Kinetic theory of Onsager's vortices in two-dimensional hydrodynamics

Abstract

Starting from the Liouville equation, and using a BBGKY-like hierarchy, we derive a kinetic equation for the point vortex gas in two-dimensional (2D) hydrodynamics, taking two-body correlations and collective effects into account. This equation is valid at the order 1/N where N>>1 is the number of point vortices in the system (we assume that their individual circulation scales like γ~ 1/N). It gives the first correction, due to graininess and correlation effects, to the 2D Euler equation that is obtained for $N\to +\infty$. For axisymmetric distributions, this kinetic equation does not relax towards the Boltzmann distribution of statistical equilibrium. This implies either that (i) the "collisional" (correlational) relaxation time is larger than Nt_D, where t_D is the dynamical time, so that three-body, four-body... correlations must be taken into account in the kinetic theory, or (ii) that the point vortex gas is non-ergodic (or does not mix well) and will never attain statistical equilibrium. Non-axisymmetric distributions may relax towards the Boltzmann distribution on a timescale of the order Nt_D due to the existence of additional resonances, but this is hard to prove from the kinetic theory. On the other hand, 2D Euler unstable vortex distributions can experience a process of "collisionless" (correlationless) violent relaxation towards a non-Boltzmannian quasistationary state (QSS) on a very short timescale of the order of a few dynamical times. This QSS is possibly described by the Miller-Robert-Sommeria (MRS) statistical theory which is the counterpart, in the context of two-dimensional hydrodynamics, of the Lynden-Bell statistical theory of violent relaxation in stellar dynamics.

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Cited by in corpus (13)

• Secular diffusion in discrete self-gravitating tepid discs I : analytic solution in the tightly wound limit
• Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction
• Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit
• Classical Heisenberg spins with long-range interactions: Relaxation to equilibrium for finite systems
• Geometric microcanonical theory of two-dimensional Truncated Euler flows
• Chiral Edge Modes in Helmholtz-Onsager Vortex Systems
• Functional integral derivation of the kinetic equation of two-dimensional point vortices
• Explicit formula of energy-conserving Fokker-Planck type collision term for single species point vortex systems with weak mean flow
• Properties and evolution of anisotropic structures in collisionless plasmas
• Functional integral approach to the kinetic theory of inhomogeneous systems
• Decay of Time Correlations in Point Vortex Systems
• The Turbulent 'Mixing' Layer as a Problem in the Non-equilibrium Statistical Mechanics of a Vortex Gas
• Dynamics of point-vortex type systems near thermal equilibrium: relaxation or not?