Diffusion coefficient expression for asymmetric discrete random walk with unequal jump times, lengths, and probabilities

Random walk has wide applications in many fields, such as generative AI, biology, physics, and chemistry. Asymmetric random walk could be described by the drift-diffusion equation. Although random walk has been an extensively investigated topic, the current reported theoretical results for discrete asymmetric random walks are often limited to simple asymmetry due to unequal jump probabilities or lengths. This paper investigated more complicated asymmetric random walks. In the random walk, the particle could take three possible actions at each step: left jump, right jump, or immobile; additionally, each type of action could have different probabilities, times, and lengths. The general diffusion coefficient expressions are derived. The results indicate that the exchanges between different types of actions play critical roles in determining the diffusion coefficient. The obtained theoretical results can be reduced to reported results. The obtained results show that the diffusion coefficients are significantly different between discrete and continuous time asymmetric random walks. Additionally, discrete random walk simulations are performed to verify the obtained theoretical results. There are good agreements between the theoretical predictions and simulation results. The diffusion coefficient and probability distribution function could provide critical information for understanding dynamic processes in many systems, such as polymers or biological systems.