# Packets of Diffusing Particles Exhibit Universal Exponential Tails

## TL;DR

This paper demonstrates that packets of diffusing particles exhibit a universal exponential tail in their positional probability density, extending the Large Deviations approach to a broad class of transport problems in random media.

## Contribution

It introduces a universal exponential decay behavior for particle density in random media by extending Large Deviations theory to continuous time random walks.

## Key findings

- Exponential tails are observed in various physical systems.
- Universal behavior holds for both short and long times.
- Finite-time fluctuations cause exponential decay in particle density.

## 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.