Asymmetric discrete random walk and drift-diffusion with unequal jump times, lengths, and probabilities

TL;DR

This paper develops a comprehensive theoretical framework for asymmetric discrete random walks with unequal jump times, lengths, and probabilities, and verifies the results through simulations, extending existing models.

Contribution

It introduces new theoretical expressions for asymmetric random walks with unequal parameters, generalizing previous models that assumed equal jump times.

Findings

Theoretical expressions match simulation results accurately.

Results reduce to known models when jump times are equal.

Provides a basis for understanding biased drift-diffusion processes.

Abstract

Random walk has wide applications in many fields, such as machine learning, biology, physics, and chemistry. Random walk can be discrete or continuous in time and space. Asymmetric random walk could be described by drift-diffusion equation. The discrete asymmetric random walk provides a basis for understanding biased drift-diffusion. However, the current reported theoretical results for discrete random walks do not give a general theoretical treatment for the asymmetry due to unequal jump times. In this paper, the theoretical expressions for asymmetric random walks with unequal jump probabilities, times, and lengths are derived. The obtained theoretical results can be reduced to reported results when jump times are equal. Additionally, discrete random walk simulations are performed to verify the obtained theoretical results. There are good agreements between the theoretical predictions and simulation results.