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


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.


Large deviations theory Equivalence of ensembles Non-ergodicity Ising model 



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.


  1. 1.
    Baek, Y., Kafri, Y., Lecomte, V.: Dynamical phase transitions in the current distribution of driven diffusive channels. J. Phys. A Math. Theor. 51(10), 105001 (2018).
  2. 2.
    Barato, A.C., Chetrite, R.: A formal view on level 2.5 large deviations and fluctuation relations. J. Stat. Phys. 160(5), 1154–1172 (2015). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barré, J., Mukamel, D., Ruffo, S.: Inequivalence of ensembles in a system with long-range interactions. Phys. Rev. Lett. 87, 030,601 (2001). CrossRefGoogle Scholar
  4. 4.
    Bertini, L., Faggionato, A., Gabrielli, D.: From level 2.5 to level 2 large deviations for continuous time Markov chains. Markov Process. Relat. Fields. 20(3), 545–562 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Biroli, G., Garrahan, J.P.: Perspective: the glass transition. J. Chem. Phys. 138(12), 12A301 (2013). CrossRefGoogle Scholar
  6. 6.
    Bouchet, F., Dauxois, T., Mukamel, D., Ruffo, S.: Phase space gaps and ergodicity breaking in systems with long-range interactions. Phys. Rev. E 77(1), 011,125 (2008). CrossRefGoogle Scholar
  7. 7.
    Bouchet, F., Gupta, S., Mukamel, D.: Thermodynamics and dynamics of systems with long-range interactions. Phys. A Stat. Mech. Appl. 389(20), 4389–4405 (2010). Proceedings of the 12th International Summer School on Fundamental Problems in Statistical Physics
  8. 8.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Budini, A.A., Turner, R.M., Garrahan, J.P.: Fluctuating observation time ensembles in the thermodynamics of trajectories. J. Stat. Mech. Theory Exp. 2014(3), P03012 (2014).
  10. 10.
    Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics. Wiley (1985).
  11. 11.
    Campa, A., Dauxois, T., Ruffo, S.: Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Repo. 480(3–6), 57–159 (2009).
  12. 12.
    Chandler, D., Garrahan, J.P.: Dynamics on the way to forming glass: bubbles in space-time. Annu. Rev. Phys. Chem. 61(1), 191–217 (2010). PMID: 20055676
  13. 13.
    Chetrite, R., Touchette, H.: Nonequilibrium microcanonical and canonical ensembles and their equivalence. Phys. Rev. Lett. 111, 120,601 (2013). CrossRefGoogle Scholar
  14. 14.
    Chetrite, R., Touchette, H.: Nonequilibrium Markov processes conditioned on large deviations. Ann. Henri Poincaré 16(9), 2005–2057 (2015). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Costeniuc, M., Ellis, R.S., Touchette, H.: Complete analysis of phase transitions and ensemble equivalence for the Curie–Weiss–Potts model. J. Math. Phys. 46(6), 063301 (2005). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Csiszár, I., Shields, P.C.: Information Theory and Statistics: A Tutorial. Now Foundations and Trends (2004).
  17. 17.
    Dinwoodie, I.H.: Identifying a large deviation rate function. Ann. Probab. 21(1), 216–231 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dinwoodie, I.H., Zabell, S.L.: Large deviations for exchangeable random vectors. Ann. Probab. 20(3), 1147–1166 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ellis, R.S., Touchette, H., Turkington, B.: Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume–Emery–Griffiths model. Phys. A Stat. Mech. Appl. 335(3–4), 518–538 (2004).
  20. 20.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, Hoboken (2008)zbMATHGoogle Scholar
  21. 21.
    Ferré, G., Touchette, H.: Adaptive sampling of large deviations. J. Stat. Phys. (2018).
  22. 22.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Grundlehren der mathematischen Wissenschaften. Springer, New York (1984).
  23. 23.
    Garrahan, J.P.: Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles. J. Stat. Mech. Theory Exp. 2016(7), 073208 (2016).
  24. 24.
    Garrahan, J.P., Jack, R.L., Lecomte, V., Pitard, E., van Duijvendijk, K., van Wijland, F.: Dynamical first-order phase transition in kinetically constrained models of glasses. Phys. Rev. Lett. 98(19), 195,702 (2007). CrossRefzbMATHGoogle Scholar
  25. 25.
    Garrahan, J.P., Jack, R.L., Lecomte, V., Pitard, E., van Duijvendijk, K., van Wijland, F.: First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories. J. Phys. A Math. Theor. 42(7), 075007 (2009).
  26. 26.
    Glauber, R.J.: Time-dependant statistics of the Ising model. J. Math. Phys. 4, 294 (1963). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Perseus Books Publishing, L.L.C, New York (1992)zbMATHGoogle Scholar
  28. 28.
    Grafke, T., Grauer, R., Schäfer, T.: The instanton method and its numerical implementation in fluid mechanics. J. Phys. A Math. Theor. 48(33), 333001 (2015).
  29. 29.
    Hedges, L.O., Jack, R.L., Garrahan, J.P., Chandler, D.: Dynamic order–disorder in atomistic models of structural glass formers. Science 323(5919), 1309–1313 (2009).
  30. 30.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2012). CrossRefGoogle Scholar
  31. 31.
    Jack, R.L., Sollich, P.: Large deviations and ensembles of trajectories in stochastic models. Prog. Theor. Phys. Suppl. 184, 304–317 (2010).
  32. 32.
    Jack, R.L., Garrahan, J.P., Chandler, D.: Space-time thermodynamics and subsystem observables in a kinetically constrained model of glassy materials. J. Chem. Phys. 125(18), 184509 (2006). ADSCrossRefGoogle Scholar
  33. 33.
    Keys, A.S., Chandler, D., Garrahan, J.P.: Using the \(s\) ensemble to probe glasses formed by cooling and aging. Phys. Rev. E 92, 022304 (2015). ADSCrossRefGoogle Scholar
  34. 34.
    Lawler, G.F., Sokal, A.D.: Bounds on the \(l^2\) spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Am. Math. Soc. 309(2), 557–580 (1988).
  35. 35.
    Lebowitz, J.L., Spohn, H.: A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95(1), 333–365 (1999). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lecomte, V., Appert-Rolland, C., van Wijland, F.: Thermodynamic formalism for systems with Markov dynamics. J. Stat. Phys. 127(1), 51–106 (2007). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Levin, D.A., Yuval Peres, E.L.W.: Markov Chains and Mixing Times, vol. 58. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  38. 38.
    Maes, C., Netočný, K.: Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states. EPL (Europhys. Lett.) 82(3), 30003 (2008).
  39. 39.
    Merolle, M., Garrahan, J.P., Chandler, D.: Space-time thermodynamics of the glass transition. Proc. Natl. Acad. Sci. 102(31), 10837–10840 (2005).
  40. 40.
    Monthus, C.: Non-equilibrium steady states: maximization of the Shannon entropy associated with the distribution of dynamical trajectories in the presence of constraints. J. Stat. Mech. 2011(03), P03008 (2011).
  41. 41.
    Mukamel, D., Ruffo, S., Schreiber, N.: Breaking of ergodicity and long relaxation times in systems with long-range interactions. Phys. Rev. Lett. 95, 240,604 (2005). CrossRefGoogle Scholar
  42. 42.
    Nemoto, T.: Zon–Cohen singularity and negative inverse temperature in a trapped-particle limit. Phys. Rev. E 85, 061,124 (2012). CrossRefGoogle Scholar
  43. 43.
    Nemoto, T., Sasa, Si: Computation of large deviation statistics via iterative measurement-and-feedback procedure. Phys. Rev. Lett. 112(9), 090,602 (2014). CrossRefGoogle Scholar
  44. 44.
    Nyawo, P.T., Touchette, H.: A minimal model of dynamical phase transition. EPL (Europhys. Lett.) 116(5), 50009 (2016).
  45. 45.
    Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91(6), 1505–1512 (1953). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Pérez-Espigares, C., Carollo, F., Garrahan, J.P., Hurtado, P.I.: Dynamical criticality in driven systems: non-perturbative results, microscopic origin and direct observation. ArXiv e-prints (2018)Google Scholar
  47. 47.
    Speck, T.: Thermodynamic formalism and linear response theory for nonequilibrium steady states. Phys. Rev. E 94, 022,131 (2016). CrossRefGoogle Scholar
  48. 48.
    Speck, T., Garrahan, J.P.: Space-time phase transitions in driven kinetically constrained latticemodels. Eur. Phys. J. B 79(1), 1–6 (2011). ADSCrossRefGoogle Scholar
  49. 49.
    Szavits-Nossan, J., Evans, M.R.: Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges. J. Stat. Mech. Theory Exp. 2015(12), P12008 (2015).
  50. 50.
    Touchette, H.: Equivalence and nonequivalence of the microcanonical and canonical ensembles : a large deviations study. Ph.D. thesis, PhD Thesis Department of Physics, McGill University (2003). http://digitool.Library.McGill.CA:80/R/-?func=dbin-jump-full&object_id=84851&silo_library=GEN01
  51. 51.
    Touchette, H.: Simple spin models with non-concave entropies. Am. J. Phys. 76, 26–30 (2008). ADSCrossRefGoogle Scholar
  52. 52.
    Touchette, H.: The large deviation approach to statistical mechanics (2009).
  53. 53.
    Touchette, H.: Methods for calculating nonconcave entropies. J. Stat. Mech. 05008 (2010).
  54. 54.
    Touchette, H., Ellis, R.S., Turkington, B.: An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles. Phys. A Stat. Mech. Appl. 340(1–3), 138–146 (2004). News and Expectations in Thermostatistics
  55. 55.
    Tsobgni Nyawo, P., Touchette, H.: Large deviations of the current for driven periodic diffusions. Phys. Rev. E 94, 032,101 (2016). CrossRefGoogle Scholar
  56. 56.
    Van Kampen, N.: Stochastic Processes in Physics and Chemistry (Third Edition). Elsevier, Amsterdam (2007).

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Authors and Affiliations

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

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