A model of non-interacting particles in one dimension under resetting dynamics

In the framework of stochastic dynamics, resetting generally leads – since detailed balance is not enforced – to the existence of a non-equilibrium steady state (NESS). The model we studied here has already been treated in literature in the case of absence of resetting. We tried to understand how a resetting dynamics induces the reaching of the NESS and how resetting induces dynamical phase transitions. Specifically, we considered a one-dimensional system of noninteracting particles all located on the negative x-axis with a fixed density 𝜌. We make use of a general treatment to study the flux of particles through the origin. We also considered two cases: (i) diffusive dynamics with resetting and (ii) run-and-tumble dynamics with resetting. For simplicity, we enforced a Poissonian resetting rate r for all particles and supposed that each particle is reset to its initial position. The aim of the paper is to derive the probability 𝑃(𝑄, 𝑡) of having a net flux of particles through origin equal to Q at time t. In analogy with disordered systems, we computed the quantity 𝑃(𝑄, 𝑡) for both annealed and quenched initial conditions. In annealed case we can find the exact form of 𝑃𝑎𝑛(𝑄, 𝑡), which is a Poissonian distribution (both in diffusive and in run-and-tumble case) at any time, while in quenched case we were just able to find the large deviation form of 𝑃𝑞𝑢(𝑄, 𝑡). In particular, in case (i) we have a large deviation function for the probability quenched showing a discontinuity in the third derivative, symptomatic of a third order dynamical phase transition in 𝑃𝑞𝑢. On the contrary in case (ii) resetting does not lead to a phase transition.