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  • Letter
  • Open Access

Cauchy universality and random billiards

Roberto Artuso1,2,* and Dario Javier Zamora1,3,†

  • 1Center for Nonlinear and Complex Systems and Dipartimento di Scienza e Alta Tecnologia, Via Valleggio 11, 22100 Como, Italy
  • 2I.N.F.N., Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
  • 3Instituto de Fisica del Noroeste Argentino, CONICET and Universidad Nacional de Tucuman, Av. Independencia 1800, Tucuman, CP 4000, Argentina

  • *Contact author: roberto.artuso@uninsubria.it
  • †Contact author: dariojavier.zamora@uninsubria.it
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Phys. Rev. Research 6, L032029 – Published 5 August, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L032029

Abstract

Motion in bounded domains represents a paradigm in several settings: from billiard dynamics, to random walks in a finite lattice, with applications to relevant physical, ecological, and biological problems. A remarkable universal property, involving the average of return times to the boundary, has been theoretically proposed and experimentally verified in quite different contexts. We discuss here mechanisms that lead to violations of universality, induced by boundary effects and we also emphasize the role played by replacing straight lines with random walks in this framework. We suggest that our analysis should be relevant where nonhomogeneity appears in the stationary probability distribution in bounded domain.

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References (53)

  1. A.-L. Cauchy, Mémoire sur la Rectification des Courbes et la Quadrature des Surfaces Courbées (Gauthier-Villars, Paris, 1850).
  2. It is interesting to remark that in the physical literature other quantities that depend on the ratio between volume and surface (irrespective of the shape) appear: for example Weisskopf's estimate for resonance mean width, which in turn is (surprisingly) related to Sabine law of room acoustics: see D. V. Savin, O. Legrand, and F. Mortessagne, Inhomogeneous losses and complexness of wave functions in chaotic cavities, Europhys. Lett. 76, 774 (2006).
  3. A. Santalò, Integral Geometry and Geometric Probability (Addison-Wesley, Reading MA, 1976).
  4. A. Mazzolo, On the generalization of the average chord length, Ann. Nucl. Energy 35, 503 (2008).
  5. S. Hidalgo-Caballero, A. Cassinelli, M. Labousse, and E. Fort, Mean arc theorem for exploring domains with randomly distributed arbitrary closed trajectories, Eur. Phys. J. Plus 137, 501 (2022).
  6. A. Mazzolo, B. Roesslinger, and W. Gille, Properties of chord length distributions of nonconvex bodies, J. Math. Phys. 44, 6195 (2003).
  7. P. A. M. Dirac, Approximate rate of neutron multiplication for a solid of arbitrary shape and uniform density, Declassified British Report MS-D-5, Part I (1943).
  8. K. M. Case, Introduction to the Theory of Neutron Diffusion (Los Alamos scientific Laboratory, NM, 1953).
  9. W. J. M. de Kruijf, and J. L. Kloosterman, On the average chord length in reactor physics, Ann. Nucl. Energy 30, 549 (2003).
  10. J. N. Bardsley and A. Dubi, The average transport path in scattering media, SIAM J. Appl. Math. 40, 71 (1981).
  11. S. Blanco and R. Fournier, An invariance property of diffusive random walks, Europhys. Lett. 61, 168 (2003).
  12. R. Jeanson, S. Blanco, R. Fournier, J. L. Deneubourg, V. Fourcassie, and G. Theraulaz, A model of animal movements in a bounded space, J. Theor. Biol. 225, 443 (2003).
  13. S. Blanco and R. Fournier, Short-path statistics and the diffusion approximation, Phys. Rev. Lett. 97, 230604 (2006).
  14. A. Mazzolo, Properties of diffusive random walks in bounded domains, Europhys. Lett. 68, 350 (2004).
  15. O. Bénichou, M. Coppey, M. Moreau, P. H. Suet, and R. Voituriez, Average residence times of stochastic motion in bounded domains, Europhys. Lett. 70, 42 (2005).
  16. A. Mazzolo, C. de Mulatier, and A. Zoia, Cauchy's formula for random walks in bounded domains, J. Math. Phys. 55, 083308 (2014).
  17. G. Frangipane, G. Vizsnyiczai, C. Maggi, R. Savo, A. Sciortino, S. Gigan, and R. Di Leonardo, Invariance properties of bacterial random walks in complex structures, Nat. Commun. 10, 2442 (2019).
  18. R. Pierrat, P. Ambicht, S. Gigan, A. Haber, R. Carminati, and S. Rotter, Invariance property of wave scattering through disordered media, Proc. Natl. Acad. Sci. USA 111, 17765 (2014).
  19. R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, Observation of mean path length invariance in light-scattering media, Science 358, 765 (2017).
  20. L. Caprini, and U. Marini Bettolo Marconi, Active chiral particles under confinement: surface currents and bulk accumulation phenomena, Soft Matter 15, 2627 (2019).
  21. G. Volpe, S. Gigan, and G. Volpe, Simulation of the active Brownian motion of a microswimmer, Am. J. Phys. 82, 659 (2014).
  22. X. Yang, M. L. Manning, and M. C. Marchetti, Aggregation and segregation of confined active particles, Soft Matter 10, 6477 (2014).
  23. M. Souzy, A. Allard, J.-F. Louf, M. Contino, I. Touval, M. Polin, Microbial narrow-escape is facilitated by wall interactions, Phys. Rev. Res. 4, L022029 (2022).
  24. V. F. Lazutkin, The existence of caustics for the billiard problem in a convex domain, Izv. Akad. Nauk SSSR Ser. Mat. 37, 186 (1973) [Mathematics of the USSR - Izvestija 7, 185 (1973)].
  25. S. Tabachnikov, Billiards (Société Mathématique de France, Paris, 1995).
  26. N. Chernov and R. Markarian, Chaotic Billiards (AMS, Providence RI, 2006).
  27. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1995).
  28. The relationship between the invariant measure of the flow and that of the map can be used to verify Cauchy result for chaotic billiards in any dimension: see N. Chernov, Entropy, Lyapunov exponents, and mean free path for billiards, J. Stat. Phys. 88, 1 (1997); N. I. Chernov, New proof of Sinai's formula for entropy of hyperbolic billiards. Application to Lorentz gas and Bunimovich stadium, Funct. Anal. Appl. 25, 204 (1991).
  29. J. F. C. Kingman, Mean free path in a convex reflecting region, J. Appl. Probab. 2, 162 (1965).
  30. R. Feres and G. Yablonsky, Knudsen's cosine law and random billiards, Chem. Eng. Sci. 59, 1541 (2004).
  31. F. Comets, S. Popov, G. M. Schütz, and M. Vachkovskaia, Billiards in general domain with random reflections, Arch. Rational Mech. Anal. 191, 497 (2009).
  32. F. Celestini and F. Mortessagne, Cosine law at the atomic scale: towards realistic simulations of Knudsen diffusion, Phys. Rev. E 77, 021202 (2008).
  33. D. J. Zamora, and R. Artuso, Many other choices for the distribution of paths and new directions may be devised: a thorough check whether Cauchy universality will be preserved is under study (unpublished).
  34. D. S. Broomhead and E. Gutkin, The dynamics of billiards with no-slip collisions, Physica D 67, 188 (1993).
  35. G. Arya, H.-C. Chang, and E. J. Maginn, Knudsen diffusivity of a hard sphere in a rough slit pore, Phys. Rev. Lett. 91, 026102 (2003).
  36. E. D. Leonel, Corrugated waveguide under scaling investigation, Phys. Rev. Lett. 98, 114102 (2007).
  37. F. Bonetto, N. I. Chernov, and J. L. Lebowitz, (Global and local) fluctuations of phase space contraction in deterministic stationary nonequilibrium, Chaos 8, 823 (1998).
  38. S. N. Evans, Stochastic billiards on general tables, Ann. Appl. Probab. 11, 419 (2001).
  39. N. Fétique, Explicit speed of convergence of the stochastic billiard in a convex set, in Séminaire de Probabilités L., edited by C. Donati-Martin et al. (Springer, Switzerland, 2019).
  40. A. B. Dieker, and S. V. Vempala, Stochastic billiards for sampling from the boundary of a convex set, Math. Oper. Res. 40, 888 (2015).
  41. M. Horowitz, Probability of random paths across elementary geometrical shapes, J. Appl. Probab. 2, 169 (1965).
  42. R. Coleman, Random paths through convex bodies, J. Appl. Probab. 6, 430 (1969).
  43. J. F. C. Kingman, Random secants of a convex body, J. Appl. Probab. 6, 660 (1969).
  44. A. Moroz, Electron mean-free path in metal-crated nanowires, J. Opt. Soc. Am. B 28, 1130 (2011).
  45. In all the cases we consider such a density is known to exist, see for instance [30, 38, 39]; moreover mixing is exponentially fast, which supports the reliability of our numerical simulations.
  46. E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature (London) 349, 630 (1991).
  47. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  48. R. Markarian, L. T. Rolla, V. Sidoravicius, F. A. Tal, and M. E. Vares, Stochastic perturbations of convex billiards, Nonlinearity 28, 4425 (2015).
  49. Actually, the behavior of the perturbation near the boundaries is crucial: while the form we are considering avoids sticking to 0 or π, it is possible to choose other stochastic perturbations for which trajectories get captured at the boundary, see [48].
  50. D. Holcman and Z. Schuss, Stochastic Narrow Escape in Molecular and Cellular Biology: Analysis and Applications (Springer, New York, 2016).
  51. D. S. Grebenkov, First passage times for multiple particles with reversible target-binding kinetics, J. Chem. Phys. 147, 134112 (2017).
  52. D. S. Grebenkov and A. Kumar, First-passage times of multiple diffusing particles with reversible target-binding kinetics, J. Phys. A 55, 325002 (2022).
  53. D. S. Grebenkov, Diffusion-controlled reactions with non-Markovian binding/unbinding kinetics, J. Chem. Phys. 158, 214111 (2023).

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Phys. Rev. Research 6, L032029– Published 5 August, 2024
  • Received 25 March 2024
  • Accepted 25 June 2024
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