Ro Sunghan, Kim Yong Woon
Department of Physics, Technion-Israel Institute of Technology, Haifa 3200003, Israel.
Department of Physics, Korea Advanced Institute of Science and Technology, Deajeon 34141, Korea and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
Phys Rev E. 2022 Aug;106(2-1):024101. doi: 10.1103/PhysRevE.106.024101.
We consider a problem of finding a target located in a finite d-dimensional domain, using N independent random walkers, when partial information about the target location is given as a probability distribution. When N is large, the first-passage time sensitively depends on the initial searcher distribution, which invokes the question of the optimal searcher distribution that minimizes the first-passage time. Here, we analytically derive the equation for the optimal distribution and explore its limiting expressions. If the target volume can be ignored, the optimal distribution is proportional to the target distribution to the power of one third. If we consider a target of a finite volume and the probability of the initial overlapping of searchers with the target cannot be ignored in the large N limit, the optimal distribution has a weak dependence on the target distribution, with its variation being proportional to the logarithm of the target distribution. Using Langevin dynamics simulations, we numerically demonstrate our predictions in one and two dimensions.
当关于目标位置的部分信息以概率分布的形式给出时，使用(N)个独立的随机漫步者在有限的(d)维区域中寻找目标。当(N)很大时，首次通过时间敏感地依赖于初始搜索者分布，这就引出了使首次通过时间最小化的最优搜索者分布的问题。在这里，我们通过解析推导得出最优分布的方程，并探索其极限表达式。如果目标体积可以忽略不计，最优分布与目标分布的三分之一次幂成正比。如果我们考虑一个有限体积的目标，并且在(N)很大的极限情况下搜索者与目标初始重叠的概率不能忽略，那么最优分布对目标分布的依赖性较弱，其变化与目标分布的对数成正比。使用朗之万动力学模拟，我们在一维和二维中通过数值验证了我们的预测。
