论文标题
随机重置下的随机加速过程
Random acceleration process under stochastic resetting
论文作者
论文摘要
我们考虑在随机重置机制下，在一个维度中随机加速粒子的运动。 Denoting the position and velocity by $x$ and $v$ respectively, we consider two different resetting protocols - (i) complete resetting: here both $x$ and $v$ reset to their initial values $x_0$ and $v_0$ at a constant rate $r$, (ii) partial resetting: here only $x$ resets to $x_0$ while $v$ evolves without interruption.为了完全重置，我们发现粒子在$ x $和$ v $中都达到固定状态。我们计算了$ x $和$ v $的非平衡联合固定状态，还研究了分配功能的较晚时间。另一方面，对于部分重置，联合分布始终处于瞬态状态。在很大的$ t $上，位置分配具有缩放行为$（x/ \ sqrt {t}）$，我们严格地得出。接下来，我们研究了原始壁的第一个通行时间属性。为了进行完整的重置，我们发现平均第一个通道时间是通过重置机制渲染有限的。我们明确地得出了平均第一个通道时间的表达方式，并且以$ t $的价格得出了生存概率。但是，与之形成鲜明对比的是，对于部分重置，我们发现重置并没有使有限的第一个通行时间。这是因为，即使将$ x $带到$ x_0 $，但$ v $（$ \ sim \ sqrt {t} $）中的大波动可能会使粒子远离原点。我们所有的分析结果都通过数值模拟来证实。
We consider the motion of a randomly accelerated particle in one dimension under stochastic resetting mechanism. Denoting the position and velocity by $x$ and $v$ respectively, we consider two different resetting protocols - (i) complete resetting: here both $x$ and $v$ reset to their initial values $x_0$ and $v_0$ at a constant rate $r$, (ii) partial resetting: here only $x$ resets to $x_0$ while $v$ evolves without interruption. For complete resetting, we find that the particle attains stationary state in both $x$ and $v$. We compute the non-equilibrium joint stationary state of $x$ and $v$ and also study the late time relaxation of the distribution function. On the other hand, for partial resetting, the joint distribution is always in the transient state. At large $t$, the position distribution possesses a scaling behaviour $(x/ \sqrt{t})$ which we rigorously derive. Next, we study the first passage time properties with an absorbing wall at the origin. For complete resetting, we find that the mean first passage time is rendered finite by the resetting mechanism. We explicitly derive the expressions for the mean first passage time and the survival probability at large $t$. However, in stark contrast, for partial resetting, we find that resetting does not render finite mean first passage time. This is because even though $x$ is brought to $x_0$, the large fluctuation in $v$ ($\sim \sqrt{t}$) can take the particle substantially far from the origin. All our analytic results are corroborated by the numerical simulations.
