Abstract
This work provides an analytical model for the diffusive motion of particles in a heterogeneous environment where the diffusivity varies with position. The model for diffusivity describes the environment as being homogeneous with randomly positioned pockets of larger diffusivity. This general framework for heterogeneity is amenable to a systematic expansion of the Green's function, and we employ a diagrammatic approach to identify common terms in this expansion. Upon collecting a common family of these diagrams, we arrive at an analytical expression for the particle Green's function that captures the spatially varying diffusivity. The resulting Green's function is used to analyze anomalous diffusion and kurtosis for varying levels of heterogeneity, and we compare these results with numerical simulations to confirm their validity. These results act as a basis for analysis of a range of diffusive phenomena in heterogeneous materials and living cells.
Highlights
Brownian motion of microscopic objects [1–3] is a ubiquitous phenomenon that plays a significant role in virtually all molecular processes
We explore the statistical behavior of particle motion in a heterogeneous environment, as defined by our random diffusivity model (Equation 10)
To validate our analytical theory, we perform numerical simulations of particle diffusion in a heterogeneous environment based on our model for microscopic heterogeneity
Summary
Brownian motion of microscopic objects [1–3] is a ubiquitous phenomenon that plays a significant role in virtually all molecular processes. Our work provides an analytical approach to determining the Green’s function for diffusion in a spatially varying environment. We develop a systematic approach to determining the diffusivity-averaged Green’s function over a range of degrees of heterogeneity.
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