Scaling study of diffusion in dynamic crowded spaces

TL;DR

This paper develops a scaling theory for long-time diffusion in crowded spaces with moving obstacles, revealing how obstacle density and mobility influence effective diffusion, with implications across multiple dimensions.

Contribution

It introduces a novel scaling framework that accounts for obstacle mobility and density, extending previous models with frozen obstacles to dynamic environments.

Findings

Effective diffusion constant depends on obstacle density and diffusivity.

Two critical exponents, μ and ψ, characterize the scaling behavior.

Diffusive motion exhibits anomalous behavior before reaching steady state.

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.