论文标题
在广义运行和翻滚过程中的位置分布
Position distribution in a generalised run and tumble process
论文作者
论文摘要
我们研究了类型$ \ frac {d^n x} {dt^n} = v_0 \，σ（t）$的随机过程，其中$ n> 0 $是一个正整数，$σ（t）= \ pm 1 $代表从一个状态到一个状态的电视噪声，从一个状态到另一个状态到另一个状态到另一个状态$ $ $ $。对于$ n = 1 $，它可以减少到一个维度颗粒的标准运行和翻滚过程。可以在分析上继续进行此过程，包括任何$ n> 0 $，包括非整数值。我们准确地计算了所有$ n> 0 $的时间$ t $的平均平方位移，并在后期增长为$ \ sim t^{2n-1} $，对于$ n> 1/2 $，它将接近$ n <1/2 $的常数。在边缘情况下，$ n = 1/2 $，随着时间的流逝，它的生长非常缓慢，为$ \ sim \ ln t $。因此，该过程在$ n = 1/2 $下进行{\ em局部化}过渡。我们还表明，即使在$ n \ ge 1/2 $的后期，位置分配$ p_n（x，t）$也仍然取决于时间，但是接近$ n <1/2 $的固定时间独立的形式。后期分布的尾部显示出较大的偏差形式，$ p_n（x，t）\ sim \ exp \ left [-γ\，t \，φ_n\ left（\ frac {x} {x} {x^*（x^*（t）} \ right）我们对所有$ n> 0 $分析计算速率函数$φ_n（z）$，并使用重要性采样方法来计算，在它们之间找到了极好的一致性。对于三个特殊值$ n = 1 $，$ n = 2 $和$ n = 1/2 $，我们始终计算位置分布的确切累积生成函数。
We study a class of stochastic processes of the type $\frac{d^n x}{dt^n}= v_0\, σ(t)$ where $n>0$ is a positive integer and $σ(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to the other with a constant rate $γ$. For $n=1$, it reduces to the standard run and tumble process for active particles in one dimension. This process can be analytically continued to any $n>0$ including non-integer values. We compute exactly the mean squared displacement at time $t$ for all $n>0$ and show that at late times while it grows as $\sim t^{2n-1}$ for $n>1/2$, it approaches a constant for $n<1/2$. In the marginal case $n=1/2$, it grows very slowly with time as $\sim \ln t$. Thus the process undergoes a {\em localisation} transition at $n=1/2$. We also show that the position distribution $p_n(x,t)$ remains time-dependent even at late times for $n\ge 1/2$, but approaches a stationary time-independent form for $n<1/2$. The tails of the position distribution at late times exhibit a large deviation form, $p_n(x,t)\sim \exp\left[-γ\, t\, Φ_n\left(\frac{x}{x^*(t)}\right)\right]$, where $x^*(t)= v_0\, t^n/Γ(n+1)$. We compute the rate function $Φ_n(z)$ analytically for all $n>0$ and also numerically using importance sampling methods, finding excellent agreement between them. For three special values $n=1$, $n=2$ and $n=1/2$ we compute the exact cumulant generating function of the position distribution at all times $t$.
