Diffusion with resetting in a logarithmic potential

We study the effect of resetting on diffusion in a logarithmic potential. In this model, a particle diffusing in a potential U(x) = U₀ log |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 transitions as a function of a single parameter, βU₀, the ratio of the strength of the potential to the thermal energy. For βU₀ < -1, the potential is strongly repulsive, preventing the particle from reaching the origin. Resetting then generates a non-equilibrium steady state, which is exactly characterized and thoroughly analyzed. In contrast, for βU₀ > -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 βU₀ = 5. Namely, we find that resetting can expedite arrival to the origin when -1 < βU₀ < 5, but not when βU₀ > 5. The results presented herein generalize the 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.