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Journal of Statistical Physics

, Volume 174, Issue 2, pp 404–432 | Cite as

Non-equivalence of Dynamical Ensembles and Emergent Non-ergodicity

  • Hadrien VroylandtEmail author
  • Gatien Verley
Article

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.

Keywords

Large deviations theory Equivalence of ensembles Non-ergodicity Ising model 

Notes

Acknowledgements

We acknowledge Massimiliano Esposito for his advice on this project that started when GV was a post-doctoral fellow in his research team. We thank V. Lecomte and A. Lazarescu for the insightful discussions in connection with this work.

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Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique (UMR8627), CNRSUniv. Paris-Sud, Université Paris-SaclayOrsayFrance

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