Packets of Diffusing Particles Exhibit Universal Exponential Tails

Abstract

Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function P (X, t) of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the Large Deviations approach for a continuous time random walk we uncover a general universal behavior for the decay of the density. It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to exponential decay (with logarithmic corrections) of P (X, t). This universal behavior holds also for short times, a fact that makes experimental observations readily achievable. *

Large Deviations for Continuous Time Random Walks

“…As expected, in the limit , we find the exponential decay again where we used the asymptotic behavior of the Lambert function, i.e., with . The claim in [ 28 ] is much more general as a similar behavior is found for a large class of waiting times and jump lengths PDFs, see details below.…”

“…Generally, the solution of the model is given by Here the sum is over the possible outcomes of the number of jumps in the process, is the probability of attaining n jumps [ 31 ], while is the probability of finding the particle on x conditioned it made n jumps. In Laplace space [ 28 ], where s is the Laplace pair of t . In particular, when , is called the survival probability.…”

“…Here the goal is to consider special choices of jump length distributions and waiting time PDFs which allow us to express the solution as an infinite sum over n explicitly, i.e., without restoring to inverse Fourier and Laplace transforms and/or numerical simulations. This allows us to compare between exact solutions of the problem, and the large deviation theory we have promoted previously [ 28 ]. Further, with the specific choices of the input distributions and we can derive large deviations theory in a straight forward way.…”

“…In this regard, two of us have found a general solution of the problem [ 28 ]. The large n limit of is controlled by the small limit of the waiting time PDF, since to have many jumps within a finite fixed time t , the time intervals between jumps must be made small.…”

“…In [ 4 ] it was suggested that a specific type of continuous time random walk (CTRW) model could explain the physics of the observed behavior. Two of us have recently shown that the phenomenon is indeed universal as it holds in general [ 28 ] and under very mild conditions (see below). The goal of this paper is to produce further evidence for the phenomenon, present exact solutions of the model and compare it with the new theory.…”