On the basis of statistical mechanics. The Liouville equation for systems with an infinite countable number of degrees of freedom

Abstract

In this paper statistical properties of physical systems with a phase that is an infinite-dimensional separable Hilbert space are considered. For such systems it is possible to define the basic concepts of classical mechanics as well as statistical mechanics. In particular as an initial probability measure we assume a quasi-invariant measure. Based on this idea we derived the Liouville equation in an infinite-dimensional case. The obtained Liouville equation contains an additional term dependent also on the assumed initial measure which vanishes when we pass to a system with finite number of degrees of freedom but which is not equal to zero when we study a system with an infinite number of degrees of freedom. This term is in a sense a compensation for avoiding complications due to boundary conditions. The formalism given in this paper is still valid even in more general situations, when the phase space is not Hilbert space. This fact may play an essential role in statistical mechanics.