Studies in Nonequilibrium Rate Processes. V. The Relaxation of Moments Derived from a Master Equation

Abstract

A study has been made of the relaxation of the moments of probability distributions whose time evolution are governed by a master equation. The necessary and sufficient condition for the first moment, M 1(t), to undergo a simple exponential relaxation is found to be ∑ n=0 ∞ nA nm =βm+γ, where Anm is the transition probability per unit time for transitions from state m to n, and where β and γ are constants. The necessary and sufficient condition under which the first k moments, M 1(t), M 2(t), ···, Mk (t), satisfy a closed system of linear equations is found to be ∑ n=0 ∞ n r A nm = ∑ i=0 k β ri m i . Near equilibrium, i.e., as t → ∞, all the moments Mr (t) obey, to a good approximation, a simple exponential relaxation law irrespective of the form of the Anm . For systems described by the Fokker‐Planck equation ∂P(x, t) ∂t =− ∂ ∂x [b 1 (x)P(x, t)]+ 1 2 ∂ 2 ∂x 2 [b 2 (x)P(x, t)], the necessary and sufficient condition that the first moment M 1(t) undergo a simple exponential relaxation is found to be b 1(x) = βx + γ and the necessary and sufficient condition for the 2nd moment, M 2(t) to have a simple exponential relaxation is 2xb 1 (x)+b 2 =β 22 x 2 +γ 2 . It is shown that these conditions are equivalent to the conditions on the Anm stated above.