Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows

Bouchet, Freddy; Barré, Julien; Venaille, Antoine

doi:10.1063/1.2839113

Cited by 2 publications

(8 citation statements)

References 91 publications

(222 reference statements)

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“…Examples of such systems include galaxies and globular clusters [56][57][58][59][60][61][62][63][64], two-dimensional and geophysical flows and vortex models [18,[65][66][67][68][69][70], quantum spin models [71], dipolar excitons [72], cold atom models [73], as well as magnetically confined plasmas [23,[74][75][76]. In order to predict the behavior of systems with short-range forces we can rely on thermodynamics and statistical mechanics both of which, however, fail for systems with LR interactions.…”

Section: Systems With Long Range Forcesmentioning

confidence: 99%

Nonequilibrium statistical mechanics of systems with long-range interactions

et al. 2014

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Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary state (qSS), the lifetime of which diverges with the number of particles. Therefore, in thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle-wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.

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Section: Systems With Long Range Forcesmentioning

confidence: 99%

Nonequilibrium statistical mechanics of systems with long-range interactions

et al. 2014

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“…The study of the statistical mechanics of systems with long-range interactions has been a very active branch of statistical mechanics over the past ten years (see articles in proceedings and reviews [59,44,38,17,37], among others). In two-dimensional and geophysical flows, unusual thermodynamic properties related to long range interactions have also been observed [178,109,33,69,70,197] and their consequence for the stability theory [70] and related phase transitions [16] has been discussed.…”

Section: Long Range Interactions and Possible Statistical Ensemble Nomentioning

confidence: 99%

“…We know from observations of the associated physical phenomena (coarsening, phase separations, and so on) that the system has a tendency to form larger and larger sub-domains. We thus assume that such sub-domains are delimited by interfaces, with typical radius of curvature r much larger than R 17 . Actually the term R 2 (∇φ ) 2 is negligible except on an interface of width R separating the sub-domains.…”

Section: First Order Phase Transition and Phase Separationmentioning

confidence: 99%

“…The conservation of the distribution γ (σ ) is equivalent to the conservation of all Casimir's functionals (17). The domain averaged potential vorticity Γ, the enstrophy G 2 and the other moments of the potential vorticity G n are Casimirs of a particular interest…”

Section: Casimir's Conservation Lawsmentioning

confidence: 99%

“…The first observation of ensemble inequivalence have been made in the astrophysical context[128,92], and then observed for two dimensional flows[178,109,75,69,197]. Thorough study of ensemble inequivalence in the broad class of systems with long range interactions has been addressed during the last decade by many others, see for instance[59,44,38,17,37,16] and references therein.…”

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Statistical mechanics of two-dimensional and geophysical flows

Bouchet¹,

Venaille²

2012

Physics Reports

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256

329

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The theoretical study of the self-organization of two-dimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a self-contained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to applications to Jupiter's troposphere and ocean vortices and jets. Emphasize has been placed on examples with available analytical treatment in order to favor better understanding of the physics and dynamics.After a brief presentation of the 2D Euler and quasi-geostrophic equations, the specificity of two-dimensional and geophysical turbulence is emphasized. The equilibrium microcanonical measure is built from the Liouville theorem. Important statistical mechanics concepts (large deviations, mean field approach) and thermodynamic concepts (ensemble inequivalence, negative heat capacity) are briefly explained and described.On this theoretical basis, we predict the output of the long time evolution of complex turbulent flows as statistical equilibria. This is applied to make quantitative models of twodimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the GulfStream, and ocean vortices. A detailed comparison between these statistical equilibria and real flow observations is provided.We also present recent results for non-equilibrium situations, for the studies of either the relaxation towards equilibrium or non-equilibrium steady states. In this last case, forces and dissipation are in a statistical balance; fluxes of conserved quantity characterize the system and microcanonical or other equilibrium measures no longer describe the system.

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Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows

Cited by 2 publications

References 91 publications

Nonequilibrium statistical mechanics of systems with long-range interactions

Nonequilibrium statistical mechanics of systems with long-range interactions

Statistical mechanics of two-dimensional and geophysical flows

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