Abstract

The motion of active matter, like bacteria and other tiny particles, has been studied extensively using mathematical models such as persistent and run-and-tumble random walks. These models incorporate a memory element, making it more likely for the walker to move in the same direction as their previous step. In this talk, I will explore various random-walk models, with a specific focus on calculating the large-deviation function. This function helps us understand how the end-to-end distance scales with the number of steps taken. Additionally, I will consider its Laplace transform, the so-called Langevin function providing insight into the relationship between force and extension. When persistence is present, there are some unexpected phenomena that are absent in walks without memory. Firstly, in on-lattice random walks with persistence in spatial dimension three or larger, two new inflexion points appear in the Langevin function. This suggests an initial softening phase before the usual stiffening, which occurs beyond a critical force level. For off-lattice random walks with persistence and run-and-tumble walks in spatial dimension larger than four, the large deviation function undergoes a first-order phase transition. In the corresponding force-versus-extension relation, this transition manifests as the attainment of complete extension at a finite force magnitude. Analytically, the origin of this phenomenology bears many similarities with the calculation of the partition function of an ideal quantum boson gas, and the phase transitions found have the same mathematical origin than the Bose-Einstein condensation.