Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges

Abstract

We study stochastic processes in which the trajectories are constrained so\nthat the process realises a large deviation of the unconstrained process. In\nparticular we consider stochastic bridges and the question of inequivalence of\npath ensembles between the microcanonical ensemble, in which the end points of\nthe trajectory are constrained, and the canonical or s ensemble in which a bias\nor tilt is introduced into the process. We show how ensemble inequivalence can\nbe manifested by the phenomenon of temporal condensation in which the large\ndeviation is realised in a vanishing fraction of the duration (for long\ndurations). For diffusion processes we find that condensation happens whenever\nthe process is subject to a confining potential, such as for the\nOrnstein-Uhlenbeck process, but not in the borderline case of dry friction in\nwhich there is partial ensemble equivalence. We also discuss continuous-space,\ndiscrete-time random walks for which in the case of a heavy tailed step-size\ndistribution it is known that the large deviation may be achieved in a single\nstep of the walk. Finally we consider possible effects of several constraints\non the process and in particular give an alternative explanation of the\ninteraction-driven condensation in terms of constrained Brownian excursions.\n