Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows

Abstract

Statistical equilibrium models of coherent structures in two-dimensional and barotropic quasi-geostrophic turbulence are formulated using canonical and microcanonical ensembles, and the equivalence or nonequivalence of ensembles is investigated for these models. The main results show that models in which the global invariants are treated microcanonically give richer families of equilibria than models in which they are treated canonically. Such global invariants are those conserved quantities for ideal dynamics which depend on the large scales of the motion; they include the total energy and circulation. For each model a variational principle that characterizes its equilibrium states is derived by invoking large deviations techniques to evaluate the continuum limit of the probabilistic lattice model. An analysis of the two different variational principles resulting from the canonical and microcanonical ensembles reveals that their equilibrium states coincide only when the microcanonical entropy function is concave. These variational principles also furnish Lyapunov functionals from which the nonlinear stability of the mean flows can be deduced. While in the canonical model the well-known Arnold stability theorems are reproduced, in the microcanonical model more refined theorems are obtained which extend known stability criteria when the microcanonical and canonical ensembles are not equivalent. A numerical example pertaining to geostrophic turbulence over topography in a zonal channel is included to illustrate the general results.

An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

“…The works of these authors represent also the primary sources of information for the theory of macrostate nonequivalence of ensembles. Various illustrations of this theory, dealing with statistical models of turbulence, can be found in [26,27]. We mention finally our recent work [28] on the mean-field BlumeEmery-Griffiths spin model which can be consulted as an easily accessible introduction to the material surveyed in this paper.…”

Dynamics and thermodynamics of axisymmetric flows: Theory

“…Such functionals are exactly conserved by the axisymmetric equations (they are particular Casimirs). Therefore, as in 2D hydrodynamics [28], the maximization of S at fixed energy E, helicity H, circulation Γ and angular momentum I determines a nonlinearly dynamically stable stationary solution of the axisymmetric Euler equations. This refined stability criterion is stronger than the maximization of J = S − βE − µH − νΓ − αI which just provides a sufficient condition of formal nonlinear dynamical stability [11].…”

“…The difference between these two criteria is similar to a notion of ensemble inequivalence in thermodynamics (where S plays the role of an entropy and J the role of a free energy) [20,29]. We shall not prove the nonlinear dynamical stability result in this paper and refer to [28] for a precise discussion in 2D hydrodynamics. In Sec.…”

“…They are determined by substituting (32) in the constraints (27) and (28). Plugging the optimal currents in (26) we get…”

“…(27)- (28). Following the MEPP, we maximizė S withĖ =Ḣ = 0, the normalization constraint (79) and the additional constraints…”