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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , which depends on the obstacle diffusivity and density. The scaling of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , above and below a critical regime, is characterized by two independent critical parameters: the conductivity exponent µ , also found in models with frozen obstacles, and an exponent ψ , which quantifies the effect of obstacle diffusivity.