- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Solutions of first-passage time problems: A biscaling approach

Talia Baravi1, David A. Kessler2, and Eli Barkai1

  • 1Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
  • 2Department of Physics, Br-Ilan University, Ramat Gan 52900, Israel

PDF

Phys. Rev. E 111, 044103 – Published 1 April, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.044103

Abstract

We study the first-passage time (FPT) problem for widespread recurrent processes in confined though large systems and present a comprehensive framework for characterizing the FPT distribution over many timescales. We find that the FPT statistics can be described by two scaling functions: one corresponds to the solution for an infinite system, and the other describes a scaling that depends on system size. We find a universal scaling relationship for the FPT moments 〈tq〉 with respect to the domain size and the source-target distance. This scaling exhibits a transition at qc=θ, where θ is the persistence exponent. For low-order moments with q<qc, convergence occurs towards the moments of an infinite system. In contrast, the high-order moments, q>qc, can be derived from an infinite density function. The presented uniform approximation, connecting the two scaling functions, provides a description of the first-passage time statistics across all timescales. We extend the results to include diffusion in a confining potential in the high-temperature limit, where the potential strength takes the place of the system's size as the relevant scale. This study has been applied to various mediums, including a particle in a box, two-dimensional wedge, fractal geometries, non-Markovian processes, and the nonequilibrium process of resetting.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (73)

  1. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, England, 2001).
  2. R. Grebenkov, D. Metzler, and G. Oshanin, Target Search Problems (Springer International Publishing, Cham, 2024).
  3. A. J. Bray, S. N. Majumdar, and G. Schehr, Persistence and first-passage properties in nonequilibrium systems, Adv. Phys. 62, 225 (2013).
  4. R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen. 37, R161 (2004).
  5. B. Meyer, C. Chevalier, R. Voituriez, and O. Bénichou, Universality classes of first-passage-time distribution in confined media, Phys. Rev. E 83, 051116 (2011).
  6. O. Bénichou and R. Voituriez, From first-passage times of random walks in confinement to geometry-controlled kinetics, Phys. Rep. 539, 225 (2014).
  7. U. Basu, S. N. Majumdar, A. Rosso, and G. Schehr, Active Brownian motion in two dimensions, Phys. Rev. E 98, 062121 (2018).
  8. D. Boyer, O. Miramontes, G. Ramos-Fernandez, J. L. Mateos, and G. Cocho, Modeling the searching behavior of social monkeys, Physica A 342, 329 (2004).
  9. P. Jolakoski, A. Pal, T. Sandev, L. Kocarev, R. Metzler, and V. Stojkoski, A first passage under resetting approach to income dynamics, Chaos Solitons Fractals 175, 113921 (2023).
  10. Y. Lanoiselée, N. Moutal, and D. S. Grebenkov, Diffusion-limited reactions in dynamic heterogeneous media, Nat. Commun. 9, 4398 (2018).
  11. M. F. Shlesinger, Search research, Nature (London) 443, 281 (2006).
  12. T. Guérin, N. Levernier, O. Bénichou, and R. Voituriez, Mean first-passage times of non-Markovian random walkers in confinement, Nature (London) 534, 356 (2016).
  13. D. S. Grebenkov and A. T. Skvortsov, Boundary homogenization for target search problems, in Target Search Problems, edited by D. S. Grebenkov (Springer, Cham, 2024), pp. 247–279.
  14. Y. Scher, S. Reuveni, and D. S. Grebenkov, Escape of a sticky particle, Phys. Rev. Res. 5, 043196 (2024).
  15. A. Padash, T. Sandev, H. Kantz, R. Metzler, and A. V. Chechkin, Asymmetric Lévy flights are more efficient in random search, Fractal Fract. 6, 260 (2022).
  16. A. A. Dubkov, C. Guarcello, and B. Spagnolo, Enhancement of stability of metastable states in the presence of Lévy noise, SciPost Phys. 18, 006 (2025).
  17. A. Godec and R. Metzler, Universal proximity effect in target search kinetics in the few-encounter limit, Phys. Rev. X 6, 041037 (2016).
  18. A. Godec and R. Metzler, First passage time distribution in heterogeneity controlled kinetics: Going beyond the mean first passage time, Sci. Rep. 6, 20349 (2016).
  19. D. S. Grebenkov, R. Metzler, and G. Oshanin, Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control, Commun. Chem. 1, 96 (2018).
  20. Z. Schuss, A. Singer, and D. Holcman, The narrow escape problem for diffusion in cellular microdomains, Proc. Natl. Acad. Sci. USA 104, 16098 (2007).
  21. L. Mirny, Cell commuters avoid delays, Nat. Phys. 4, 93 (2008).
  22. O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Optimizing intermittent reaction paths, Phys. Chem. Chem. Phys. 10, 7059 (2008).
  23. L. Giuggioli and V. M. Kenkre, Consequences of animal interactions on their dynamics: Emergence of home ranges and territoriality, Movement Ecol. 2, 20 (2014).
  24. O. Bénichou, M. Coppey, M. Moreau, P.-H. Suet, and R. Voituriez, Optimal search strategies for hidden targets, Phys. Rev. Lett. 94, 198101 (2005).
  25. O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Two-dimensional intermittent search processes: An alternative to Lévy flight strategies, Phys. Rev. E 74, 020102(R) (2006).
  26. S. Sugaya and V. M. Kenkre, Analysis of transmission of infection in epidemics: Confined random walkers in dimensions higher than one, Bull. Math. Biol. 80, 3106 (2018).
  27. L. K. Gallos, C. Song, S. Havlin, and H. A. Makse, Scaling theory of transport in complex biological networks, Proc. Natl. Acad. Sci. USA 104, 7746 (2007).
  28. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, First-passage times in complex scale-invariant media, Nature (London) 450, 77 (2007).
  29. B. O'Shaughnessy and I. Procaccia, Diffusion on fractals, Phys. Rev. A 32, 3073 (1985).
  30. B. O'Shaughnessy and I. Procaccia, Analytical solutions for diffusion on fractal objects, Phys. Rev. Lett. 54, 455 (1985).
  31. T. Baravi, D. A. Kessler, and E. Barkai, First passage times in compact domains exhibit biscaling, Phys. Rev. Lett. 134, 127101 (2025).
  32. P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, On strong anomalous diffusion, Physica D 134, 75 (1999).
  33. S. Havlin and D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 36, 695 (1987).
  34. G. Afek, N. Davidson, D. A. Kessler, and E. Barkai, Colloquium: Anomalous statistics of laser-cooled atoms in dissipative optical lattices, Rev. Mod. Phys. 95, 031003 (2023).
  35. E. Aghion, D. A. Kessler, and E. Barkai, From non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory, Phys. Rev. Lett. 122, 010601 (2019).
  36. N. Korabel and E. Barkai, Pesin-type identity for intermittent dynamics with a zero Lyapunov exponent, Phys. Rev. Lett. 102, 050601 (2009).
  37. E. Barkai, G. Radons, and T. Akimoto, Transitions in the ergodicity of subrecoil-laser-cooled gases, Phys. Rev. Lett. 127, 140605 (2021).
  38. E. Schrödinger, Zur Theorie der Fall- und Steigversuche an Teilchen mit brownscher Bewegung, Phys. Z. 16, 289 (1915).
  39. M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Rivista Nuovo Cimento 1, 161 (1971).
  40. W. G. Glöckle and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68, 46 (1995).
  41. V. Balakrishnan, Anomalous diffusion in one dimension, Physica A 132, 569 (1985).
  42. E. Barkai, Fractional Fokker-Planck equation, solution, and application, Phys. Rev. E 63, 046118 (2001).
  43. J-H. Jeon, A. V. Chechkin, and R. Metzler, First passage behaviour of fractional Brownian motion in two-dimensional wedge domains, Europhys. Lett. 94, 20008 (2011).
  44. M. E. Fisher and M. P. Gelfand, The reunions of three dissimilar vicious walkers, J. Stat. Phys. 53, 175 (1988).
  45. T. G. Mattos, C. Mejía-Monasterio, R. Metzler, and G. Oshanin, First passages in bounded domains: When is the mean first passage time meaningful? Phys. Rev. E 86, 031143 (2012).
  46. The parameters k and ck were obtained straightforwardly using the Python Scipy library.
  47. R. Metzler, W. G. Glöckle, and T. F. Nonnenmacher, Fractional model equation for anomalous diffusion, Physica A 211, 13 (1994).
  48. V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger, Generalized master equations for continuous-time random walks, J. Stat. Phys. 9, 45 (1973).
  49. E. Martin, U. Behn, and G. Germano, First-passage and first-exit times of a Bessel-like stochastic process, Phys. Rev. E 83, 051115 (2011).
  50. A. J. Bray, Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensional XY model, Phys. Rev. E 62, 103 (2000).
  51. D. Ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, England, 2000).
  52. K. S. Fa and E. K. Lenzi, Anomalous diffusion, solutions, and first passage time: Influence of diffusion coefficient, Phys. Rev. E 71, 012101 (2005).
  53. C. Zunke, J. Bewerunge, F. Platten, S. U. Egelhaaf, and A. Godec, First-passage statistics of colloids on fractals: Theory and experimental realization, Sci. Adv. 8, eabk0627 (2022).
  54. K. S. Fa and E. K. Lenzi, Power law diffusion coefficient and anomalous diffusion: Analysis of solutions and first passage time, Phys. Rev. E 67, 061105 (2003).
  55. O. Bénichou, B. Meyer, V. Tejedor, and R. Voituriez, Zero constant formula for first-passage observables in bounded domains, Phys. Rev. Lett. 101, 130601 (2008).
  56. B. M. Hambly and T. J. Lyons, Stochastic area for Brownian motion on the Sierpinski gasket, Ann. Probab. 26, 132 (1998).
  57. B. M. Hambly, Brownian motion on a homogeneous random fractal, Probab. Theory Relat. Fields 94, 1 (1992).
  58. B. M. Hambly, Brownian motion on a random recursive Sierpinski gasket, Ann. Probab. 25, 1059 (1997).
  59. A. M. Teguia and A. P. Godbole, Sierpinski gasket graphs and some of their properties, Australasian J. Comb. 35, 181192 (2006).
  60. H. Risken, Fokker-Planck Equation (Springer, 1996).
  61. U. Bhat, C. De Bacco, and S. Redner, Stochastic search with Poisson and deterministic resetting, J. Stat. Mech. (2016) 083401.
  62. S. Gupta and A. M. Jayannavar, Stochastic resetting: A (very) brief review, Front. Phys. 10, 789097 (2022).
  63. M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601 (2011).
  64. R. Yin and E. Barkai, Restart expedites quantum walk hitting times, Phys. Rev. Lett. 130, 050802 (2023).
  65. S. Ray and S. Reuveni, Diffusion with resetting in a logarithmic potential, J. Chem. Phys. 152, 234110 (2020).
  66. A. Dubey and A. Pal, First-passage functionals for Ornstein-Uhlenbeck process with stochastic resetting, J. Phys. A: Math. Theor. 56, 435002 (2023).
  67. P. Singh and A. Pal, First-passage Brownian functionals with stochastic resetting, J. Phys. A: Math. Theor. 55, 234001 (2022).
  68. M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, J. Phys. A: Math. Theor. 53, 193001 (2020).
  69. P. S. Lovely and F. W. Dahlquist, Statistical measures of bacterial motility and chemotaxis, J. Theor. Biol. 50, 477 (1975).
  70. H. C. Berg and D. A. Brown, Chemotaxis in Escherichia coli analysed by three-dimensional tracking, Nature (London) 239, 500 (1972).
  71. U. Basu, S. Sabhapandit, and I. Santra, Target search by active particles, Cham: Springer Nat. Switzerland 463 (2024).
  72. F. Mori, P. Le Doussal, S. N. Majumdar, and G. Schehr, Universal survival probability for a D-dimensional run-and-tumble particle, Phys. Rev. Lett. 124, 090603 (2020).
  73. M. R. Evans and S. N. Majumdar, Run and tumble particle under resetting: A renewal approach, J. Phys. A: Math. Theor. 51, 475003 (2018).

Outline

Information

Phys. Rev. E 111, 044103– Published 1 April, 2025
  • Received 6 November 2024
  • Accepted 4 March 2025
Crossmark - Check for updates button

©2025 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation