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Article Dans Une Revue Physica A: Statistical Mechanics and its Applications Année : 2008

Two-dimensional Brownian vortices

Résumé

We introduce a stochastic model of two-dimensional Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (plasma case) and at negative temperatures, like-sign vortices attract each other (gravity case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N-body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N, where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for $N\to +\infty$. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.

Dates et versions

hal-00419412 , version 1 (23-09-2009)

Identifiants

Citer

Pierre-Henri Chavanis. Two-dimensional Brownian vortices. Physica A: Statistical Mechanics and its Applications, 2008, 387 (28), pp.6917-6942. ⟨10.1016/j.physa.2008.09.019⟩. ⟨hal-00419412⟩
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