Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

TL;DR

This paper develops a kinetic theory for systems with long-range interactions, deriving a general inhomogeneous equation that accounts for memory effects and finite N correlations, extending classical plasma kinetic equations.

Contribution

It introduces a novel kinetic formalism in physical space for spatially inhomogeneous long-range systems, including collective effects and memory, beyond traditional Fourier space approaches.

Findings

Derived a general kinetic equation for inhomogeneous systems with long-range interactions.

Reproduced classical Landau and Lenard-Balescu equations in homogeneous plasma limits.

Analyzed relaxation times and collisional processes in weak long-range interacting systems.

Abstract

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit $N\to +\infty$. These correlations are responsible for the ``collisional'' evolution of the system beyond the Vlasov regime due to finite $N$ effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size and provide a coherent discussion of all the numerical results obtained for these systems.