School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, and The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel.
Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan.
Phys Rev E. 2019 Oct;100(4-1):040101. doi: 10.1103/PhysRevE.100.040101.
The canonical Evans-Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a nontrivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the steady state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely nontrivial, our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.
经典的 Evans-Majumdar 扩散模型带有随机原点重置，假设重置时间为零：在重置时，扩散粒子被瞬移回原点，重新开始运动。然而，在现实中，从一个地方到另一个地方需要花费有限的时间，这必须被考虑到，因为扩散与重置已经成为物理和其他领域无数过程的模型。在这里，我们考虑在重置时，扩散粒子以有限（而不是无限）速度回到原点的情况。这会在粒子在重置时刻的随机位置与其返回时间之间产生耦合，进一步导致运动的两个不同阶段之间的非平凡交叉对话：扩散阶段和返回阶段。我们表明，这些阶段中的每一个都以独特的方式达到稳态；尽管这也可能使总弛豫动力学变得非常复杂，但我们的分析令人惊讶地揭示了相反的情况。事实上，描述粒子在我们模型中的位置的时变分布对返回速度完全不变。因此，无论返回速度是慢还是快，我们始终恢复对原点进行瞬时返回的扩散的原始结果。
