Kinetic theory of stellar systems and two-dimensional vortices

We discuss the kinetic theory of stellar systems and two-dimensional vortices and stress their analogies. We recall the derivation of the Landau and Lenard–Balescu equations from the Klimontovich formalism. These equations take into account two-body correlations and are valid at the order 1/N, where N is the number of particles in the system. They have the structure of a Fokker–Planck equation involving a diffusion term and a drift term. The systematic drift of a vortex is the counterpart of the dynamical friction experienced by a star. At equilibrium, the diffusion and the drift terms balance each other establishing the Boltzmann distribution of statistical mechanics. We discuss the problem of kinetic blocking in certain cases and how it can be solved at the order 1/N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/N^2$$\end{document} by the consideration of three-body correlations. We also consider the behaviour of the system close to the critical point following a recent suggestion by Hamilton and Heinemann (2023). We present a simple calculation, valid for spatially homogeneous systems with long-range interactions described by the Cauchy distribution, showing how the consideration of the Landau modes regularizes the divergence of the friction by polarization at the critical point. We mention, however, that fluctuations may be very important close to the critical point and that deterministic kinetic equations for the mean distribution function (such as the Landau and Lenard–Balescu equations) should be replaced by stochastic kinetic equations.