From diffusion in compartmentalized media to non-Gaussian random walks

Stanislav Burov,Jakub Ślęzak

doi:10.1038/s41598-021-83364-0

Abstract

In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the non-Gaussian diffusion that exhibits linear or power-law growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly placed, identical barriers. Using this model of a heterogeneous medium we derive a general class of random walks with simple jump rules that are dictated by the geometry of the compartments. Exponential decay of positional probability density is observed and we also quantify the significant decrease of the long time diffusion constant. Our results suggest that the observed exponential decay is a general feature of the transient regime in compartmentalized media.

Highlights

In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the non-Gaussian diffusion that exhibits linear or power-law growth of the mean square displacement joined by the exponential shape of the positional probability density
The Langevin theory of diffusion investigates the time scales lower than those required by the central limit theorem[8] which leads to the motions in which the deviations from Brownian motion appear in the memory structure; the probability density of motion is still Gaussian, only with a different scale
The methods which we used in the former section were successful in revealing the broad tails of the studied diffusion, but it would be beneficial to show the exponential decay of the probability density function (PDF) d irectly[78]

Summary

Transiently confined diffusion with locally Brownian dynamics

The solution itself can be split into symmetric and antisymmetric terms which evolve independently with reflective (Neumann), ∂pXS (0+, t)/∂x = 0 , and radiation (Robin), ∂pXA(0+, t)/∂x = 2κpXA(0+, t) boundary conditions Both can be solved using multiple standard methods. ΤI2n,d.e.e.d=d, LEejxapy(sκh)otwheedptrhoacteisfswwehtiackhe a is series of independent and identically distributed (i.i.d.) random times the reflected Brownian motion on the one side of the barrier until lt (x0) > τ1 , the reflected Brownian motion on the other side until lt (x0) > τ1 + τ2 and so on, has a PDF which solves (6) and (7) This makes sense if we come back to the derivation: for a barrier with a finite thickness x the particle must explore the inside of the barrier for the time long enough it will reach the other side.

From the transient confinement to a random walk

Jk generated by jumps

Bessel and gamma waiting times CTRWs

Compound Poisson approximation

Using the tail asymptotic

Discussion

Additional information