Journal of Statistical Physics

, Volume 101, Issue 5–6, pp 999–1064 | Cite as

Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

  • Richard S. Ellis
  • Kyle Haven
  • Bruce Turkington
Article

Abstract

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the following question in global optimization. What are the relationships between the set of solutions of the constrained minimization problem that characterizes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and sufficient condition for equivalence of ensembles to hold at the level of equilibrium macrostates is that it holds at the level of thermodynamic functions, which is the case if and only if the microcanonical entropy is concave. The necessity of this condition is new and has the following striking formulation. If the microcanonical entropy is not concave at some value of its argument, then the ensembles are nonequivalent in the sense that the corresponding set of microcanonical equilibrium macrostates is disjoint from any set of canonical equilibrium macrostates. We point out a number of models of physical interest in which nonconcave microcanonical entropies arise. We also introduce a new class of ensembles called mixed ensembles, obtained by treating a subset of the dynamical invariants canonically and the complementary set microcanonically. Such ensembles arise naturally in applications where there are several independent dynamical invariants, including models of dispersive waves for the nonlinear Schrödinger equation. Complete equivalence and nonequivalence results are presented at the level of equilibrium macrostates for the pure canonical, the pure microcanonical, and the mixed ensembles.

large deviation principle equilibrium macrostates equivalence of ensembles microcanonical entropy 

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REFERENCES

  1. 1.
    M. Aizenman, S. Goldstein, and J. L. Lebowitz, Conditional equilibrium and the equivalence of microcanonical and grand canonical ensembles in the thermodynamic limit, Commun. Math. Phys. 62:279–302 (1978).Google Scholar
  2. 2.
    V. I. Arnold, Mathematical Methods of Classical Mechanics, K. Vogtmann and A. Weinstein, Transl. (Springer-Verlag, New York, 1978).Google Scholar
  3. 3.
    V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics (Springer-Verlag, New York, 1998).Google Scholar
  4. 4.
    R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (John Wiley & Sons, New York, 1975).Google Scholar
  5. 5.
    T. Bodineau and A. Guionnet, About the stationary states of vortex systems, Ann. Inst. Henri Poincaré 35:205–237 (1999).Google Scholar
  6. 6.
    C. Boucher, R. S. Ellis, and B. Turkington, Derivation of maximum entropy principles in two-dimensional turbulence via large deviations, J. Statist. Phys. 98:1235–1278 (2000).Google Scholar
  7. 7.
    C. Boucher, R. S. Ellis, and B. Turkington, Spatializing random measures: Doubly indexed processes and the large deviation principle, Ann. Probab. 27:297–324 (1999).Google Scholar
  8. 8.
    E. Caglioti, P. L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanical description, Commun. Math. Phys. 143:501–525 (1992).Google Scholar
  9. 9.
    E. Caglioti, P. L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanical description. Part II. Commun. Math. Phys. 174:229–260 (1995).Google Scholar
  10. 10.
    A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed. (Springer-Verlag, New York, 1998).Google Scholar
  11. 11.
    J.-D. Deuschel and D. W. Stroock, Large Deviations (Academic Press, Boston, 1989).Google Scholar
  12. 12.
    J.-D. Deuschel, D. W. Stroock, and H. Zessin, Microcanonical distributions for lattice gases, Commun. Math. Phys. 139:83–101 (1991).Google Scholar
  13. 13.
    M. DiBattista and A. Majda, An equilibrium statistical theory for large-scale features of open-ocean convection, to appear in J. Phys. Oceanography (2000).Google Scholar
  14. 14.
    M. DiBattista, A. Majda, and B. Turkington, Prototype geophysical vortex structures via large-scale statistical theory, Geophys. Astrophys. Fluid Dyn. 89:235–283 (1998).Google Scholar
  15. 15.
    P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations (John Wiley & Sons, New York, 1997).Google Scholar
  16. 16.
    R. S. Ellis, K. Haven, and B. Turkington, Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows, submitted for publication (2000).Google Scholar
  17. 17.
    R. S. Ellis, R. Jordan, and B. Turkington, A large deviation approach to soliton turbulence for the nonlinear Schrö dinger equation, in preparation (2000).Google Scholar
  18. 18.
    R. E. Ellis and J. S. Rosen, Laplace's method for Gaussian integrals with an application to statistical mechanics, Ann. Probab. 10:47–66 (1982). Correction in Ann. Probab. 11:456 (1983).Google Scholar
  19. 19.
    G. L. Eyink and H. Spohn, Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Statist. Phys. 70:833–886 (1993).Google Scholar
  20. 20.
    H. Fö llmer and S. Orey, Large deviations for the empirical field of a Gibbs measure, Ann. Probab. 16:961–977 (1987).Google Scholar
  21. 21.
    H.-O. Georgii, Large deviations and maximum entropy principle for interacting random fields on ℤd, Ann. Probab. 21:1845–1875 (1993).Google Scholar
  22. 22.
    A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems (Elsevier North-Holland, New York, 1979).Google Scholar
  23. 23.
    R. Jordan and C. Josserand, Self-organization in nonlinear wave turbulence, Phys. Rev. E 61:1527–1539 (2000).Google Scholar
  24. 24.
    R. Jordan, B. Turkington, and C. L. Zirbel, A mean-field statistical theory for the non-linear Schro- dinger equation, Physica D 137:353–378 (2000).Google Scholar
  25. 25.
    G. Joyce and D. C. Montgomery, Negative temperature states for the two-dimensional guiding center plasma, J. Plasma Phys. 10:107–121 (1973).Google Scholar
  26. 26.
    M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math. 46:27–56 (1993).Google Scholar
  27. 27.
    M. K.-H. Kiessling, Statistical Mechanics of Weakly Dissipative Current-Carrying Plasma, Habilitationsschrift (Ruhr-Universitä t Bochum, 1995).Google Scholar
  28. 28.
    M. K.-H. Kiessling and Joel L. Lebowitz, The micro-canonical point vortex ensemble: beyond equivalence, Lett. Math. Phys. 42:43–56 (1997).Google Scholar
  29. 29.
    M. K.-H. Kiessling and T. Neukirch, Negative specific heat of toroidally confined plasma, Technical report, 1997.Google Scholar
  30. 30.
    R. Kraichnan, Statistical dynamics of two-dimensional flow, J. Fluid Mech. 67:155–175 (1975).Google Scholar
  31. 31.
    O. E. Lanford, Entropy and equilibrium states in classical statistical mechanics, Statistical Mechanics and Mathematical Problems, A. Lenard, ed., Lecture Notes in Physics (Springer-Verlag, Berlin, 1973), Vol. 20, pp. 1–113.Google Scholar
  32. 32.
    J. T. Lewis, C.-E. Pfister, and W. G. Sullivan, Entropy, concentration of probability and conditional limit theorems, Markov Proc. Related Fields 1:319–386 (1995).Google Scholar
  33. 33.
    J. T. Lewis, C.-E. Pfister, and W. G. Sullivan, The equivalence of ensembles for lattice systems: Some examples and a counterexample, J. Statist. Phys. 77:397–419 (1994).Google Scholar
  34. 34.
    J. Lynch and J. Sethuraman, Large deviations for processes with independent increments, Ann. Probab. 15:610–627 (1987).Google Scholar
  35. 35.
    C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids (Springer-Verlag, New York, 1994).Google Scholar
  36. 36.
    J. Messer and H. Spohn, Statistical mechanics of the Lane-Emden equation, J. Statist. Phys. 29:561–578 (1982).Google Scholar
  37. 37.
    J. Michel and R. Robert, Large deviations for Young measures and statistical mechanics of infinite dimensional dynamical systems with conservation law, Commun. Math. Phys. 159:195–215 (1994).Google Scholar
  38. 38.
    J. Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett. 65: 2137–2140 (1990).Google Scholar
  39. 39.
    J. Miller, P. Weichman, and M. C. Cross, Statistical mechanics, Euler's equations, and Jupiter's red spot, Phys. Rev. A 45:2328–2359 (1992).Google Scholar
  40. 40.
    S. Olla, Large deviations for Gibbs random fields, Probab. Th. Related Fields 77:343–359 (1988).Google Scholar
  41. 41.
    L. Onsager, Statistical hydrodynamics, Suppl. Nuovo Cim. 6:279–287 (1949).Google Scholar
  42. 42.
    R. Robert, Concentration et entropie pour les mesures d'Young, C. R. Acad. Sci. Paris, Sé r. I 309:757–760 (1989).Google Scholar
  43. 43.
    R. Robert, A maximum-entropy principle for two-dimensional perfect fluid dynamics, J. Statist. Phys. 65:531–553 (1991).Google Scholar
  44. 44.
    R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid Mech. 229:291–310 (1991).Google Scholar
  45. 45.
    R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970).Google Scholar
  46. 46.
    S. Roelly and H. Zessin, The equivalence of equilibrium principles in statistical mechanics and some applications to large particle systems, Expositiones Mathematicae 11:384–405 (1993).Google Scholar
  47. 47.
    J. Schrö ter and R. Wegener, The problem of equivalence in statistical mechanics of equilibrium, Math. Methods Appl. Sci. 14:319–331 (1991).Google Scholar
  48. 48.
    R. A. Smith and T. M. O'Neil, Nonaxisymmetric thermal equilibria of a cylindrically bounded guiding center plasma or discrete vortex system, Phys. Fluids B 2:2961–2975 (1990).Google Scholar
  49. 49.
    W. Thirring, A Course in Mathematical Physics 4: Quantum Mechanics of Large Systems, E. M. Harrell, Transl. (Springer-Verlag, New York, 1983).Google Scholar
  50. 50.
    B. Turkington, Statistical equilibrium measures and coherent states in two-dimensional turbulence, Commun. Pure Appl. Math 52:781–809 (1999).Google Scholar
  51. 51.
    B. Turkington and N. Whitaker, Statistical equilibrium computations of coherent structures in turbulent shear layers, SIAM J. Sci. Comput. 17:1414–1433 (1996).Google Scholar
  52. 52.
    N. Whitaker and B. Turkington, Maximum entropy states for rotating vortex patches, Phys. Fluids A 6:3963–3973 (1994).Google Scholar
  53. 53.
    E. Zeidler, Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization (Springer-Verlag, New York, 1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Richard S. Ellis
    • 1
  • Kyle Haven
    • 2
  • Bruce Turkington
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherst
  2. 2.Department of Mathematics and StatisticsUniversity of MassachusettsAmherst
  3. 3.Department of Mathematics and StatisticsUniversity of MassachusettsAmherst

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