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