  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.
