Conservative random walks in confining potentials

Abstract

Lévy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version Lévy walks move with a finite speed. Here, we present an extension of the Lévy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of Lévy walk processes and will be of use in a variety of systems, for which the particles are externally confined.

Deterministic and randomized motions in single-well potentials

“…Densities given by Eqs. (5) and (6) are similar to the arcsine distribution with modes located at V (x) = E for p(x) or mv 2 /2 = E for p(v), see Fig. 1.…”

“…We study similarities and differences between 1D and 2D models with particular attention to the consequences of the fact that orbits in 2D single-well potentials do not need to be closed. (2) and [5]. Solid lines present exact results, see Eqs.…”

“…(5) and (6) were derived in [5] using semi-analytical and numerical methods. The detailed, universal, analytical derivation of general formulas (5) and (6) is presented in Appendix A. In Dybiec et al [5], the 1D deterministic motion governed by Eq.…”

“…The detailed, universal, analytical derivation of general formulas (5) and (6) is presented in Appendix A. In Dybiec et al [5], the 1D deterministic motion governed by Eq. (1) with fixed initial condition is perturbed by hard velocity reversals which do not break the energy conservation.…”

“…The action of additive noise, which is not counterbalanced by some other mechanism, e.g., dissipation, breaks both mean and instantaneous energy conservation [3,4]. Nevertheless, a special class of disturbances in dynamical systems can still preserve the energy of the system, see Dybiec et al [5]. For instance, the disturbance which reverses the velocity vector at random instants of time does not break the energy conservation at all.…”