Majumdar Satya N, Sabhapandit Sanjib, Schehr Grégory
LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.
Raman Research Institute, Bangalore 560080, India.
Phys Rev E. 2016 Dec;94(6-1):062131. doi: 10.1103/PhysRevE.94.062131. Epub 2016 Dec 21.
We study the probability density function (PDF) of the cover time t_{c} of a finite interval of size L by N independent one-dimensional Brownian motions, each with diffusion constant D. The cover time t_{c} is the minimum time needed such that each point of the entire interval is visited by at least one of the N walkers. We derive exact results for the full PDF of t_{c} for arbitrary N≥1 for both reflecting and periodic boundary conditions. The PDFs depend explicitly on N and on the boundary conditions. In the limit of large N, we show that t_{c} approaches its average value of 〈t_{c}〉≈L^{2}/(16DlnN) with fluctuations vanishing as 1/(lnN)^{2}. We also compute the centered and scaled limiting distributions for large N for both boundary conditions and show that they are given by nontrivial N independent scaling functions.
我们研究了由(N)个独立的一维布朗运动覆盖长度为(L)的有限区间的覆盖时间(t_{c})的概率密度函数（PDF），每个布朗运动的扩散常数为(D)。覆盖时间(t_{c})是使得整个区间的每个点至少被(N)个漫步者中的一个访问所需的最短时间。对于任意(N\geq1)，我们推导了反射和周期边界条件下(t_{c})的完整PDF的精确结果。这些PDF明确依赖于(N)和边界条件。在(N)很大的极限情况下，我们表明(t_{c})趋近于其平均值(\langle t_{c}\rangle\approx L^{2}/(16D\ln N))，波动以(1/(\ln N)^{2})的速度消失。我们还计算了两种边界条件下(N)很大时的中心化和缩放极限分布，并表明它们由非平凡的与(N)无关的缩放函数给出。
