Diffusion with resetting in a logarithmic potential
Abstract
We study the effect of resetting on diffusion in a logarithmic potential. In this model, a particle diffusing in a potential U(x) = U0|x| is reset, i.e., taken back to its initial position, with a constant rate r. We show that this analytically tractable model system exhibits a series of phase transitions as a function of a single parameter, β U0, the ratio of the strength of the potential to the thermal energy. For β U0<-1 the potential is strongly repulsive, preventing the particle from reaching the origin. Resetting then generates a non-equilibrium steady state which is characterized exactly and thoroughly analyzed. In contrast, for β U0>-1 the potential is either weakly repulsive or attractive and the diffusing particle eventually reaches the origin. In this case, we provide a closed form expression for the subsequent first-passage time distribution and show that a resetting transition occurs at β U0=5. Namely, we find that resetting can expedite arrival to the origin when -1<β U0<5, but not when β U0>5. The results presented herein generalize results for simple diffusion with resetting -- a widely applicable model that is obtained from ours by setting U0=0. Extending to general potential strengths, our work opens the door to theoretical and experimental investigation of a plethora of problems that bring together resetting and diffusion in logarithmic potential.