Eur. Phys. J. B 52, 47-59 (2006)
https://doi.org/10.1140/epjb/e2006-00267-9

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

Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France

Corresponding author: ^a chavanis@irsamc.ups-tlse.fr

Received: 4 October 2005
Published online: 6 July 2006

Abstract

We present first elements of kinetic theory appropriate to the inhomogeneous phase of the Hamiltonian Mean Field (HMF) model. In particular, we investigate the case of strongly inhomogeneous distributions for T→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.