Kinetic theory of two dimensional point vortices from a BBGKY-like hierarchy

Abstract

Starting from the Liouville equation, we derive the exact hierarchy of equations satisfied by the reduced distribution functions of the single species point vortex gas in two dimensions. Considering an expansion of the solution in powers of 1/N in a proper thermodynamic limit $N\to +\infty$, and neglecting some collective effects, we derive a kinetic equation satisfied by the smooth vorticity field which is valid at order $O(1/N)$. This equation was obtained previously [P.H. Chavanis, Phys. Rev. E, 64, 026309 (2001)] from a more abstract projection operator formalism. If we consider axisymmetric flows and make a markovian approximation, we obtain a simpler kinetic equation which can be studied in great detail. We discuss the properties of these kinetic equations in regard to the $H$-theorem and the convergence (or not) towards the statistical equilibrium state. We also study the growth of correlations by explicitly calculating the time evolution of the two-body correlation function in the linear regime. In a second part of the paper, we consider the relaxation of a test vortex in a bath of field vortices and obtain the Fokker-Planck equation by directly calculating the second (diffusion) and first (drift) moments of the increment of position of the test vortex. A specificity of our approach is to obtain general equations, with a clear physical meaning, that are valid for flows that are not necessarily axisymmetric and that take into account non-Markovian effects. A limitations of our approach, however, is that it ignores collective effects.

References in corpus (5)

• Statistics of 3-dimensional Lagrangian turbulence
• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• On the properties of steady states in turbulent axisymmetric flows
• Statistical mechanics of two-dimensional vortices and stellar systems
• Slow relaxation in the two dimensional electron plasma under the strong magnetic field

Cited by in corpus (6)

• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Two-dimensional Brownian vortices
• Decay of Time Correlations in Point Vortex Systems