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Two-particle problem in comblike structures

Elena Agliari, Davide Cassi, Luca Cattivelli, and Fabio Sartori
Phys. Rev. E 93, 052111 – Published 5 May 2016
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  • INTRODUCTION
  • TWO RANDOM WALKERS ON COMBS
  • FURTHER CHARACTERIZATIONS AND EXTENSIONS
  • CHECKING THE ROBUSTNESS VIA TOPOLOGICAL…
  • MAPPING THE TWO-PARTICLE PROBLEM INTO A…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comblike structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comblike architecture can strongly differ according to whether both reactants are moving (no matter their relative diffusivities) or only one is moving and according to the density of shortcuts among the branches.

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    • Received 16 May 2015
    • Revised 15 February 2016

    DOI:https://doi.org/10.1103/PhysRevE.93.052111

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Random walks
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Elena Agliari1,*, Davide Cassi2,†, Luca Cattivelli3,‡, and Fabio Sartori4,§

    • 1Dipartimento di Matematica, Sapienza Università di Roma, P. le Aldo Moro 5, 00185 Roma, Italy
    • 2Dipartimento di Fisica e Scienze della Terra, Parco Area delle Scienze 7/A, 43124 Parma, Italy
    • 3Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
    • 4Max Planck Institute for Brain Research, Max-von-Laue-Straße 4, 60438 Frankfurt am Main, Germany

    • *agliari@mat.uniroma1.it
    • †davide.cassi@fis.unipr.it
    • ‡luca.cattivelli@sns.it
    • §fabio.sartori@brain.mpg.de

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    Issue

    Vol. 93, Iss. 5 — May 2016

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    Images

    • Figure 1
      Figure 1

      Upper panel: Generic branched structure obtained by attaching to each site of the arbitrary base B an arbitrary fiber F. Middle panel: Generic comb obtained by attaching to each site of the arbitrary base B a linear chain. Lower panel: Simple two-dimensional comb obtained by taking as base a linear chain and attaching a linear chain to each site of the base.

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    • Figure 2
      Figure 2

      Two particles (named A and B, respectively) on the comb sharing the same tooth. In this picture, when walker A reaches the tooth already occupied by walker B, the latter is at position Y along the side chain, namely at that time Δx=0 and Δy=Y.

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    • Figure 3
      Figure 3

      In order to check the scaling in Eq. (1) it is convenient to look at the complementary of the cumulative distribution, namely at the quantity 1−∫1zψ(t)dt, which represents the probability that two random walkers have not yet shared the same tooth after a time z. This quantity is obtained via numerical simulations (bullets) and successfully compared with the power law ∼z−1/4 (solid line) resulting from the estimate in Eq. (1). Numerical simulations are performed on an “infinite” comb, where the walkers are initially set on the backbone with relative distance of two sites (Δx=2). At each time step (t=1,2,3,⋯) the two walkers change synchronously their position toward a nearest site selected with equal probability. The underlying “infinite” comb is mimicked by not imposing any boundary conditions and by using a data type for the instantaneous positions whose maximum cannot be reached in the considered time interval. The latter is fixed by a cutoff in time corresponding to 106. Thus, a simulation stops upon the walkers find themselves on the same tooth at a certain time t<106 or whenever the time cutoff is reached. The results shown here have been averaged over 107 replicas.

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    • Figure 4
      Figure 4

      Probability Pbackbone(t) that the two walkers encounter in a site belonging to the backbone (upper panel) and probability Ptooth(t) that the two walkers encounter in a site belonging to teeth (lower panel). Results from numerical simulations (bullets) are successfully compared with analytical estimates (solid line) according to Eqs. (7) and (8), respectively. In the numerical simulations the walkers are initially set on the backbone with relative distance Δx=2 and at each time step (t=1,2,3,⋯) they change synchronously their position toward a nearest site selected with equal probability. A simulation stops as a time threshold 4×103 is reached and the size of the comb is taken large enough that, for this temporal cutoff, the walkers do not realize its finiteness. We repeat the simulation 107 times and for each realization we keep track of the time step τ when walkers possibly occur to encounter on the backbone (upper panel) or on a tooth (lower panel). The final distributions are then obtained as histograms over τ. Note that in a single realization, there may be more than one encounters and therefore a single realization may return several values for τ.

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    • Figure 5
      Figure 5

      Main plot: Mean encounter time τ for two random walkers moving on a finite two-dimensional comb and started at the same point on the backbone to first meet. The comb has a backbone of size L and side chains of length L each, in such a way that, overall, the comb counts L(L+1) nodes. Data points (bullets) are obtained via numerical simulations (averaged over 105 realizations) and are fitted (solid line) according to the theoretical predictions (13). Upper inset: In order to check the goodness of the theoretical prediction, we plotted the ratio between the experimental value from simulations and the expected value from the analytical estimate. This is done for the analytical prediction given by (13) and given by a purely power law (in this case, the best-fit exponent α is α=2.754). In both cases the ratio is approximately 1, with fluctuations which are less broadened for the former. Lower inset: We plotted the ratio between the experimental value from simulations and the expected value from the analytical estimate (14) for d-dimensional combs (d=2, d=3, d=4, as shown by the legend). As expected, the ratio fluctuates around 1.

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    • Figure 6
      Figure 6

      Panels (a)–(e): Ratio between the numerical estimate of the encounter probability P¯enc and the value provided by the fitting function; each panel corresponds to a different choice of k (i.e., a different choice of α). For k=1,2,3,4, the fitting function is y(t)=Penc∞(α)−at−b while for k=5 it is y1(t)=Penc∞(α)−at−b−c/log(t). In each panel we compare results for three different system size: L=211 (bright blue), L=213 (blue), and L=215 (black). The best-fit coefficient Penc∞(α) is used in panel (f) to show how the probability of never meeting varies as the number of links inserted is progressively increased. For the cases analyzed, the encounter is certain only for α=2. The best-fit coefficient Penc∞(α) is also used in panel (g), where we plot, in a log-log scale, the difference Penc∞(α)−P¯enc(α,t) pertaining to the cases k=1,⋯,4. The dashed black lines have slope −b. The linear outline versus time corroborates the expected power-law behavior for the related encounter probabilities.

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    • Figure 7
      Figure 7

      Upper panel: The two-particle problem on a two-dimensional comb can be mapped into a one-particle comb embedded in a structure as the one shown here. Every point of this structure is univocally associated to a triplet (Δx,Δy,Yc.m.) and the encounter between the two walkers on the comb corresponds to the single particle being in any point of the straight line Δx=Δy=0 denoted in red. Lower panel: The plane Δx=0 (called the “plane of encounters”) is shown alone to highlight the encounter line along Yc.m..

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    • Figure 8
      Figure 8

      The return to the origin v for a walker, passing through w.

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    • Figure 9
      Figure 9

      Two walkers start form v and collide in w.

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    • Figure 10
      Figure 10

      In this schematic representation of M, we distinguish the “pages” (in dark color) and the “bookbindings” (in bright color).

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