Test particles in a gas: Markovian and non-Markovian Langevin dynamics

The description of the behaviour of test particles in a gas through Markovian and non-Markovian Langevin dynamics is critically examined. In the Markovian case, in the Maxwell interaction model, it is shown that the correct test-particle mean-energy relaxation can be obtained from the (exact) Langevin equation only through a rather complicated fluctuating-force autocorrelation function exactly reducing to the standard one only in the Rayleigh-gas limit. In addition, except that in this limit, the second fluctuation-dissipation theorem should then be modified. These results pose the non-easy problem of the appropriate choice of fluctuating-force autocorrelation function and memory kernel in the more general non-Markovian case in non-equilibrium conditions. So, the effective possibility of an alternative Markovian approach avoiding such difficulties is also considered. Moreover, the evolution (Fokker-Planck) equations in velocity-, phase-, and position-space, both in Markovian and in non-Markovian cases, are briefly discussed. Furthermore, given the importance of the memory kernel in the non-Markovian dynamics, the average test-particle motion is discussed when an exponentially decaying memory kernel is used in the generalized Langevin equation. Finally, the exigence of further theoretical and/or experimental investigations is stressed.