标题： 在限制势中的偏好重新定位扩散 显示英文标题

标题： Diffusion with preferential relocation in a confining potential

摘要： 我们研究了一个被限制在任意外部势场中的扩散粒子，该粒子受到非马尔可夫重置协议的影响。 以恒定速率$r$，从给定的概率分布$K(\tau,t)$中选择一个之前的时间$\tau$，该时间介于初始时间与当前时间$t$之间，并将粒子重置到它在时间$\tau$所处的位置。 根据$K(\tau,t)$的形状，粒子要么趋向于吉布斯-玻尔兹曼分布，要么趋向于一种非平凡的平稳分布，该分布破坏了遍历性，并依赖于初始位置和重置协议。 从一般的渐近理论中，我们发现如果核$K(\tau,t)$在$\tau=0$附近足够局部化，即，轨迹的初始部分被记住并重新访问，稳态是非吉布斯-玻尔兹曼的。相反，如果$K(\tau,t)$足够缓慢地衰减或随着$\tau$增加，即，最近的位置更可能被重新访问，粒子的概率分布会在长时间后趋向于吉布斯-玻尔兹曼状态。然而在后一种情况下，到达稳态的时间过程通常异常缓慢，例如遵循反幂律或拉伸指数，如果$K(\tau,t)$在当前时间$t$处不是太尖锐地集中的话。这些发现通过几个可精确求解的情况分析和数值模拟得到了验证。 显示英文摘要

摘要： 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.