Eur. Phys. J. Plus (2013) 128: 126
https://doi.org/10.1140/epjp/i2013-13126-9

Regular Article

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

Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse, F-31062, Toulouse, France

^* e-mail: chavanis@irsamc.ups-tlse.fr

Received: 27 March 2013
Accepted: 8 September 2013
Published online: 25 October 2013

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 N² (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.