Exploring run-and-tumble movement in confined settings through simulation

Motion in bounded domains is a fundamental concept in various fields, including billiard dynamics and random walks on finite lattices, with important applications in physics, ecology and biology. An important universal property related to the average return time to the boundary, the Mean Path Length Theorem (MPLT), has been proposed theoretically and confirmed experimentally in various contexts. In this discussion, we investigate a wide range of mechanisms that lead to deviations from this universal behavior, such as boundary effects, reorientation and memory processes. In particular, this study investigates the dynamics of run-and-tumble particles within a confined two-dimensional circular domain. Through a combination of theoretical approaches and numerical simulations, we validate the MPLT under uniform and isotropic particle inflow conditions. The research demonstrates that although the MPLT is generally applicable for different step length distributions, deviations occur for non-uniform angular distributions, non-elastic boundary conditions or memory processes. These results underline the crucial influence of boundary interactions and angular dynamics on the behaviour of particles in confined spaces. Our results provide new insights into the geometry and dynamics of motion in confined spaces and contribute to a better understanding of a broad spectrum of phenomena ranging from the motion of bacteria to neutron transport. This type of analysis is crucial in situations where inhomogeneity occurs, such as multiple real-world scenarios within a limited domain. This bridges the gap between theoretical models and practical applications in biological and physical systems since the study of the statistics of movement in confined settings can bring some light in explaining the mobility mechanism of active agents.