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, M1(t), to undergo a simple exponential relaxation is found to be ∑ n=0∞nAnm=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, M1(t), M2(t), ···, Mk(t), satisfy a closed system of linear equations is found to be ∑ n=0∞nrAnm= ∑ i=0kβrimi.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 [b1(x)P(x, t)]+12∂2∂x2 [b2(x)P(x, t)],the necessary and sufficient condition that the first moment M1(t) undergo a simple exponential relaxation is found to be b1(x) = βx + γ and the necessary and sufficient condition for the 2nd moment, M2(t) to have a simple exponential relaxation is 2xb1(x)+b2=β22x2+γ2. It is shown that these conditions are equivalent to the conditions on the Anm stated above.