  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.
