Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edifici Cc. Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain.
Phys Rev E. 2019 Jan;99(1-1):012141. doi: 10.1103/PhysRevE.99.012141.
Stochastic resets have lately emerged as a mechanism able to generate finite equilibrium mean-square displacement (MSD) when they are applied to diffusive motion. Furthermore, walkers with an infinite mean first-arrival time (MFAT) to a given position x may reach it in a finite time when they reset their position. In this work we study these emerging phenomena from a unified perspective. On one hand, we study the existence of a finite equilibrium MSD when resets are applied to random motion with 〈x^{2}(t)〉_{m}∼t^{p} for 0<p≤2. For exponentially distributed reset times, a compact formula is derived for the equilibrium MSD of the overall process in terms of the mean reset time and the motion MSD. On the other hand, we also test the robustness of the finiteness of the MFAT for different motion dynamics which are subject to stochastic resets. Finally, we study a biased Brownian oscillator with resets with the general formulas derived in this work, finding its equilibrium first moment and MSD and its MFAT to the minimum of the harmonic potential.
随机重置最近已成为一种机制，当应用于扩散运动时，能够产生有限的平衡均方位移（MSD）。此外，当到达给定位置 x 的平均首次到达时间（MFAT）无限的 walker 重置其位置时，它们可能会在有限的时间内到达该位置。在这项工作中，我们从统一的角度研究这些新出现的现象。一方面，我们研究了当重置应用于 〈x^{2}(t)〉_{m}∼t^{p} （0<p≤2）的随机运动时，是否存在有限的平衡 MSD。对于指数分布的重置时间，我们推导出了整个过程的平衡 MSD 的紧凑公式，该公式以平均重置时间和运动 MSD 为参数。另一方面，我们还测试了不同的运动动力学在受到随机重置时，MFAT 的有限性的稳健性。最后，我们使用这项工作中推导的一般公式研究了具有重置的有偏布朗振荡器，找到了它的平衡第一矩和 MSD 以及到调和势最小值的 MFAT。
