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We present first elements of kinetic theory appropriate to the inhomogeneousphase of the HMF model. In particular, we investigate the case of stronglyinhomogeneous distributions for $T\to 0$ and exhibit curious behaviour of theforce auto-correlation function and friction coefficient. The temporalcorrelation function of the force has an oscillatory behaviour which averagesto zero over a period. By contrast, the effects of friction accumulate withtime and the friction coefficient does not satisfy the Einstein relation. Onthe contrary, it presents the peculiarity to increase linearly with time.Motivated by this result, we provide analytical solutions of a simplifiedkinetic equation with a time dependent friction. Analogies withself-gravitating systems and other systems with long-range interactions arealso mentioned

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