Diffusion with preferential relocation in a confining potential (2411.00641v2)

Abstract: We study the relaxation of a diffusive particle confined in an arbitrary external potential and subject to a non-Markovian resetting protocol. With a constant rate $r$, a previous time $\tau$ between the initial time and the present time $t$ is chosen from a given probability distribution $K(\tau,t)$, and the particle is reset to the position that it occupied at time $\tau$. Depending on the shape of $K(\tau,t)$, the particle either relaxes toward the Gibbs-Boltzmann distribution or toward a non-trivial stationary distribution that breaks ergodicity and depends on the initial position and the resetting protocol. From a general asymptotic theory, we find that if the kernel $K(\tau,t)$ is sufficiently localized near $\tau=0$, i.e., mostly the initial part of the trajectory is remembered and revisited, the steady state is non-Gibbs-Boltzmann. Conversely, if $K(\tau,t)$ decays slowly enough or increases with $\tau$, i.e., recent positions are more likely to be revisited, the probability distribution of the particle tends toward the Gibbs-Boltzmann state at large times. In the latter case, however, the temporal approach to the stationary state is generally anomalously slow, following for instance an inverse power law or a stretched exponential, if $K(\tau,t)$ is not too strongly peaked at the current time $t$. These findings are verified by the analysis of several exactly solvable cases and by numerical simulations.