Renewal stochastic processes with correlated events: Phase transitions along time evolution

We consider renewal stochastic processes generated by nonindependent events from the perspective that their basic distribution and associated generating functions obey the statistical-mechanical structure of systems with interacting degrees of freedom. Based on this fact we look briefly into the less-known case of processes that display phase transitions along time. When the density distribution ${\psi }_n(t)$ 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 $z$ 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 deviations theory, while the entropy, or Massieu potential, derived from ${\psi }_n(t)$ satisfies an Euler relation. We illustrate this scheme first for an ordinary renewal process of events generated by a simple exponential waiting-time distribution $\psi (t)$. Then we examine a process modeled after the so-called Hamiltonian mean-field model that is representative of agents that perform a repeated task with an associated outcome, such as an opinion poll. When a sequence of (many) events takes place in a sufficiently short time the process exhibits clustering of the outcome, but for larger times the process resembles that of independent events. The two regimes are separated by a sharp transition, technically of the second order. Finally we point out the existence of a similar scheme for random-walk processes.

Figure 1

Dependence of $t/n$, $t,n\gg 1$, on the Laplace variable $\epsilon$ for the renewal process model of correlated events designed to be analogous to the HMF model. The figure is equivalent to the caloric equation of the HMF model and shows the two-phase behavior described in the text.

Figure 2

Dependence of the active time per unit event $\langle W_n\rangle /n$ for $n\gg 1$ on $t/n$ for the renewal process model of correlated events designed to be analogous to the HMF model. The figure shows the two-phase behavior described in the text.

Figure 3

Dependence of ${\psi }_n(t;\tau ,\theta )$, $n\gg 1$, on the $\theta$ when $t<t_c$. When $t>t_c$ the function ${\psi }_n(t;\tau ,\theta )$ is $\theta$-independent.

Figure 4

Dependence of $\partial (\langle W_n\rangle /n)/\partial (t/n)$, $t,n\gg 1$, on $t/n$.