论文标题
随机重置与空间相关的扩散：第一步研究
Space-dependent diffusion with stochastic resetting: A first-passage study
论文作者
论文摘要
我们探讨了随机重置对存在恒定偏置的空间依赖性扩散的第一步性能的影响。在我们的分析可处理的模型系统中，一种粒子在线性电位$ u（x）\proptoμ| x | $中的扩散，具有空间变化的扩散系数$ d（x）= d_0 | x | $进行随机重置，即，随机的$ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ r $ $ $ $。考虑到放置在$ x_a <x_0 $的吸收边界，我们首先得出了在拉普拉斯空间中扩散粒子的生存概率的精确表达，然后探索其对原点的第一步，作为该一般结果的限制情况。在限制$ x_a \ to0 $中，我们得出了基础过程的首次邮递时间分布的精确分析表达式。一旦引入重置，就会观察到系统以唯一参数为$ν=（1+μd_0^{ - 1}）$显示一系列动态跃迁，从而捕获了漂移和扩散的相互作用。根据$ν$构建完整的图表，我们表明，对于$ν<0 $，即当电位强烈排斥时，粒子永远无法达到原点。相比之下，对于弱势或有吸引力的潜力（$ν> 0 $），它最终达到了起源。当$ν<3 $时，重置会加速此类第一盘，但阻碍了$ν> 3 $的完成。因此，在$ν= 3 $下观察到重置过渡，我们提供了相同的全面分析。本研究为一系列理论和实验著作铺平了道路，这些著作将随机重置与保守的力场中的不均匀扩散相结合。
We explore the effect of stochastic resetting on the first-passage properties of space-dependent diffusion in presence of a constant bias. In our analytically tractable model system, a particle diffusing in a linear potential $U(x)\proptoμ|x|$ with a spatially varying diffusion coefficient $D(x)=D_0|x|$ undergoes stochastic resetting, i.e., returns to its initial position $x_0$ at random intervals of time, with a constant rate $r$. Considering an absorbing boundary placed at $x_a<x_0$, we first derive an exact expression of the survival probability of the diffusing particle in the Laplace space and then explore its first-passage to the origin as a limiting case of that general result. In the limit $x_a\to0$, we derive an exact analytic expression for the first-passage time distribution of the underlying process. Once resetting is introduced, the system is observed to exhibit a series of dynamical transitions in terms of a sole parameter, $ν=(1+μD_0^{-1})$, that captures the interplay of the drift and the diffusion. Constructing a full phase diagram in terms of $ν$, we show that for $ν<0$, i.e., when the potential is strongly repulsive, the particle can never reach the origin. In contrast, for weakly repulsive or attractive potential ($ν>0$), it eventually reaches the origin. Resetting accelerates such first-passage when $ν<3$, but hinders its completion for $ν>3$. A resetting transition is therefore observed at $ν=3$, and we provide a comprehensive analysis of the same. The present study paves the way for an array of theoretical and experimental works that combine stochastic resetting with inhomogeneous diffusion in a conservative force-field.
