On the Decay of Initial Correlations in Stochastic Processes

We study the relaxation of a noninteracting, initially correlated many‐particle system in contact with an infinite reservoir. We use the master equation to study the time development of the r‐particle distribution function P_r(n;t) and assume that the relaxation process is Markovian. We study the decay of the correlations by investigating the time development of the r‐particle Ursell functions, U_r(n;t). We show that the correlation function U_r(n;t) goes to zero much more rapidly with time than the r‐particle distribution function approaches its equilibrium value $$P_r(\mathbf{n}\mathbf{;}\mathrm{\hspace{0.17em}\infty )=\Pi }{\displaystyle {lim}_{\text{i=1}}^r}{\mathrm{P(n}}_i\mathbf{;}\mathrm{\hspace{0.17em}\infty ).}$$ The exact forms of the relaxation of P_r(n;t) and U_r(n;t) depend upon the eigenvalue spectrum of the transition rate matrix of the master equation. The general theory is developed and then applied to a number of examples.