Relaxation and fluctuation of macrovariables

Abstract

An extensive macrovariable X of a system of a large size Ω may have an extensive property of its probability distribution; namely the time-dependent probability function has the form P (x, t) = C exp Ωϕ (x, t) , x = X/Ω.

This ansatz has been proved by assuming a Markoffian process with transition probability satisfying a homogeneity condition. The function ϕ(x,t) can be identified with the action integral and the problem can be formulated by the Hamilton-Jacobi method, which is naturally. related to a path-integral formalism. In normal cases, the distribution is Gaussian corresponding to a central limit theorem. Evolution of the mean value and the variance is determined by simple equations which contains the first and second moments of the basic transition probability.