Non equivalence of dynamical ensembles and emergent non ergodicity

TL;DR

This paper investigates the conditions under which dynamical ensembles are equivalent, highlighting that ergodicity guarantees equivalence, while non-ergodic systems exhibit emergent non-ergodicity and initial condition dependence, with implications demonstrated on the infinite range Ising model.

Contribution

It extends the understanding of ensemble equivalence to non-ergodic systems and introduces methods to analyze initial condition effects on fluctuations.

Findings

Ergodicity guarantees ensemble equivalence in Markov jump processes.

Non-ergodic systems show initial condition dependence and emergent non-ergodicity.

In the Ising model, initial conditions are forgotten exponentially slowly with system size.

Abstract

Dynamical ensembles have been introduced to study constrained stochastic processes. In the microcanonical ensemble, the value of a dynamical observable is constrained to a given value. In the canonical ensemble a bias is introduced in the process to move the mean value of this observable. The equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result. In this paper, we study the physical conditions associated with ensemble equivalence and the consequences of non-equivalence. For continuous time Markov jump processes, we show that ergodicity guarantees ensemble equivalence. For non-ergodic systems or systems with emergent ergodicity breaking, we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition. We illustrate our results on the infinite range Ising model by characterizing the fluctuations of magnetization and activity. We discuss the emergence of non ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins.