Large deviations of ergodic counting processes: a statistical mechanics approach

TL;DR

This paper applies a statistical mechanics framework to analyze large deviations in ergodic counting processes, revealing new measurement interpretations and properties like scale invariance and intermittence.

Contribution

It introduces a thermodynamic approach to ergodic counting processes, including renewal and non-renewal types, with a novel auxiliary process and measurement interpretation.

Findings

Auxiliary process obtained via conditional measurement scheme.

Scale invariance and intermittence phenomena identified.

Results extend to non-renewal processes with memory effects.

Abstract

The large-deviation method allows to characterize an ergodic counting process in terms of a thermodynamic frame where a free energy function determines the asymptotic non-stationary statistical properties of its fluctuations. Here, we study this formalism through a statistical mechanics approach, i.e., with an auxiliary counting process that maximizes an entropy function associated to the thermodynamic potential. We show that the realizations of this auxiliary process can be obtained after applying a conditional measurement scheme to the original ones, providing is this way an alternative measurement interpretation of the thermodynamic approach. General results are obtained for renewal counting processes, i.e., those where the time intervals between consecutive events are independent and defined by a unique waiting time distribution. The underlying statistical mechanics is controlled by the same waiting time distribution, rescaled by an exponential decay measured by the free energy function. A scale invariance, shift closure, and intermittence phenomena are obtained and interpreted in this context. Similar conclusions apply for non-renewal processes when the memory between successive events is induced by a stochastic waiting time distribution.