Statistical mechanics of a neutral point-vortex gas at low energy

Esler, J. G.; Ashbee, T. L.; McDonald, N. Robb

doi:10.1103/physreve.88.012109

Cited by 17 publications

(18 citation statements)

References 27 publications

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“…To ensure that our initial vortex distribution corresponds to a positive temperature, we calculated the statistical weights by sampling M = 100, 000 realizations of a neutral vortex gas consisting of N v vortices. For each realization, the interaction energy per vortex was calculated and a probability distribution constructed from [38]…”

Section: Point Vortex Modelmentioning

confidence: 99%

Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium

Salman

Maestrini

2016

Phys. Rev. A

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We study the relaxation of a two-dimensional (2D) ultracold Bose gas from a nonequilibrium initial state containing vortex excitations in experimentally realizable square and rectangular traps. We show that the subsystem of vortex gas excitations results in the spontaneous emergence of a coherent superfluid flow with a nonzero coarse-grained vorticity field. The stream function of this emergent quasiclassical 2D flow is governed by a Poisson-Boltzmann equation. This equation reveals that maximum entropy states of a neutral vortex gas that describe the spectral condensation of energy can be classified into types of flow depending on whether or not the flow spontaneously acquires angular momentum. Numerical simulations of a neutral point vortex model and a Bose gas governed by the 2D Gross-Pitaevskii equation in a square reveal that a large-scale monopole flow field with net angular momentum emerges that is consistent with predictions of the Poisson-Boltzmann equation. The results allow us to characterize the spectral energy condensate in a 2D quantum fluid that bears striking similarity to similar flows observed in experiments of 2D classical turbulence. By deforming the square into a rectangular region, the resulting maximum entropy state switches to a dipolar flow field with zero net angular momentum.By deforming the square into a rectangular region, the resulting maximum entropy state switches to a dipolar flow field with zero net angular momentum

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Section: Point Vortex Modelmentioning

confidence: 99%

Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium

Salman

Maestrini

2016

Phys. Rev. A

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“…While nearly symmetric for small n, the scaled density converges rapidly to a skewed distribution as n increases. The scaled inverse temperature asymptotes to a fixed negative value at large positive energies [11,19,20]. There is little difference in either the density of states or the temperature when n increases beyond 200.…”

Section: Equilibrium Statisticsmentioning

confidence: 99%

Ergodicity and spectral cascades in point vortex flows on the sphere

2015

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We present results for the equilibrium statistics and dynamic evolution of moderately large [n = O(10 2 −10 3 )] numbers of interacting point vortices on the sphere under the constraint of zero mean angular momentum. For systems with equal numbers of positive and negative identical circulations, the density of rescaled energies, p(E), converges rapidly with n to a function with a single maximum with maximum entropy. Ensemble-averaged wave-number spectra of the nonsingular velocity field induced by the vortices exhibit the expected k −1 behavior at small scales for all energies. Spectra at the largest scales vary continuously with the inverse temperature of the system. For positive temperatures, spectra peak at finite intermediate wave numbers; for negative temperatures, spectra decrease everywhere. Comparisons of time and ensemble averages, over a large range of energies, strongly support ergodicity in the dynamics even for highly atypical initial vortex configurations. Crucially, rapid relaxation of spectra toward the microcanonical average implies that the direction of any spectral cascade process depends only on the relative difference between the initial spectrum and the ensemble mean spectrum at that energy, not on the energy, or temperature, of the system.

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“…(42) holds if ψ 2n+1 ef f = 0 for all n ∈ N but this condition is not always satisfied. In particular, some important solutions may be forgotten by using the sinh-Poisson equation (42) instead of the Boltzmann-Poisson equation (40), as discussed in Sec. 4.…”

Section: The Sinh-poisson Equationmentioning

confidence: 99%

“…The strong mixing (or low energy) limit allows us to make a connection between the rigorous maximum entropy principle of statistical mechanics [14] and the phenomenological minimum enstrophy principle [32] introduced for a slightly viscous flow. 17 Indeed, in this limit, the statistical theory leads to a linear ω − ψ relationship (110), like when we minimize the enstrophy at fixed energy and circulation, writing the first variations as 16 In their original paper, Debye and Hückel [6] explicitly write the multi-species Boltzmann-Poisson equation (19) and the sinh-Poisson equation (42) with β > 0 and Aa = na. They linearize these equations at high temperature leading to Eq.…”

Section: The Minimum Enstrophy Principlementioning

confidence: 99%

Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

Chavanis

2014

Eur. Phys. J. B

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We complement the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of Virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and Hückel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-Hückel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-sign vortices is attractive instead of repulsive. This leads to an organization at large scales presenting geometry-induced phase transitions, instead of Debye shielding. We compare the results obtained with point vortices to those obtained in the context of the statistical mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results but differences appear at the next order. In particular, the MRS theory predicts a transition between sinh and tanh-like ω − ψ relationships depending on the sign of Ku − 3 (where Ku is the Kurtosis) while there is no such transition for point vortices which always show a sinh-like ω − ψ relationship. We derive the form of the relaxation equations in the strong mixing limit and show that the enstrophy plays the role of a Lyapunov functional. PACS. 2 D hydrodynamics; point vortices; kinetic theory3 We generalize previous works [14,19,20,21,22] that consider, for simplicity, only one type of point vortices with circulation γ or two types of point vortices with circulation +γ and −γ. This generalization, although straightforward, is important in order to compare, in Sec. 4, the limit of strong mixing for point vortices and continuous vorticity fields. 4 The MEPP was introduced in the context of the MRS theory [29] for continuous vorticity fields. We apply it here to the point vortex gas in order to see the similarities and the differences.

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Statistical mechanics of a neutral point-vortex gas at low energy

Cited by 17 publications

References 27 publications

Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium

Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium

Ergodicity and spectral cascades in point vortex flows on the sphere

Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

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