Scaling study of 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 study diffusive motion in a space with a high density of moving obstacles in one, two and three dimensions. We employ a generalisation of the commonly used Swiss-cheese model, which we study with extensive numerical simulations. We find that 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 characterise the critical point governing the anomalous diffusion using two exponents, one of which is the traditional conductivity exponent µ, also found in models with frozen obstacls, and the other of which we call ψ. For dimensions one through three we find ψ = 0, ψ = 0.274(2) and ψ = 0.520(3), respectively.

Together, µ and ψ can be used to encode the quantitative behavior of the system for a wide range of obstacle densities and diffusivities. This scaling is expected to be universal and covers several orders of magnitude in the mobility of the obstacles, so it should be directly relevant to the interpretation of cellular transport and other experiments on crowded spaces.