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Contact statistics in populations of noninteracting random walkers in two dimensions

Mark Peter Rast
Phys. Rev. E 105, 014103 – Published 3 January 2022
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  • INTRODUCTION
  • RANDOM-WALK MODEL
  • CONTACT INTERVAL
  • CONTACT DURATION
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    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 interarrival-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 ballistic mean-free-collision time but not for times shorter than that, and the contact duration distribution is strongly peaked 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 overdispersion of the contact-count probability density function for all time intervals. The paper demonstrates that for populations of small particles (with a walker radius that is small compared to the mean-separation or random-walk step size), the ballistic 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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    • Received 30 June 2021
    • Revised 11 August 2021
    • Accepted 9 December 2021

    DOI:https://doi.org/10.1103/PhysRevE.105.014103

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Population dynamicsRandom walksTransport phenomena
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Mark Peter Rast

    • Department of Astrophysical and Planetary Sciences, Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80309, USA

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    Vol. 105, Iss. 1 — January 2022

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    • Figure 1
      Figure 1

      Normalized probability densities p(τ) of the time interval τ between two successive contacts for any individual random walker in a population of walkers. Time is measured in units of mean step duration Δt. Distributions below the uppermost one are offset vertically by factors of one-tenth for clarity. Cases differ only in walker number density (n=[4,10,25,100,400,1600] top to bottom, blue to brown). They share the same mean step-size (Δr=0.0251) and walker radii (a=0.0005). The nearly indistinguishable overlapping distributions second from top are from simulations with identical n conducted separately in 1×1 and 3×3 domains. These were undertaken to test the waiting-time distribution sensitivity to domain periodicity. Overlapping distributions fifth from the top result from populations having the same product of nΔr but differing in n and Δr individually (see the main text). The exponential ballistic mean-free-collision interval distribution is overplotted with solid lines for the three cases with highest walker number densities [nearly indistinguishable from the actual distribution in the highest number density case (bottommost, shown brown)]. An analytic approximation (see Sec. 3b) to the low number density waiting-time distribution (uppermost blue plot) is indicated with a solid black curve.

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    • Figure 2
      Figure 2

      Probability density p(τ/τ¯b) of the waiting time τ (with time measured in units of mean step duration) scaled by the ballistic mean-free-collision interval τ¯b=1/(4nav¯), with relative velocity v¯=2Δr/Δt. Cases differ in walker number density n=[4,10,25,100,400,1600], but share the same mean step-size Δr=0.0251 and walker radius a=0.0005. Color scheme and underlying simulations match those of Fig. 1, but here they are difficult to distinguish after the τ scaling. An analytic approximation (Sec. 3b) to the waiting-time distribution is indicated with an underlying solid black curve. Note that while plotted over the full abscissal range, this solution is only valid for times very short compared to τ¯b. The inset replicates the plots for two pairs of random-walk simulations. The lower pair illustrates the dependence on domain periodicity at small number densities, and the upper pair (offset vertically) displays the small mismatch between simulations with the same value of nΔr but with differing number density n and mean step length Δr individually (see the main text).

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    • Figure 3
      Figure 3

      Probability density p(Nt) of the number of contacts Nt experienced by a walker in time interval t. Simulation runs and color coding are the same as those of Figs. 1 and 2, with cases differing in the walker number density n=[4,10,25,100,400,1600] (shown blue and brown, left to right in the upper panel) but sharing the same mean step-size Δr=0.0251 and walker radii a=0.0005. In the upper panel, count distributions are plotted for an interval of 2000 steps. In the lower panel, count distributions are plotted for four different time intervals scaled by the ballistic mean-free-collision time: approximately 11, 5.7, 2.8, and 0.57τ¯b (offset top to bottom). For reference these correspond to 40 000, 20 000, 10 000, and 2000 random-walk steps in the n=4 simulation (shown blue and purple in 3×3 and 1×1 domains, respectively) and 100, 50, 25, and 5 steps in the n=1600 run (shown brown). Poisson distributions based on the mean Nt values are overplotted with solid-dotted curves in both panels.

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    • Figure 4
      Figure 4

      Waiting-time distributions for simulations that share the same step size and number density but differ in particle size (interaction distance). Colors indicate distributions (top to bottom, blue to brown) obtained from simulations with walker radii a≈[0.004,0.04,0.08,0.2,0.4]Δr or equivalently about [0.00071,0.0071,0.014,0.035,0.071] times the mean nearest-neighbor distance between walkers. The inset shows waiting-time distributions for simulations with a≈[0.6,2.8]Δr or about [0.1,0.5] times the mean nearest-neighbor distance, gray and yellow, respectively. Waiting time in the upper panel is scaled by the ballistic mean-free-collision time τ¯b and in the lower panel by τ¯btb, where tb is the ballistic-crossing time for head-on collisions. The later scaling is independent of a, but it does not account for multiparticle overlap (gray and yellow in the inset).

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    • Figure 5
      Figure 5

      Contact duration tc (in units of the mean step time) distributions (upper panel) for simulations differing in particle size only. Colors indicate distributions (left to right, blue to yellow) obtained from simulations with walker radii a≈[0.004,0.04,0.08,0.2,0.4,0.6,2.8]Δr or equivalently about [0.00071,0.0071,0.014,0.035,0.071,0.1,0.5] times the mean nearest-neighbor distance between walkers (the same simulations and the same color scheme as Fig. 4). Contact duration distribution for randomly oriented ballistic intersections (lower panel) between objects of the same size as the walkers in the simulations yielding the distributions in the upper panel. Vertical fiducial lines indicate the ballistic-crossing times for head-on collisions, tb=2a/Δr. Note that, as discussed in the main text, the distributions in the lower panel collapse identically into one when the contact duration is rescaled by tb. Scaled simulation distributions are shown in Fig. 6.

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    • Figure 6
      Figure 6

      Contact duration tc distributions for simulations differing in particle size only, scaled by the ballistic-crossing time tb=2a/Δr. Colors indicate distributions obtained from simulations with walker radii a≈[0.004,0.04,0.08,0.2,0.4]Δr or equivalently about [0.00071,0.0071,0.014,0.035,0.071] times the mean nearest-neighbor distance between walkers (the same simulations and color scheme as Figs. 4 and 5). Contact duration distribution for randomly oriented ballistic intersections is indicated with the underlying solid black curve. The inset shows waiting-time distributions for simulations with a≈[0.6,2.8]Δr or about [0.1,0.5] times the mean nearest-neighbor distance (gray, rightmost peak, and yellow, leftmost peak, respectively).

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    • Figure 7
      Figure 7

      Standard deviation of the pair separation distribution as a function of time (in units of the number of steps taken) for random walkers of varying initial separation and step length. Curves of solid (gray) colors indicate pair separation results for walks with step lengths (left to right) Δr=[0.0041,0.0021] and an initial separation of r0=0.7. The black curve plots standard deviation of the separation for a pair with step length Δr=0.0251 and initial separation r0=0.002. The latter are values typical of those immediately after contact between pairs in the walker population simulations discussed in the main text. Dotted curves of the same colors indicate the expected standard deviations in the diffusion limit as discussed in this Appendix. Dark orange and dark blue dashed curves plot the pair separation distribution standard deviations for two pairs of random walkers with step length Δr=[0.0041,0.0021] and initial separation r0=0.02. Small dotted vertical fiducial lines indicate the time of first possible contact between those pairs and the approximate time of transition to the long-time Rayleigh distribution limit (see Fig. 8 and discussion in this Appendix).

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    • Figure 8
      Figure 8

      Snapshots of the pair-separation probability density as a function of time for pairs of walkers whose separation variance is plotted with the dashed dark blue curve in Fig. 7. Distributions are shown for t−t0=[0.5,1.0,3.0,r0/(2Δr)=4.76] (narrow to wider PDFs, respectively) in the upper panel, and for t−t0=[10.0,60.0,4r02/Δr2=363] (narrow to wider PDFs, respectively) in the lower panel. In the upper panel, black curves indicate best-fit Gaussian distributions, and in the lower panel, black curves indicate best-fit Rice distributions. Best-fit Rayleigh distributions are shown in red in the lower panel.

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