论文标题
当地时间奔跑和翻滚粒子
Local time for run and tumble particle
论文作者
论文摘要
我们研究了当地时间$（t_ {loc}）$统计量，并在一个维度不均匀的介质中滚动粒子。通过考虑$ r（x）=γ\ frac {| x |^α} {l^α} $带有$α\ geq 0 $的位置{| x |^α} $的位置依赖性率来引入不均匀性。对于$α= 0 $，我们得出了$ t_ {loc} $的概率分布，该分布完全表示为一系列$δ$符合词，其中系数可以解释为粒子对粒子对来源的多次重新审视的概率，从原点开始。对于一般$α$，我们表明，$ t_ {loc} $的典型波动随时间缩放为$ t_ {loc} \ sim t^{\ frac {\ frac {1+α} {2+α} {2+α}} $对于大$ t $，其概率分布及其概率分布具有按比例分配功能所描述的缩放函数，我们计算出了我们已经计算出的分析。在第二部分中，我们研究$ t_ {loc} $的统计信息，直到RTP首次通过$ x = m〜（> 0）$。在这种情况下，我们还表明，对于所有$ α〜（\ geq 0）$的$δ$符合的概率分布可以表示为$δ$符合的串联总和，并且来自适当的退出问题的系数。我们所有的分析结果都得到了数值模拟的支持。
We investigate the local time $(T_{loc})$ statistics for a run and tumble particle in an one dimensional inhomogeneous medium. The inhomogeneity is introduced by considering the position dependent rate of the form $R(x) = γ\frac{|x|^α}{l^α}$ with $α\geq 0$. For $α=0$, we derive the probability distribution of $T_{loc}$ exactly which is expressed as a series of $δ$-functions in which the coefficients can be interpreted as the probability of multiple revisits of the particle to the origin starting from the origin. For general $α$, we show that the typical fluctuations of $T_{loc}$ scale with time as $T_{loc} \sim t^{\frac{1+α}{2+α}}$ for large $t$ and their probability distribution possesses a scaling behaviour described by a scaling function which we have computed analytically. In the second part, we study the statistics of $T_{loc}$ till the RTP makes a first passage to $x=M~(>0)$. In this case also, we show that the probability distribution can be expressed as a series sum of $δ$-functions for all values of $α~(\geq 0)$ with coefficients appearing from appropriate exit problems. All our analytical findings are supported with the numerical simulations.
