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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2411.00641 |
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| _version_ | 1866910825402335232 |
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| author | Boyer, Denis Evans, Martin R. Majumdar, Satya N. |
| author_facet | Boyer, Denis Evans, Martin R. Majumdar, Satya N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diffusion with preferential relocation in a confining potential Boyer, Denis Evans, Martin R. Majumdar, Satya N. Statistical Mechanics 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. |
| title | Diffusion with preferential relocation in a confining potential |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2411.00641 |