Distributional Behavior of Diffusion Coefficients Obtained by Single Trajectories in Annealed Transit Time Model

TL;DR

This paper provides a rigorous theoretical analysis of the distributional behavior of diffusion coefficients in disordered systems, focusing on the annealed transit time model relevant to living cells, and offers analytical solutions for key diffusion metrics.

Contribution

It introduces analytical solutions for the mean square displacement and the variability of the time-averaged MSD in both equilibrium and non-equilibrium states of the annealed transit time model.

Findings

Time-averaged MSD grows linearly with time.

Diffusion coefficients are inherently random in non-equilibrium.

Theoretical framework for distributional behavior of diffusion in disordered systems.

Abstract

Local diffusion coefficients in disordered systems such as spin glass systems and living cells are highly heterogeneous and may change over time. Such a time-dependent and spatially heterogeneous environment results in irreproducibility of single-particle-tracking measurements. Irreproducibility of time-averaged observables has been theoretically studied in the context of weak ergodicity breaking in stochastic processes. Here, we provide rigorous descriptions of equilibrium and non-equilibrium diffusion processes for the annealed transit time model, which is a heterogeneous diffusion model in living cells. We give analytical solutions for the mean square displacement (MSD) and the relative standard deviation of the time-averaged MSD for equilibrium and non-equilibrium situations. We find that the time-averaged MSD grows linearly with time and that the diffusion coefficients are intrinsically random in non-equilibrium situations. Our findings pave the way for a theoretical understanding of distributional behavior of the diffusion coefficients in disordered systems.