Classical Motion in Force Fields with Short Range Correlations

Abstract

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 p 2 (t) /2 and mean-squared displacement q 2 (t) 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, p 2 (t) ∼ t 2/5 independently of the details of the potential and of the space dimension. Motion is then superballistic in one dimension, with q 2 (t) ∼ t 12/5 , and ballistic in higher dimensions, with q 2 (t) ∼ 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: p 2 (t) ∼ t 2/3 and q 2 (t) ∼ t 8/3 in all dimensions d ≥ 1.

Equilibration, Generalized Equipartition, and Diffusion in Dynamical Lorentz Gases

“…Note that n * (p) in (5.6) behaves asα −1 . This is as expected, since it was shown in [ABLP10] that, whenα = 0, not only does the particle's speed not decrease, it actually increases without bound as p n ∼ n 1/6 as a function of the collision number n. But, as shown in [ABLP10], for initially fast particles of momentum p, this stochastic acceleration begins to manifest itself only after n s (p) ∼ p 6 collisions. Thus, the time scale n * (p) ∼ p 4 over which dynamical friction manifests itself by slowing down fast particles, and which characterizes the equilibration process, is always much shorter than is required by the relatively slow process of stochastic acceleration.…”

“…A more careful computation, carried out in [ABLP10] verifies that this is indeed the leading order contribution to the correlation function. Hence, we write…”

“…Our analysis closely follows the approach of [ABLP10], where the caseα = 0 was treated. At the end of the section we compare the behavior of the mean-squared displacement in these two very different situations.…”

“…Starting from the first equation of (2.6) and (5.1), a simple computation [ABLP10] yields the result e n+1 = e n + δ n , (5.3) where…”

“…Approach to equilibrium, although observed numerically, was not addressed in that paper. Our work here benefits from the insights gained since in [ABLP10], where the related but very different problem of the motion of fast particles in a random time-dependent potential was studied. The potentials considered in that paper, while also generated by an array of isolated time-evolving scatterers, are non-reactive (or inert), in that they do not respond to the particle's passage.…”