Stochastic particle acceleration and statistical closures

Abstract

In a recent paper, Maasjost and Elsässer (ME) concluded, from the results of numerical experiments and heuristic arguments, that the Bourret and the direct-interaction approximation (DIA) are “of no use in connection with the stochastic acceleration problem” because (1) their predictions were equivalent to that of the simpler Fokker-Planck (FP) theory, and (2) either all or none of the closures were in good agreement with the data. Here some analytically tractable cases are studied and used to test the accuracy of these closures. The cause of the discrepancy (2) is found to be the highly non-Gaussian nature of the force used by ME, a point not stressed by them. For the case where the force is a position-independent Ornstein-Uhlenbeck (i.e., Gaussian) process, an effective Kubo numberK can be defined. ForK≪1 an FP description is adequate, and conclusion (1) of ME follows; however, forK>1 the DIA behaves much better qualitatively than the other two closures. For the non-Gaussian stochastic force used by ME, all common approximations fail, in agreement with (2).