二维无交互作用随机游动种群的接触统计
Contact statistics in populations of noninteracting random walkers in two dimensions
摘要：
生物种群中个体之间的相互作用、化学系统中稀释组分之间的相互作用或由湍流输送的颗粒之间的相互作用，关键取决于它们的接触统计。这项工作澄清了这些统计的简化假设下，基本的运动近似布朗随机行走，粒子可以被视为非相互作用。我们测量接触间隔（也称为等待时间或到达时间间隔），接触计数，接触持续时间分布在人口中的个人经历非相互作用的连续时空随机行走在一个周期性的二维平面（一个环面），作为人口数密度，步行半径和随机行走步长的函数。的接触间隔是指数分布的时间比平均自由碰撞时间长，但不是时间短于，和接触持续时间分布强烈的峰值在弹道交叉时间的迎面碰撞，当弹道交叉时间是短的相比，平均步骤持续时间。虽然个体之间的连续接触是独立的，但重复接触的概率在前一次接触后随着时间的推移而降低。这导致等待时间间隔的负持续时间依赖性和所有时间间隔的接触计数概率密度函数的过度分散。该文件表明，人口的小颗粒（步行者半径小的平均分离或随机行走步长）的平均自由碰撞间隔，弹道交叉时间，和随机行走的步骤持续时间可以用来构建时间尺度，允许共同的等待时间，接触计数，接触持续时间分布在不同的人口。半解析近似的等待时间和接触持续时间的分布。
英文摘要：
The interaction between individuals in biological populations, dilute components of chemical systems, or particles transported by turbulent flows depends critically on their contact statistics. This work clarifies those statistics under the simplifying assumptions that the underlying motions approximate a Brownian random walk and that the particles can be treated as noninteracting. We measure the contact-interval (also called the waiting-time or inter-arrival-time), contact-count, and contact-duration distributions in populations of individuals undergoing noninteracting continuous-space-time random walks on a periodic two-dimensional plane (a torus), as functions of the population number density, walker radius, and random-walk step size. The contact-interval is exponentially distributed for times longer than the mean-free-collision time but not for times shorter than that, and the contact duration distribution is strongly peak at the ballistic-crossing time for head-on collisions, when the ballistic-crossing time is short compared to the mean step duration. While successive contacts between individuals are independent, the probability of repeat contact decreases with time after a previous contact. This leads to a negative duration dependence of the waiting-time interval and over dispersion of the contact-count probability density function for all time intervals. The paper demonstrates that for populations of small particles (walker radius small compared to the mean-separation or random-walk step size) the mean-free-collision interval, the ballistic-crossing time, and the random-walk-step duration can be used to construct temporal scalings which allow for common waiting-time, contact-count, and contact-duration distributions across different populations. Semi-analytic approximations for both the waiting-time and contact-duration distributions are also presented.
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