Diffusion with preferential relocation in a confining potential

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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 τ between the initial time and the present time t is chosen from a given probability distribution K ( τ , t ) , and the particle is reset to the position that it occupied at time τ. Depending on the shape of K ( τ , 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 ( τ , t ) is sufficiently localized near τ = 0, i.e. mostly the initial part of the trajectory is remembered and revisited, the steady state is non-Gibbs–Boltzmann. Conversely, if K ( τ , t ) decays slowly enough or increases with τ, 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 ( τ , 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.

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