Hamiltonian and Brownian systems with long-range interactions

Abstract

We discuss the dynamics and thermodynamics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system to the canonical description of a stochastically forced Brownian system. We show that the mean-field approximation is exact in a proper thermodynamic limit. The equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKY-like hierarchy. It also optimizes a thermodynamical potential (entropy or free energy) under appropriate constraints. We discuss the kinetic theory of these systems. In the $N\to +\infty$ limit, a Hamiltonian system is described by the Vlasov equation. To order 1/N, the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. We also consider the motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. This can lead to anomalous diffusion. For Brownian systems, in the $N\to +\infty$ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit $\xi \to +\infty$, or for large times $t\gg {\xi }^{-1}$, it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.

The original paper has been split in two parts with some new material and corrections

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• Chapman-Enskog derivation of the generalized Smoluchowski equation
• The stochastic process of equilibrium fluctuations, of a system with long range interactions
• Post-collapse dynamics of self-gravitating Brownian particles in D dimensions
• Generalized Fokker-Planck equations and effective thermodynamics
• Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations

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