 We examine the contact statistics of noninteracting random walkers on a two-dimensional plane  . We find that the waiting time between successive contacts for an individual (the interarrival time ) is non-exponentially distributed, with consequent non-Poison distributed contact counts. This manifests as over-dispersion of the contact-count probability mass function, caused by negative duration dependence of the waiting time interval when the walkers are in close proximity. While successive contacts  are independent, the probability of repeat contact between individuals decreases with time after a previous contact. Further, contact duration depends sensitively on walker radius. For walker radii smaller than both the mean step length and the mean nearest neighbor separation, the probability density of the contact duration is strongly peaked at the ballistic crossing time for head-on collisions. The contact statistics of individuals in biological populations, dilute components of chemical systems , or Lagrangian particles in turbulent flows depends critically on the behavior at close proximity , whether it is attractive, repulsive, or neutral. This work clarifies those statistics for the neutral case, under the assumption that the underlying motions approximate a collection of random walkers, and we suggest it may be useful in understanding contagion in human populations .
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 weakly interacting. We examine the contact interval, count, and duration in populations of individuals undergoing noninteracting continuous-space-time random walks on a periodic two-dimensional plane , as a function of the population number density, walker radius, and random-walk step size. The mean-free-collision and the ballistic-crossing times are key parameters. 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 results presented are most relevant to systems in which particle proximity is critical and contact results in no change (or a low probability of change) in the particles' motions, or to systems in which multiple contact encounters are required before interaction .
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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 weakly interacting.
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statistics of noninteracting random walkers on a
interval, count, and duration in populations of individuals undergoing noninteracting continuous-space-time random walks on a periodic
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, as a function of the population number density, walker radius, and random-walk step size. The mean-free-collision and the ballistic-crossing times are key parameters. 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
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We find that the waiting time between successive contacts for an individual (the interarrival time ) is non-exponentially distributed, with consequent non-Poison distributed contact counts. This manifests as over-dispersion of the contact-count probability mass function, caused by negative duration dependence of the waiting time interval when the walkers are in close proximity.
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between individuals
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between individuals
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Further, contact duration depends sensitively on walker radius. For walker radii smaller than both the mean step length and the mean nearest neighbor separation, the probability density of the contact duration is strongly peaked at the ballistic crossing time for head-on collisions. The contact statistics of individuals in biological populations, dilute components of chemical systems , or Lagrangian particles in turbulent flows depends critically on the behavior at close proximity , whether it is attractive, repulsive, or neutral. This work clarifies those statistics for the neutral case, under the assumption that the underlying motions approximate a collection of random walkers, and we suggest it may be useful in understanding contagion in human populations
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 results presented are most relevant to systems in which particle proximity is critical and contact results in no change (or a low probability of change) in the particles' motions, or to systems in which multiple contact encounters are required before interaction