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

TL;DR

This paper investigates the kinetic behavior of the inhomogeneous HMF model at low temperatures, revealing unusual oscillatory force correlations and a linearly increasing friction coefficient that defies Einstein relation expectations.

Contribution

It introduces a kinetic theory framework for the inhomogeneous phase of the HMF model and uncovers novel behaviors of force correlations and friction in strongly inhomogeneous conditions.

Findings

Force auto-correlation oscillates and averages to zero

Friction coefficient increases linearly with time

Friction does not satisfy Einstein relation

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.