A memory-based random walk model to understand diffusion in crowded heterogeneous environment

We study memory-based random walk models to understand diffusive motion in crowded heterogeneous environments. The models considered are non-Markovian as the current move of the random walk is determined by randomly selecting a move from history. At each step, particle can take right, left or stay moves which is correlated with the randomly selected past step. There is a perfect stay–stay correlation which ensures that the particle does not move if the randomly selected past step is a stay move. The probability of traversing the same direction as the chosen history or reversing it depends on the current time and the time or position of the history selected. The time- or position-dependent biasing in moves implicitly corresponds to the heterogeneity of the environment and dictates the long-time behavior of the dynamics that can be diffusive, sub or superdiffusive. A combination of analytical solution and Monte Carlo (MC) simulation of different random walk models gives rich insight on the effects of correlations on the dynamics of a system in heterogeneous environment.