Kinetic equations for systems with long-range interactions: a unified description

Abstract

We complete the existing literature on the kinetic theory of systems with long-range interactions. Starting from the BBGKY hierarchy, or using projection operator technics or a quasilinear theory, a general kinetic equation can be derived when collective effects are neglected. This equation (which is not well-known) applies to possibly spatially inhomogeneous systems, which is specific to systems with long-range interactions. Interestingly, the structure of this kinetic equation bears a clear physical meaning in terms of generalized Kubo relations. Furthermore, this equation takes a very similar form for stellar systems and two-dimensional point vortices providing therefore a unified description of the kinetic theory of these systems. If we assume that the system is spatially homogeneous (or axisymmetric for point vortices), this equation can be simplified and reduces to the Landau equation (or its counterpart for point vortices). Our formalism thus offers a simple derivation of Landau-type equations. We also use this general formalism to derive a kinetic equation, written in angle-action variables, describing spatially inhomogeneous systems with long-range interactions. This new derivation solves the shortcomings of our previous derivation [P.H. Chavanis, Physica A 377, 469 (2007)]. Finally, we consider a test particle approach and derive general expressions for the diffusion and friction (or drift) coefficients of a test particle evolving in a bath of field particles. We make contact with the expressions previously obtained in the literature. As an application of the kinetic theory, we argue that the relaxation time is shorter for inhomogeneous (or high-dimensional) systems than for homogeneous (or low-dimensional) systems because there are potentially more resonances.

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Cited by in corpus (27)

• Kinetic theory of spatially inhomogeneous stellar systems without collective effects
• Quasi-stationary states in the self-gravitating sheet model
• Scaling of the dynamics of homogeneous states of one-dimensional long-range interacting systems
• Inhomogeneous Tsallis distributions in the HMF model
• Self-consistent inhomogeneous steady states in Hamiltonian mean field dynamics
• Out-of-equilibrium phase transitions in the HMF model: a closer look
• Dynamics and physical interpretation of quasi-stationary states in systems with long-range interactions
• Noise-induced dynamical phase transitions in long-range systems
• Kinetic theory of two-dimensional point vortices with collective effects
• Spectral and formal stability criteria of spatially inhomogeneous stationary solutions to the Vlasov equation for the Hamiltonian mean-field model
• Kinetic theory of $1D$ homogeneous long-range interacting systems sourced by $1/N^2$ effects
• Action diffusion and lifetimes of quasistationary states in the Hamiltonian Mean Field model
• Caloric curves fitted by polytropic distributions in the HMF model
• Scaling quasi-stationary states in long range systems with dissipation
• Thermodynamics of the HMF model with a magnetic field
• Attractor non-equilibrium stationary states in perturbed long-range interacting systems
• Strange Scaling and Temporal Evolution of Finite-Size Fluctuation in Thermal Equilibrium
• Stability of inhomogeneous states in mean-field models with a local potential
• Formation of disks with long-lived spiral arms from violent gravitational dynamics
• Stochastic treatment of finite-N effects in mean-field systems and its application to the lifetimes of coherent structures
• Generalized vortex-model for the inverse cascade of two-dimensional turbulence
• Explicit formula of energy-conserving Fokker-Planck type collision term for single species point vortex systems with weak mean flow
• Finite $N$ corrections to Vlasov dynamics and the range of pair interactions
• Second-order solutions of the equilibrium statistical mechanics for self-gravitating systems
• Quantum kinetic theory of flux-carrying Brownian particles
• Violent relaxation in one-dimensional self-gravitating system: deviation from the Vlasov limit due to finite-$N$ effect
• Expansion into the vacuum of stochastic gases with long-range interactions