Scaling study of diffusion in dynamic crowded spaces

Abstract

We formulate a scaling theory for the long-time diffusive motion in a space occluded by a high density of moving obstacles in dimensions 1, 2 and 3. Our tracers diffuse anomalously over many decades in time, before reaching a diffusive steady state with an effective diffusion constant $D_\mathrm{eff}$, which depends on the obstacle diffusivity and density. The scaling of $D_\mathrm{eff}$, above and below a critical regime, is characterized by two independent critical parameters: the conductivity exponent $\mu$, also found in models with frozen obstacles, and an exponent $\psi$, which quantifies the effect of obstacle diffusivity.