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Classical motion in force fields with short range correlations. (English) Zbl 1187.82106

Summary: We study the long time motion of fast particles moving through time-dependent random force fields with correlations that decay rapidly in space, but not necessarily in time. The time dependence of the averaged kinetic energy \(\langle p ^{2}(t)\rangle /2\) and mean-squared displacement \(\langle q ^{2}(t)\rangle \) is shown to exhibit a large degree of universality; it depends only on whether the force is, or is not, a gradient vector field. When it is, \(\langle p ^{2}(t)\rangle \sim t ^{2/5}\) independently of the details of the potential and of the space dimension. The stochastically accelerated particle motion is then superballistic in one dimension, with \(\langle q ^{2}(t)\rangle \sim t ^{12/5}\), and ballistic in higher dimensions, with \(\langle q ^{2}(t)\rangle \sim t ^{2}\). These predictions are supported by numerical results in one and two dimensions. For force fields not obtained from a potential field, the power laws are different: \(\langle p ^{2}(t)\rangle \sim t ^{2/3}\) and \(\langle q ^{2}(t)\rangle \sim t ^{8/3}\) in all dimensions \(d\geq 1\).

MSC:

82C70 Transport processes in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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