Statistical-mechanical structure for renewal stochastic processes

Abstract

We consider ordinary renewal stochastic processes (generated by independent events) and indicate that their basic distribution and associated generating functions obey the statistical-mechanical structure of noninteracting systems. Based on this fact we look briefly into the less known case of correlated renewal events. When the density distribution ψn(t) for the occurrence of the n-th event at time t is considered to be a partition function, of a ‘microcanonical’ type for n ‘degrees of freedom’ at fixed ‘energy’ t, one obtains a set of four partition functions of which that for the generating function variable μ and Laplace transform variable 'epsilon' (conjugate to n and t, respectively) plays a central role. These partition functions relate to each other in the customary way and in accordance to the precepts of large deviation theory, while the entropy, or Massieu potential, derived from ψn(t) satisfies an Euler relation. We illustrate this scheme first for a renewal process of events generated by a simple exponential waiting time distribution ψ(t). Then we examine a specially modified variety of process in which a power law ψ(t) leads to closed-form expressions analogous to those for the exponential ψ(t). Finally we mention the benefits of this identification to the generalization of renewal processes to the case of correlated events, and point out the existence of a similar scheme for random walk processes.