Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model

Abstract

We present first elements of kinetic theory appropriate to the inhomogeneous phase of the HMF model. In particular, we investigate the case of strongly inhomogeneous distributions for $T\to 0$ and exhibit curious behaviour of the force auto-correlation function and friction coefficient. The temporal correlation function of the force has an oscillatory behaviour which averages to zero over a period. By contrast, the effects of friction accumulate with time and the friction coefficient does not satisfy the Einstein relation. On the contrary, it presents the peculiarity to increase linearly with time. Motivated by this result, we provide analytical solutions of a simplified kinetic equation with a time dependent friction. Analogies with self-gravitating systems and other systems with long-range interactions are also mentioned.

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• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• Lynden-Bell and Tsallis distributions for the HMF model
• Kinetic equations for systems with long-range interactions: a unified description
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Kinetic theory with angle-action variables
• Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
• Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model
• Kinetic theory of two dimensional point vortices from a BBGKY-like hierarchy
• Kinetic theory of two-dimensional point vortices with collective effects
• Classical Goldstone modes in Long-Range Interacting Systems