Diffusion in dynamic crowded spaces

Brownian motion in disordered media is now well understood in the case of immobile hard obstacles. In many practical applications, however, the space itself can be dynamic. An important example is transport inside the cell, a very crowded environment with obstacles of varying sizes and complicated shapes that are constantly being rearranged. This situation has received comparatively little attention. With the ever-increasing quality of microscopy techniques, allowing for the tracking of particles inside living cells, the need for a quantitative model is clear.

Here we propose an extension of commonly used "Swiss-cheese" models to include moving obstacles and study it with numerical simulations in one, two and three dimensions. The motion of our tracer particles is anomalous over many decades in time, before reaching a diffusive steady state with an effective diffusion constant that depends on the obstacle density and diffusivity. Moreover, we find that the scaling behaviour of the effective diffusivity, above and below a critical regime at the percolation point for void space, can be characterised by two critical exponents: the conductivity μ, also found in models with frozen obstacles, and a new exponent ψ that quantifies the effect of the obstacle diffusivity.