论文标题
一维运行的极端统计和带有吸收墙的滚动粒子
Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
论文作者
论文摘要
我们在一个维度上研究运行粒子（RTP）的极端价值统计数据，直到首次从位置开始$ x_0〜（> 0）$开始到原点。由于其在对某些细菌的运动建模中的生物学应用中，该模型最近引起了很多兴趣。本文中，我们在有限的间隔中分析了单个RTP的确切时间依赖性传播器，其两端具有吸收条件。通过利用路径分解技术，我们适当地使用这些繁殖器来计算最大排量$ m $的联合分配$ \ mathscr {p}（m，t_m）$，直到第一通量和时间$ t_m $。相应的边缘分布$ \ mathbb {p} _m（m）$和$ p_m（t_m）$被分开研究并进行数值验证。特别是，我们发现边际分配$ p_m（t_m）$具有有趣的大小$ t_m $的有趣渐近形式。对于小$ t_m $，分布$ p_m（t_m）$敏感地取决于初始速度方向$σ_i$，并且与布朗尼运动完全不同，大的$ t_m $ $ $ $ $ p_m（t_m）$与布朗尼运动相同，尽管良好的储蓄率最初取决于最初的条件$ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ =我们通过数值模拟将所有分析结果验证为高精度。
We study the extreme value statistics of a run and tumble particle (RTP) in one dimension till its first passage to the origin starting from the position $x_0~(>0)$. This model has recently drawn a lot of interest due to its biological application in modelling the motion of certain species of bacteria. Herein, we analytically study the exact time-dependent propagators for a single RTP in a finite interval with absorbing conditions at its two ends. By exploiting a path decomposition technique, we use these propagators appropriately to compute the joint distribution $\mathscr{P}(M,t_m)$ of the maximum displacement $M$ till first-passage and the time $t_m$ at which this maximum is achieved exactly. The corresponding marginal distributions $\mathbb{P}_M(M)$ and $P_M(t_m)$ are studied separately and verified numerically. In particular, we find that the marginal distribution $P_M(t_m)$ has interesting asymptotic forms for large and small $t_m$. While for small $t_m$, the distribution $P_M(t_m)$ depends sensitively on the initial velocity direction $σ_i$ and is completely different from the Brownian motion, the large $t_m$ decay of $P_M(t_m)$ is same as that of the Brownian motion although the amplitude crucially depends on the initial conditions $x_0$ and $σ_i$. We verify all our analytical results to high precision by numerical simulations.
