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 [1]. 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 also holds for short times, a fact that makes experimental observations readily achievable. Time permitting, we will discuss the Hitchhiker model [2] which gives a microscopical description of the observed behavior, in terms of a model describing aggregation processes, that leads to a distribution of molecule sizes and to spatial exponential decay of the spreading particles.
[1] E, Barkai, Stas Burov, Packets of diffusing particles exhibit universal exponential tails, Physical Review Letters 124, 060603 (2020).
[2] M. Hidalgo-Soria, and E. Barkai The Hitchhiker model for Laplace diffusion processes in the cell environment, Physical Review E 102, 012109 (2020).