- 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
  • Open Access

Stochastically perturbed billiards: Fingerprints of chaos and universality classes

Roberto Artuso1,2,* and Matteo Burlo1,†

  • *Contact author: roberto.artuso@uninsubria.it
  • †Contact author: matteo.burlo46@gmail.com
PDF

Phys. Rev. Research 8, 033312 – Published 15 September, 2026

DOI: https://doi.org/10.1103/rxth-ltgq

Abstract

Billiards tables—a minimal model for particles moving in a confined region—are known to present different classical (and quantum) features according to their shape, ranging from strongly chaotic to integrable dynamics. Here, we consider the role of a stochastic perturbation of the elastic reflection law and show that while chaotic billiards maintain their key statistical feature, the behavior for integrable billiard tables is completely different: It can be linked, for tiny perturbations, to the Evans stochastic billiard, where at each collision the reflected angle is a uniformly distributed stochastic variable on (−π/2,π/2). The resulting spatial stationary measure has peculiar aspects, like being typically nonuniform along the boundary, differently from any chaotic billiard table.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (64)

  1. S. Tabachnikov, Billiards (Société Mathématique de France, Paris, 1995).
  2. N. Chernov and R. Markarian, Chaotic Billiards (American Mathematical Society, Providence, RI, 2006).
  3. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1995).
  4. L. A. Bunimovich, Mechanisms of chaos in billiards: Dispersing, defocusing and nothing else, Nonlinearity 31, R78 (2018).
  5. F. Pène, Chaotic behavior in billiards, J. Math. Phys. 66, 042712 (2025).
  6. V. Kaloshin and A. Sorrentino, On the integrability of Birkhoff billiards, Philos. Trans. R. Soc. A 376, 20170419 (2018).
  7. O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984).
  8. F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 2001).
  9. K. Nakamura and T. Harayama, Quantum Chaos and Quantum Dots (Oxford University Press, Oxford, 2003).
  10. N. I. Chernov and J. L. Lebowitz, Stationary nonequilibrium states for boundary driven Hamiltonian systems, J. Stat. Phys. 86, 953 (1997).
  11. 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).
  12. M. Knudsen, Kinetic Theory of Gases—Some Modern Aspects (Methuen, London, 1952).
  13. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  14. M. von Smoluchowski, Zur kinetischen theorie der transpiration und diffusion verdünnter gase, Ann. Phys. 338, 1559 (1910).
  15. 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).
  16. F. Celestini and F. Mortessagne, Cosine law at the atomic scale: Towards realistic simulations of Knudsen diffusion, Phys. Rev. E 77, 021202 (2008).
  17. R. Feres and G. Yablonsky, Knudsen's cosine law and random billiards, Chem. Eng. Sci. 59, 1541 (2004).
  18. R. Feres, Random walks derived from billiards, Recent Prog. Dyn. 54, 179 (2007).
  19. F. Comets, S. Popov, G. M. Schütz, and M. Vachkovskaia, Billiards in general domain with random reflections, Arch. Ration. Mech. Anal. 191, 497 (2009).
  20. We remark that Knudsen law leads to a microcanonical invariant density [or Eq. (10) for the collision Markov chain]: This guarantees the validity of Cauchy mean chord theorem [see Ref. [22] and references herein]. We will briefly comment on this in our conclusions.
  21. S. N. Evans, Stochastic billiards on general tables, Ann. Appl. Probab. 11, 419 (2001).
  22. R. Artuso and D. J. Zamora, Cauchy universality and random billiards, Phys. Rev. Res. 6, L032029 (2024).
  23. T. Ostapenko, F. J. Schwarzendahl, T. J. Böddeker, C. T. Kreis, J. Cammann, M. G. Mazza, and O. Bäumchen, Curvature-guided motility of microalgae in geometric confinement, Phys. Rev. Lett. 120, 068002 (2018).
  24. S. Williams, R. Jeanneret, I. Tuval, and M. Polin, Confinement-induced accumulation and de-mixing of microscopic active-passive mixtures, Nat. Commun. 13, 4776 (2022).
  25. R. Di Leonardo, A. Búzas, L. Kelemen, D. Tóth, S. Z. Tóth, P. Ormos, and G. Vizsnyiczai, Active billiards: Engineering boundary shape for the spatial control of confined active particles, Proc. Natl. Acad. Sci. USA 122, e2426715122 (2025).
  26. N. Razin, Entropy production of an active particle in a box, Phys. Rev. E 102, 030103(R) (2020).
  27. A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, Hydrodynamic attraction of swimming microorganisms by surfaces, Phys. Rev. Lett. 101, 038102 (2008).
  28. T. Pietrangeli, R. Foffi, R. Stocker, C. Ybert, C. Cottin-Bizonne, and F. Detcheverry, Universal law for the dispersal of motile microorganisms in porous media, Phys. Rev. Lett. 134, 188303 (2025).
  29. T. Bhattacharjee and S. S. Datta, Bacterial hopping and trapping in porous media, Nat. Commun. 10, 2075 (2019).
  30. D. Gao, Z. Wang, M. Jain, A. J. T. M. Mathijssen, and R. Tao, Selective trapping of bacteria in porous media by cell length, Integr. Comp. Biol. 66, 1 (2026).
  31. R. Markarian, L. T. Rolla, V. Sidoravicius, F. A. Tal, and M. E. Vares, Stochastic perturbations of convex billiards, Nonlinearity 28, 4425 (2015).
  32. N. Chernov, Entropy, Lyapunov exponents, and mean free path for billiards, J. Stat. Phys. 88, 1 (1997).
  33. N. I. Chernov, New proof of Sinai's formula for the entropy of hyperbolic billiard systems. Application to Lorentz gases and Bunimovich stadium, Funct. Anal. Appl. 25, 204 (1991).
  34. M. Burlo and R. Artuso (unpublished).
  35. 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), pp. 519–560.
  36. Y. Kifer, General random perturbations of hyperbolic and expanding transformations, J. Anal. Math. 47, 111 (1986).
  37. L.-S. Young, Stochastic stability of hyperbolic attractors, Ergodic Theory Dyn. Syst. 6, 311 (1986).
  38. S. E. Troubetzkoy, Stochastic stability of dispersing billiards, Theor. Math. Phys. 86, 151 (1991).
  39. R. Artuso, G. Casati, and I. Guarneri, Numerical experiments on billiards, J. Stat. Phys. 83, 145 (1996).
  40. J. Machta, Power law decay of correlations in billiard problems, J. Stat. Phys. 32, 555 (1983).
  41. E. J. Heller and S. Tomsovic, Postmodern quantum mechanics, Phys. Today 46(7), 38 (1993).
  42. S. Ree and L. E. Reichl, Classical and quantum chaos in a circular billiard with a straight cut, Phys. Rev. E 60, 1607 (1999).
  43. J. Chen, L. Mohr, H.-K. Zhang, and P. Zhang, Ergodicity of the generalized lemon billiards, Chaos 23, 043137 (2013).
  44. L. Bunimovich, H.-K. Zhang, and P. Zhang, On another edge of defocusing: Hyperbolicity of asymmetric lemon billiards, Commun. Math. Phys. 341, 781 (2016).
  45. Č. Lozej, D. Lukman, and M. Robnik, Effect of stickiness in the classical and quantum ergodic lemon billiard, Phys. Rev. E 103, 012204 (2021).
  46. Č. Lozej, Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach, Phys. Rev. E 101, 052204 (2020).
  47. R. V. A. Quirino, F. C. B. Leal, J. V. A. Vasconcelos, M. S. Palmero, A. R. de C. Romaguera, A. L. R. Barbosa, and T. Araújo Lima, Mixed-phase space of an active particle in experimental lemon billiards, Phys. Rev. E 112, 044218 (2025).
  48. X. Jin and P. Zhang, Homoclinic and heteroclinic intersections for lemon billiards, Adv. Math. 442, 109588 (2024).
  49. L. A. Bunimovich, On ergodic properties of certain billiards, Funct. Anal. Its Appl. 8, 254 (1974).
  50. L. A. Bunimovich, On the ergodic properties of nowhere dispersing billiards, Commun. Math. Phys. 65, 295 (1979).
  51. S. Tabachnikov, Geometry and Billiards (American Mathematical Society, Providence, RI, 2005).
  52. S.-J. Chang and R. Friedberg, Elliptical billiards and Poncelet's theorem, J. Math. Phys. 29, 1537 (1988).
  53. V. F. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Math. USSR-Izv. 7, 185 (1973).
  54. See Ref. [21]: Notice a misprint in the last equation or Remark 6.3.
  55. Notice that the density becomes constant in the circle limit (a=b).
  56. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1996).
  57. An example of a perturbation leading to absorbing end points is mentioned in Ref. [31]: The Fokker-Planck calculation we outlined in the text leads to a divergence for the stationary solution at the boundary.
  58. A.-L. Cauchy, Mémoire sur la Rectification des Courbes et la Quadrature des Surfaces Courbées (Gauthier-Villars, Paris, 1850).
  59. A. Santalò, Integral Geometry and Geometric Probability (Addison-Wesley, Reading MA, 1976).
  60. A. Mazzolo, On the generalization of the average chord length, Ann. Nucl. Energy 35, 503 (2008).
  61. 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).
  62. T. Jakuzeit, O. A. Croze, and S. Bell, Diffusion of active particles in a complex environment: Role of surface scattering, Phys. Rev. E 99, 012610 (2019).
  63. D. J. Zamora and R. Artuso, Exploring run-and-tumble movement in confined settings through simulation, J. Chem. Phys. 161, 114107 (2024).
  64. T. D. Drivas, D. Glukhovskiy, and B. Khesin, Pensive billiards, point vortices, and the silver ratio, Forum Math. Sigma 13, e171 (2025).

Outline

Information

Phys. Rev. Research 8, 033312– Published 15 September, 2026
  • Received 12 May 2026
  • Accepted 12 August 2026
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