  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.
