- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Loss of percolation transition in the presence of simple tracer-media interactions

Ofek Lauber Bonomo* and Shlomi Reuveni†

  • School of Chemistry, Center for the Physics & Chemistry of Living Systems, Ratner Institute for Single Molecule Chemistry, and the Sackler Center for Computational Molecular & Materials Science, Tel Aviv University, 6997801 Tel Aviv, Israel

  • *ofekzvil@mail.tau.ac.il
  • †shlomire@tauex.tau.ac.il
PDF

Phys. Rev. Research 5, L042015 – Published 19 October, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L042015

Abstract

Random motion in disordered media is sensitive to the presence of obstacles which prevent atoms, molecules, and other particles from moving freely in space. When obstacles are static, a transition between confined motion and free diffusion occurs at a critical obstacle density: the percolation threshold. To test if this conventional wisdom continues to hold in the presence of simple tracer-media interactions from the type seen in recent experiments, we introduce the Sokoban random walk. Akin to the protagonist of an eponymous video game, the Sokoban has some ability to push away obstacles that block its path. While one expects this will allow the Sokoban to venture further away, we surprisingly find that this is not always the case. Indeed, as it moves—pushing obstacles around—the Sokoban always confines itself after traveling a characteristic distance that is set by the initial density of obstacles. Consequently, the percolation transition is lost. This finding breaks from the ruling ant in a labyrinth paradigm, vividly illustrating that even weak and localized tracer-media interactions cannot be neglected when coming to understand transport phenomena.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (58)

  1. K. Pearson, The problem of the random walk, Nature (London) 72, 294 (1905).
  2. G. F. Lawler and V. Limic, Random Walk: A Modern Introduction (Cambridge University Press, Cambridge, UK, 2010), Vol. 123.
  3. J. Klafter and I. M. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford University Press, Oxford, 2011).
  4. F. N. David, Games, Gods, and Gambling: A History of Probability and Statistical Ideas (Courier Corporation, North Chelmsford, MA, 1998).
  5. L. Bachelier, Theory of speculation, Ann. Sci. cole Normale Supérieure 17, 21 (1900).
  6. A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik 322, 549 (1905).
  7. M. Von Smoluchowski, Zur kinetischen theorie der brownschen molekularbewegung und der suspensionen, Ann. Phys. (Leipzig) 326, 756 (1906).
  8. W. Ebeling, E. Gudowska-Nowak, and I. M. Sokolov, On stochastic dynamics in physics—remarks on history and terminology, Acta Phys. Pol. B 39, 5 (2008).
  9. E. W. Montroll and G. H. Weiss, Random walks on lattices. II, J. Math. Phys. 6, 167 (1965).
  10. V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger, Generalized master equations for continuous-time random walks, J. Stat. Phys. 9, 45 (1973).
  11. G. M. Schütz and S. Trimper, Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk, Phys. Rev. E 70, 045101 (2004).
  12. A. Gabel and S. Redner, Random walk picture of basketball scoring, J. Quant. Anal. Sports 8, 1 (2012).
  13. L. Giuggioli, Exact spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions: A century after smoluchowski and pólya, Phys. Rev. X 10, 021045 (2020).
  14. E. Barkai and S. Burov, Packets of diffusing particles exhibit universal exponential tails, Phys. Rev. Lett. 124, 060603 (2020).
  15. H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12, 2455 (1975).
  16. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  17. B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys. 44, RG2003 (2006).
  18. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, First-passage times in complex scale-invariant media, Nature (London) 450, 77 (2007).
  19. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, UK, 2010).
  20. A. J. Bray, S. N. Majumdar, and G. Schehr, Persistence and first-passage properties in nonequilibrium systems, Adv. Phys. 62, 225 (2013).
  21. R. Metzler, J. H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys. 16, 24128 (2014).
  22. V. Zaburdaev, S. Denisov, and J. Klafter, Lévy walks, Rev. Mod. Phys. 87, 483 (2015).
  23. A. Szabo, G. Lamm, and G. H. Weiss, Localized partial traps in diffusion processes and random walks, J. Stat. Phys. 34, 225 (1984).
  24. K. Kang and S. Redner, Fluctuation-dominated kinetics in diffusion-controlled reactions, Phys. Rev. A 32, 435 (1985).
  25. D. Froemberg and I. M. Sokolov, Stationary fronts in an A+B→0 reaction under subdiffusion, Phys. Rev. Lett. 100, 108304 (2008).
  26. O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Geometry-controlled kinetics, Nat. Chem. 2, 472 (2010).
  27. Y. Lanoiselée, N. Moutal, and D. S. Grebenkov, Diffusion-limited reactions in dynamic heterogeneous media, Nat. Commun. 9, 4398 (2018).
  28. Y. Scher and S. Reuveni, Unified approach to gated reactions on networks, Phys. Rev. Lett. 127, 018301 (2021).
  29. H. C. Berg, Random walks in biology, in Random Walks in Biology (Princeton University Press, Princeton, NJ, 2018).
  30. A. Amir, Cell size regulation in bacteria, Phys. Rev. Lett. 112, 208102 (2014).
  31. S. Iyer-Biswas and A. Zilman, First-passage processes in cellular biology, Adv. Chem. Phys. 160, 261 (2016).
  32. O. Bénichou, U. Bhat, P. L. Krapivsky, and S. Redner, Optimally frugal foraging, Phys. Rev. E 97, 022110 (2018).
  33. H. Meyer and H. Rieger, Optimal non-Markovian search strategies with n-step memory, Phys. Rev. Lett. 127, 070601 (2021).
  34. D. Boyer, G. Ramos-Fernández, O. Miramontes, J. L. Mateos, G. Cocho, H. Larralde, H. Ramos, and F. Rojas, Scale-free foraging by primates emerges from their interaction with a complex environment, Proc. R. Soc. B. 273, 1743 (2006).
  35. D. W. Sims, E. J. Southall, N. E. Humphries, G. C. Hays, C. J. Bradshaw, J. W. Pitchford, A. James, M. Z. Ahmed, A. S. Brierley, M. A. Hindell, and D. Morritt, Scaling laws of marine predator search behaviour, Nature (London) 451, 1098 (2008).
  36. L. Giuggioli, J. R. Potts, and S. Harris, Animal interactions and the emergence of territoriality, PLoS Comput. Biol. 7, e1002008 (2011).
  37. G. M. Viswanathan, M. G. Da Luz, E. P. Raposo, and H. E. Stanley, The Physics of Foraging: An Introduction to Random Searches and Biological Encounters (Cambridge University Press, Cambridge, UK, 2011).
  38. C. Song, T. Koren, P. Wang, and A. L. Barabási, Modelling the scaling properties of human mobility, Nat. Phys. 6, 818 (2010).
  39. O. Vilk, Y. Orchan, M. Charter, N. Ganot, S. Toledo, R. Nathan, and M. Assaf, Ergodicity breaking in area-restricted search of avian predators, Phys. Rev. X 12, 031005 (2022).
  40. J. P. Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, UK, 2003).
  41. J. P. Bouchaud, J. Bonart, J. Donier, and M. Gould, Trades, Quotes and Prices: Financial Markets under the Microscope (Cambridge University Press, Cambridge, UK, 2018).
  42. P. G. de Gennes, La percolation: un concept unificateur, La recherche 7, 919 (1976).
  43. S. Havlin and Ben-D. Avraham, Diffusion in disordered media, Adv. Phys. 36, 695 (1987).
  44. J. P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
  45. I. M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043 (2012).
  46. W. W. Brandt, Use of percolation theory to estimate effective diffusion coefficients of particles migrating on various ordered lattices and in a random network structure, J. Chem. Phys. 63, 5162 (1975).
  47. R. Kopelman, Exciton percolation in molecular alloys and aggregates, in Radiationless Processes in Molecules and Condensed Phases (Springer, Berlin, 1976), pp. 297–346.
  48. C. Mitescu and J. Roussenq, Ant in a labyrinth-diffusion in a percolation system, CR Acad. Sci. Paris 283, 999 (1976).
  49. D. Ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, UK, 2000).
  50. D. Stauffer and A. Aharony, Introduction to Percolation Theory (CRC Press, Boca Raton, FL, 2018).
  51. A. Altshuler, O. L. Bonomo, N. Gorohovsky, S. Marchini, E. Rosen, O. Tal-Friedman, S. Reuveni, and Y. Roichman, Environmental memory facilitates search with home returns, arXiv:2306.12126.
  52. A. Biswas, J. M. Cruz, P. Parmananda, and D. Das, First passage of an active particle in the presence of passive crowders, Soft Matter 16, 6138 (2020).
  53. C. S. Dias, M. Trivedi, G. Volpe, N. A. M. Araújo, and G. Volpe, Environmental memory boosts group formation of clueless individuals, arXiv:2306.00516.
  54. D. Dor and U. Zwick, SOKOBAN and other motion planning problems, Comput. Geom. 13, 215 (1999).
  55. J. C. Culberson, Sokoban is PSPACE-complete, Technical Report TR 97-02, Department of Computing Science, The University of Alberta Edmonton, Alberta, Canada (1997), http://sokoban.dk/wp-content/uploads/2016/02/Sokoban-is-PSPACE-complete.pdf.
  56. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.5.L042015 for supplemental figures and video (SV1.avi).
  57. J. L. Jacobsen, Critical points of Potts and O (N) models from eigenvalue identities in periodic Temperley-Lieb algebras, J. Phys. A: Math. Theor. 48, 454003 (2015).
  58. Y. Gefen, A. Aharony, and S. Alexander, Anomalous diffusion on percolating clusters, Phys. Rev. Lett. 50, 77 (1983).

Outline

Information

Phys. Rev. Research 5, L042015– Published 19 October, 2023
  • Received 29 November 2022
  • Accepted 28 September 2023
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation