Diffusion on percolating clusters

Calculating the Fickian diffusivity for a lattice-based random walk with agents and obstacles of different shapes and sizes

Random walk models are often used to interpret experimental observations of the motion of biological cells and molecules. A key aim in applying a random walk model to mimic an in vitro experiment is to estimate the Fickian diffusivity (or Fickian diffusion coefficient), D. However, many in vivo experiments are complicated by the fact that the motion of cells and molecules is hindered by the presence of obstacles. Crowded transport processes have been modeled using repeated stochastic simulations in which a motile agent undergoes a random walk on a lattice that is populated by immobile obstacles. Early studies considered the most straightforward case in which the motile agent and the obstacles are the same size. More recent studies considered stochastic random walk simulations describing the motion of an agent through an environment populated by obstacles of different shapes and sizes. Here, we build on previous simulation studies by analyzing a general class of lattice-based random walk models with agents and obstacles of various shapes and sizes. Our analysis provides exact calculations of the Fickian diffusivity, allowing us to draw conclusions about the role of the size, shape and density of the obstacles, as well as examining the role of the size and shape of the motile agent. Since our analysis is exact, we calculate D directly without the need for random walk simulations. In summary, we find that the shape, size and density of obstacles has a major influence on the exact Fickian diffusivity. Furthermore, our results indicate that the difference in diffusivity for symmetric and asymmetric obstacles is significant.

Blind and myopic ants in heterogeneous networks

The diffusion processes on complex networks may be described by different Laplacian matrices due to heterogeneous connectivity. Here we investigate the random walks of blind ants and myopic ants on heterogeneous networks: While a myopic ant hops to a neighbor node every step, a blind ant may stay or hop with probabilities that depend on node connectivity. By analyzing the trajectories of blind ants, we show that the asymptotic behaviors of both random walks are related by rescaling time and probability with node connectivity. Using this result, we show how the small eigenvalues of the Laplacian matrices generating the two random walks are related. As an application, we show how the return-to-origin probability of a myopic ant can be used to compute the scaling behaviors of the Edwards-Wilkinson model, a representative model of load balancing on networks.