Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model.

M Hidalgo-Soria,E Barkai,S Burov

doi:10.3390/e23020231

Abstract

We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants, , with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements for . Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state to a -function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in . We demonstrate how super-statistical framework is a zeroth order short time expansion of , in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Highlights

The emergence of non-Gaussian features for the positional probability density function (PDF) of particle spreading, denoted P(x, t), in a disordered environment is a common attribute that arises in many different physical and biological systems
Theoretical frameworks describing this behavior emerged from continuous time random walk (CTRW) approaches employing large deviations theory [22,23,24] and microscopical models like molecular dynamics of tracer particles in polymer networks [25,26] and interacting particles with fluctuating sizes [27,28,29], the so-called Hitchhiker model [28]
How different is the distribution of the diffusivities, extracted via time average mean squared displacement (TAMSD) techniques, in a two state model compared with the one present in single molecule experiments? As we show this will be determined by the values of D± and τ ±

Summary

Introduction

The emergence of non-Gaussian features for the positional probability density function (PDF) of particle spreading, denoted P(x, t), in a disordered environment is a common attribute that arises in many different physical and biological systems. A tent like shape of the PDF, in the semi-log scale, together with a linear time dependence of the mean square displacement (MSD) appear for diffusion in glassy system [1], biological cells [2,3,4,5,6,7], and colloidal suspensions [8,9,10,11] This tent shape, sometimes fitted with a Laplace distribution P(x, t) ∼ exp(−C|x|) with C a constant, suggests that the decay of the PDF is exponential. Theoretical frameworks describing this behavior emerged from continuous time random walk (CTRW) approaches employing large deviations theory [22,23,24] and microscopical models like molecular dynamics of tracer particles in polymer networks [25,26] and interacting particles with fluctuating sizes [27,28,29], the so-called Hitchhiker model [28]

Results

Discussion

Conclusion