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Local diffusion coefficients in disordered systems such as spin glass systemsand living cells are highly heterogeneous and may change over time. Such atime-dependent and spatially heterogeneous environment results inirreproducibility of single-particle-tracking measurements. Irreproducibilityof time-averaged observables has been theoretically studied in the context ofweak ergodicity breaking in stochastic processes. Here, we provide rigorousdescriptions of equilibrium and non-equilibrium diffusion processes for theannealed transit time model, which is a heterogeneous diffusion model in livingcells. We give analytical solutions for the mean square displacement (MSD) andthe relative standard deviation of the time-averaged MSD for equilibrium andnon-equilibrium situations. We find that the time-averaged MSD grows linearlywith time and that the diffusion coefficients are intrinsically random innon-equilibrium situations. Our findings pave the way for a theoreticalunderstanding of distributional behavior of the diffusion coefficients indisordered systems.

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