Comptes Rendus
Thermodynamical functions for a gas of point vortices
Comptes Rendus. Mécanique, Volume 336 (2008) no. 7, pp. 565-571.

We formulate nonlinear integro-differential equation for the averaged collective Hamiltonian of a gas of interacting two-dimensional vortices, derive its analytical solution, and discuss the equilibrium, axially-symmetrical, probability distributions that are possible for such a model. We also theoretically prove that the probability distribution for a system of 2D point vortices takes a form similar to the Gibbs distribution, but point out that the physical fundamentals of such a system differ from the standard theory of interacting particles. Furthermore, we find thermodynamical functions for positive and negative “temperature” of the system, and point out that the states with positive “temperature” correspond to stationary bell-shape vortex distributions, while the states with negative “temperature” correspond to distributions localized near container walls.

Nous formulons l'équation non linéaire integro-différentielle pour l'Hamiltonien collectif moyenné d'un gas de tourbillons 2D intéragissant et trouvons sa solution analytique. Nous discutons aussi de la distribution d'équilibre de probabilité axisymétrique possible d'un tel modèle. Nous montrons également que la probabilité pour un système de tourbillons ponctuels doit prendre une forme similaire à la distribution de Gibbs. Notons que les fondaments physiques d'un tel systèm différent de la théorie standard des particules en intéraction. Nous trouvons les fonctions « thermodynamiques » pour les températures positives et négatives du système et discutons le fait que les états avec température positive correspondent à une distribution ayant le maximum central, tandis que les états avec température négative correspondent aux distributions localisées au voisinage des parois du containeur.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2008.04.003
Keywords: Dynamical systems, Hamiltonian description, Gas of localized vortices, Positive and negative “temperatures”
Mot clés : Systèmes dynamiques, Description Hamiltonienne, Gaz des tourbillons localisés, « Températures » positive et négative

Emilie Bécu 1; Vadim Pavlov 2, 3; Elizabeth P. Tito 

1 Laboratoire de mécanique de Lille – UMR 8107, 59655 Villeneuve d'Ascq cedex, France
2 UFR de mathématiques pures et appliquées, Université de Lille 1, 59655 Villeneuve d'Ascq cedex, France
3 California Institute of Technology, Pasadena, CA 91125, USA
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Emilie Bécu; Vadim Pavlov; Elizabeth P. Tito. Thermodynamical functions for a gas of point vortices. Comptes Rendus. Mécanique, Volume 336 (2008) no. 7, pp. 565-571. doi : 10.1016/j.crme.2008.04.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.04.003/

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