that almost the entire volume of the multidimensional domain, for example, of the multidimensional ball, is concentrated in the neighborhood of the boundary. These phenomena have numerous manifestations and consequences. For example, any more or less regular function on the multidimensional ball or on the multidimensional sphere is practically constant from the point of view of the observer who takes two random points of the domain of the function and compares the values of the function at these points. Classical Maxwell and Gibbs distributions of thermodynamics are discussed here in the context of multidimensional geometry.

Détail

Keywords: multidimensional geometry, functions of a very large number of variables, Maxwell distribution, Gibbs distribution.