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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
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).
V. I. Arnold, Mathematical Methods of Classical Mechanics, K. Vogtmann and A. Weinstein, Transl. (Springer-Verlag, New York, 1978).
V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics (Springer-Verlag, New York, 1998).
R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (John Wiley & Sons, New York, 1975).
T. Bodineau and A. Guionnet, About the stationary states of vortex systems, Ann. Inst. Henri Poincaré 35:205–237 (1999).
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).
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).
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).
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).
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed. (Springer-Verlag, New York, 1998).
J.-D. Deuschel and D. W. Stroock, Large Deviations (Academic Press, Boston, 1989).
J.-D. Deuschel, D. W. Stroock, and H. Zessin, Microcanonical distributions for lattice gases, Commun. Math. Phys. 139:83–101 (1991).
M. DiBattista and A. Majda, An equilibrium statistical theory for large-scale features of open-ocean convection, to appear in J. Phys. Oceanography (2000).
M. DiBattista, A. Majda, and B. Turkington, Prototype geophysical vortex structures via large-scale statistical theory, Geophys. Astrophys. Fluid Dyn. 89:235–283 (1998).
P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations (John Wiley & Sons, New York, 1997).
R. S. Ellis, K. Haven, and B. Turkington, Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows, submitted for publication (2000).
R. S. Ellis, R. Jordan, and B. Turkington, A large deviation approach to soliton turbulence for the nonlinear Schrö dinger equation, in preparation (2000).
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).
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).
H. Fö llmer and S. Orey, Large deviations for the empirical field of a Gibbs measure, Ann. Probab. 16:961–977 (1987).
H.-O. Georgii, Large deviations and maximum entropy principle for interacting random fields on ℤd, Ann. Probab. 21:1845–1875 (1993).
A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems (Elsevier North-Holland, New York, 1979).
R. Jordan and C. Josserand, Self-organization in nonlinear wave turbulence, Phys. Rev. E 61:1527–1539 (2000).
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).
G. Joyce and D. C. Montgomery, Negative temperature states for the two-dimensional guiding center plasma, J. Plasma Phys. 10:107–121 (1973).
M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math. 46:27–56 (1993).
M. K.-H. Kiessling, Statistical Mechanics of Weakly Dissipative Current-Carrying Plasma, Habilitationsschrift (Ruhr-Universitä t Bochum, 1995).
M. K.-H. Kiessling and Joel L. Lebowitz, The micro-canonical point vortex ensemble: beyond equivalence, Lett. Math. Phys. 42:43–56 (1997).
M. K.-H. Kiessling and T. Neukirch, Negative specific heat of toroidally confined plasma, Technical report, 1997.
R. Kraichnan, Statistical dynamics of two-dimensional flow, J. Fluid Mech. 67:155–175 (1975).
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.
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).
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).
J. Lynch and J. Sethuraman, Large deviations for processes with independent increments, Ann. Probab. 15:610–627 (1987).
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids (Springer-Verlag, New York, 1994).
J. Messer and H. Spohn, Statistical mechanics of the Lane-Emden equation, J. Statist. Phys. 29:561–578 (1982).
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).
J. Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett. 65: 2137–2140 (1990).
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).
S. Olla, Large deviations for Gibbs random fields, Probab. Th. Related Fields 77:343–359 (1988).
L. Onsager, Statistical hydrodynamics, Suppl. Nuovo Cim. 6:279–287 (1949).
R. Robert, Concentration et entropie pour les mesures d'Young, C. R. Acad. Sci. Paris, Sé r. I 309:757–760 (1989).
R. Robert, A maximum-entropy principle for two-dimensional perfect fluid dynamics, J. Statist. Phys. 65:531–553 (1991).
R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid Mech. 229:291–310 (1991).
R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970).
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).
J. Schrö ter and R. Wegener, The problem of equivalence in statistical mechanics of equilibrium, Math. Methods Appl. Sci. 14:319–331 (1991).
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).
W. Thirring, A Course in Mathematical Physics 4: Quantum Mechanics of Large Systems, E. M. Harrell, Transl. (Springer-Verlag, New York, 1983).
B. Turkington, Statistical equilibrium measures and coherent states in two-dimensional turbulence, Commun. Pure Appl. Math 52:781–809 (1999).
B. Turkington and N. Whitaker, Statistical equilibrium computations of coherent structures in turbulent shear layers, SIAM J. Sci. Comput. 17:1414–1433 (1996).
N. Whitaker and B. Turkington, Maximum entropy states for rotating vortex patches, Phys. Fluids A 6:3963–3973 (1994).
E. Zeidler, Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization (Springer-Verlag, New York, 1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ellis, R.S., Haven, K. & Turkington, B. Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles. Journal of Statistical Physics 101, 999–1064 (2000). https://doi.org/10.1023/A:1026446225804
Issue Date:
DOI: https://doi.org/10.1023/A:1026446225804