Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges

TL;DR

This paper investigates the inequivalence of path ensembles in stochastic processes, revealing phenomena like temporal condensation and how different constraints affect large deviations in processes such as diffusion and random walks.

Contribution

It demonstrates how ensemble inequivalence manifests in stochastic bridges, especially through temporal condensation, and explores effects of various constraints on large deviations.

Findings

Temporal condensation occurs in confining potentials like Ornstein-Uhlenbeck.

Partial ensemble equivalence observed in dry friction scenarios.

Heavy-tailed step distributions can realize large deviations in a single step.

Abstract

We study stochastic processes in which the trajectories are constrained so that the process realises a large deviation of the unconstrained process. In particular we consider stochastic bridges and the question of inequivalence of path ensembles between the microcanonical ensemble, in which the end points of the trajectory are constrained, and the canonical or s ensemble in which a bias or tilt is introduced into the process. We show how ensemble inequivalence can be manifested by the phenomenon of temporal condensation in which the large deviation is realised in a vanishing fraction of the duration (for long durations). For diffusion processes we find that condensation happens whenever the process is subject to a confining potential, such as for the Ornstein-Uhlenbeck process, but not in the borderline case of dry friction in which there is partial ensemble equivalence. We also discuss continuous-space, discrete-time random walks for which in the case of a heavy tailed step-size distribution it is known that the large deviation may be achieved in a single step of the walk. Finally we consider possible effects of several constraints on the process and in particular give an alternative explanation of the interaction-driven condensation in terms of constrained Brownian excursions.