Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels

Journal of Statistical Physics

Volumn 159, Issue 5, 2015, Pages 987-1016

  Touchette, Hugo ^a,b  

^a NATIONAL INSTITUTE FOR THEORETICAL PHYSICS NITHEP   (South Africa)

^b STELLENBOSCH UNIVERSITY   (South Africa)

Author keywords

Entropy; Equivalent and nonequivalent ensembles; Large deviation theory; Long range systems; Microcanonical and canonical ensembles

Indexed keywords

EID: 84938775692     PISSN: 00224715     EISSN: 15729613     Source Type: Journal    
DOI: 10.1007/s10955-015-1212-2     Document Type: Article

References (66)

1. Boltzmann, L.: Über die eigenschaften monozyklischer und anderer damit verwandter Systeme. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. III. Chelsea, New York (1968)
2. Gibbs, J.W.: Elementary Principles in Statistical Mechanics with Especial Reference to the Rational Foundation of Thermodynamics (Yale University Press, 1902) reprinted by Dover, New York (1960)
3. Touchette, H., Ellis, R.S., Turkington, B.: An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles. Physica A 340, 138–146 (2004)
4. Ellis, R.S., Haven, K., Turkington, B.: Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101, 999–1064 (2000)
5. Lewis, J.T., Pfister, C.-E., Sullivan, W.G.: Large deviations and the thermodynamic formalism: a new proof of the equivalence of ensembles. In: Fannes, M., Maes, C., Verbeure, A. (eds.) On Three Levels, pp. 183–192. Plenum Press, New York (1994)
6. Lewis, J.T., Pfister, C.-E., Sullivan, G.W.: The equivalence of ensembles for lattice systems: some examples and a counterexample. J. Stat. Phys. 77, 397–419 (1994b)
7. Lewis, J.T., Pfister, C.-E., Sullivan, W.G.: Entropy, concentration of probability and conditional limit theorem. Markov Process. Relat. Fields 1, 319–386 (1995)
8. Touchette, H.: Ensemble equivalence for general many-body systems. Europhys. Lett. 96, 50010 (2011)
9. Ruelle, D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin, Amsterdam (1969)
10. Griffiths, R.B.: Rigorous results and theorems. In: Domb, C., Green, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. 1, pp. 7–109. Academic Press, London (1972)
11. Lynden-Bell, D., Wood, R.: The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems. Mon. Not. R. Astron. Soc. 138, 495–525 (1968)
12. Lynden-Bell, D., Lynden-Bell, R.M.: On the negative specific heat paradox. Mon. Not. R. Astron. Soc. 181, 405–419 (1977)
13. Lynden-Bell, D.: Negative specific heat in astronomy, physics and chemistry. Physica A 263, 293–304 (1999)
14. Thirring, W.: Systems with negative specific heat. Z. Phys. A 235, 339–352 (1970)
15. Chavanis, P.-H.: Phase transitions in self-gravitating systems. Int. J. Mod. Phys. B 20, 3113–3198 (2006)
16. Campa, A., Dauxois, T., Ruffo, S.: Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480, 57–159 (2009)
17. Dauxois, T., Ruffo, S., Cugliandolo, L.F. (eds.): Long-Range Interacting Systems, vol. 90. Lecture Notes of the Les Houches Summer School 2008. Oxford University Press, Oxford (2010)
18. Kastner, M.: Nonequivalence of ensembles for long-range quantum spin systems in optical lattices. Phys. Rev. Lett. 104, 240403 (2010a)
19. Kastner, M.: Nonequivalence of ensembles in the Curie–Weiss anisotropic quantum Heisenberg model. J. Stat. Mech. 2010, P07006 (2010)
20. Olivier, G., Kastner, M.: Microcanonical analysis of the Curie–Weiss anisotropic quantum Heisenberg model in a magnetic field. J. Stat. Phys. 157, 456–473 (2014)
21. Chalony, M., Barré, J., Marcos, B., Olivetti, A., Wilkowski, D.: Long-range one-dimensional gravitational-like interaction in a neutral atomic cold gas. Phys. Rev. A 87, 013401 (2013)
22. Gurarie, V.: The equivalence between the canonical and microcanonical ensembles when applied to large systems. Am. J. Phys. 75, 747–751 (2007)
23. Galgani, L., Scotti, A., Valz Gris, F.: Asymptotic equivalence of classical ensembles by the method of the maximum. Physica 47, 601–609 (1970)
24. Galgani, L., Manzoni, L., Scotti, A.: Asymptotic equivalence of equilibrium ensembles of classical statistical mechanics. J. Math. Phys. 12, 933–935 (1971)
25. Lanford III, O.E.: Entropy and equilibrium states in classical statistical mechanics. In: Lenard, A. (ed.) Statistical Mechanics and Mathematical Problems. Lecture Notes in Physics, vol. 20, pp. 1–113. Springer, Berlin (1973)
26. Deuschel, J.-D., Stroock, D.W., Zessin, H.: Microcanonical distributions for lattice gases. Commun. Math. Phys. 139, 83–101 (1991)
27. Stroock, D.W., Zeitouni, O.: Microcanonical distributions, Gibbs’ states, and the equivalence of ensembles. In: Durrett, R., Kesten, H. (eds.) Random Walks, Brownian Motion and Interacting Particle Systems: A Festschrift in Honor of Frank Spitzer, pp. 399–424. Birkhäuser, Boston (1991)
28. Stroock, D.W.: Microcanonical distributions for one dimensional lattice gases. In: Cruzeiro, A.B., Zambrini, J.-C. (eds.) Stochastic Analysis and Applications: Proceedings of the 1989 Lisbon Conference, Progress in Probability, vol. 26, pp. 190–197. Birkhäuser, Boston (1991)
29. Roelly, S., Zessin, H.: The equivalence of equilibrium principles in statistical mechanics and some applications to large particle systems. Expo. Math. 11, 385–405 (1993)
30. Georgii, H.-O.: Large deviations and maximum entropy principle for interacting random fields on $${ Z}^{d}$$Zd. Ann. Probab. 21, 1845–1875 (1993)
31. Georgii, H.-O.: Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Probab. Theory Relat. Fields 99, 171–195 (1994)
32. Georgii, H.-O.: The equivalence of ensembles for classical systems of particles. J. Stat. Phys. 80, 1341–1378 (1995)
33. Kac, M., Uhlenbeck, G.E., Hemmer, P.C.: On the van der Waals theory of the vapor–liquid equilibrium I: discussion of a one-dimensional model. J. Math. Phys. 4, 216–228 (1963)
34. Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)
35. Ellis, R.S.: An overview of the theory of large deviations and applications to statistical mechanics. Scand. Actuar. J. 1, 97–142 (1995)
36. Ellis, R.S.: The theory of large deviations: from Boltzmann’s 1877 calculation to equilibrium macrostates in 2D turbulence. Physica D 133, 106–136 (1999)
37. Touchette, H.: The large deviation approach to statistical mechanics. Phys. Rep. 478, 1–69 (2009)
38. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)
39. Pathria, R.K.: Statistical Mechanics, 2nd edn. Butterworth Heinemann, Oxford (1996)
40. Dorlas, T.C.: Statistical Mechanics: Fundamentals and Model Solutions. IOP, Bristol (1999)
41. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)
42. Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math. 19, 261–286 (1966)
43. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
44. Costeniuc, M., Ellis, R.S., Touchette, H., Turkington, B.: The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble. J. Stat. Phys. 119, 1283–1329 (2005)
45. Touchette, H.: Equivalence and nonequivalence of the microcanonical and canonical ensembles: a large deviations study. PhD Thesis, Department of Physics, McGill University (2003)
46. Touchette, H., Ellis, R.S.: Nonequivalent ensembles and metastability. In: Beck, C., Benedek, G., Rapisarda, A., Tsallis, C. (eds.) Complexity, Metastability and Nonextensivity, pp. 81–87. World Scientific, Singapore (2005)
47. Eyink, G.L., Spohn, H.: Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence. J. Stat. Phys. 70, 833–886 (1993)
48. Yukalov, V.: Fluctuations of composite observables and stability of statistical systems. Phys. Rev. E 72, 066119 (2005)
49. Csiszar, I.: $${I}$$I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3, 146–158 (1975)
50. Csiszar, I.: Sanov property, generalized $${I}$$I-projection and a conditional limit theorem. Ann. Probab. 12, 768–793 (1984)
51. Csiszar, I.: The method of types. IEEE Trans. Inf. Theory 44, 2505–2523 (1998)
52. Shields, P.C.: Two divergence-rate counterexamples. J. Theor. Probab. 6, 521–545 (1993)
53. Grosskinsky, S., Schütz, G.M., Spohn, H.: Condensation in the zero range process: stationary and dynamical properties. J. Stat. Phys. 113, 389–410 (2003)
54. Grosskinsky, S., Schütz, G.: Discontinuous condensation transition and nonequivalence of ensembles in a zero-range process. J. Stat. Phys. 132, 77–108 (2008)
55. Chleboun, P., Grosskinsky, S.: Condensation in stochastic particle systems with stationary product measures. J. Stat. Phys. 154, 432–465 (2014)
56. Föllmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab. 16, 961–977 (1988)
57. Föllmer, H.: Random fields and diffusion processes. In: Hennequin, P.-L. (ed.) École d’Été de Probabilités de Saint-Flour 1985–87. Lecture Notes in Mathematics, vol. 1362, pp. 101–203. Springer, New York (1988)
58. Georgii, H.-O.: Gibbs Measures and Phase Transitions, vol. 9. Walter de Gruyter and Co., Berlin (1988)
59. Campa, A., Ruffo, S., Touchette, H.: Negative magnetic susceptibility and nonequivalent ensembles for the mean-field $$\phi ^{4}$$ϕ4 spin model. Physica A 385, 233–248 (2007)
60. Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability. Springer, New York (2007)
61. Squartini, T., de Mol, J., den Hollander, F., Garlaschelli, D.: Breaking of ensemble equivalence in networks. arxiv:1501.00388
62. Jack, R.L., Sollich, P.: Large deviations and ensembles of trajectories in stochastic models. Prog. Theor. Phys. Suppl. 184, 304–317 (2010)
63. Chetrite, R., Touchette, H.: Nonequilibrium microcanonical and canonical ensembles and their equivalence. Phys. Rev. Lett. 111, 120601 (2013)
64. Chetrite, R., Touchette, H.: Nonequilibrium Markov processes conditioned on large deviations. Ann. Inst. Poincaré A (2014)
65. Evans, R.M.L.: Detailed balance has a counterpart in non-equilibrium steady states. J. Phys. A 38, 293–313 (2005)
66. Chandler, D., Garrahan, J.P.: Dynamics on the way to forming glass: bubbles in space–time. Ann. Rev. Chem. Phys. 61, 191–217 (2010)