Kinetic theory of spatially homogeneous systems with long-range interactions: II. Historic and basic equations

Abstract

We provide a short historic of the early development of kinetic theory in plasma physics and synthesize the basic kinetic equations describing the evolution of systems with long-range interactions derived in Paper I. We describe the evolution of the system as a whole and the relaxation of a test particle in a bath of field particles at equilibrium or out-of-equilibrium. We write these equations for an arbitrary long-range potential of interaction in a space of dimension d. We discuss the scaling of the relaxation time with the number of particles for non-singular potentials. For always spatially homogeneous distributions, the relaxation time of the system as a whole scales like N in d > 1 and like N 2 (presumably) or like e N (possibly) in d = 1. For always spatially inhomogeneous distributions, the relaxation time of the system as a whole scales like N in any dimension of space. For 1D systems undergoing a dynamical phase transition from a homogeneous to an inhomogeneous phase, we expect a relaxation time of the form N δ with 1 < δ < 2 intermediate between the two previous cases. The relaxation time of a test particle in a bath always scales like N . We also discuss the kinetic theory of systems with long-range interactions submitted to an external stochastic potential. This paper gathers basic equations that are applied to specific systems in Paper III.

Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction

“…(10), (12), and (13), we can now go back to the computation of the conserved quantities from Eq. (9). By adding 1 3 of every expression, we obtain…”

“…The derivation of kinetic equations describing the secular evolution of systems with long-range interactions has a rich history (see, e.g., the introduction of Refs. [8,9] for a short account). Landau [10] first derived a kinetic equation for Coulombian neutral plasmas by expanding the Boltzmann [11] equation in terms of a small deflection parameter, namely the velocity deviation experienced by a particle during a "collision."…”

Generalized Stochastic Fokker-Planck Equations

“…In the mean field limit (collisionless regime), valid when the number of particles N → +∞, these systems are described by the Vlasov equation. When finite N effects are taken into account (collisional regime), their evolution is governed by the Landau or by the Lenard-Balescu equation [11,12]. These equations satisfy an H-theorem for the Boltzmann entropy and relax towards the mean field Boltzmann distribution with a temperature T (E).…”

Effect of triplet correlation on the equation of pair correlation function in a weakly coupled inhomogeneous plasma system

“…If we drop the third and last term of equation ( 14), leading to equation ( 12), we get the equation previously derived by Chavanis from a BBGKY-like hierarchy for inhomogeneous and homogeneous systems with long range interactions [13,14]. If we drop the fourth and last term of equation ( 14), leading to equation (17), we get the equation discussed in section 5 of [15].…”

“…which differs from the expression given by equation (15). Equation ( 15) is an integral equation (contrary to equation ( 17)) since the function ψ 12 depends on χ itself through an integral.…”