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

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Access by Xinjiang University

Critical and tricritical transitions in diffusion with stochastic resetting on a ring

Pedro Julián-Salgado1,*, Pavel Castro-Villarreal2,†, Leonardo Dagdug1,‡, and Denis Boyer3,§

  • *Contact author: pjuliansalgado@gmail.com
  • †Contact author: pcastrov@unach.mx
  • ‡Contact author: dll@xanum.uam.mx
  • §Contact author: boyer@fisica.unam.mx
PDF

Phys. Rev. E 114, 044142 – Published 30 October, 2026

DOI: https://doi.org/10.1103/2fsc-jkq8

Abstract

Diffusion with an incorporated resetting mechanism provides a reference framework for modeling a wide range of natural phenomena. Within this framework, the optimal resetting rate is a key quantity that arises from the optimization of the mean first-passage time. While substantial work has focused on the study of the optimal resetting rate in unbounded one-dimensional domains, little is still known about the optimization of the mean first-passage time in bounded systems, in particular when multiple resetting sites are available. In this work, we consider a particle diffusing along a ring and undergoing stochastic resetting, with an absorbing target site at a fixed location. Using the appropriate free propagator for this system, we compute the Laplace transform of the survival probability when resetting occurs to multiple sites drawn from an arbitrary probability density function. We also calculate the mean first-passage time at the target site and study the dependence of the optimal resetting rate in terms of the relevant parameters of the system in a two-resetting site configuration. Depending on the arc length between one of the resetting sites and the absorbing target site, and the weight of the remaining resetting site, the optimal resetting rate can exhibit abrupt (“first-order”) and continuous (“second-order”) transitions. Moreover, the behavior of the mean first-passage time is rich enough to allow both critical and tricritical points to exist in the parameter space. All the transitions have mirror symmetry around the selected target site and its corresponding diametrically opposite site.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (78)

  1. G. V. Mechetin and D. O. Zharkov, Mechanisms of diffusional search for specific targets by DNA-dependent proteins, Biochemistry (Moscow) 79, 496 (2014).
  2. A. Tafvizi, L. A. Mirny, and A. M. van Oijen, Dancing on DNA: Kinetic aspects of search processes on DNA, ChemPhysChem 12, 1481 (2011).
  3. A. B. Kolomeisky, Physics of protein–DNA interactions: mechanisms of facilitated target search, Phys. Chem. Chem. Phys. 13, 2088 (2011).
  4. S. E. Halford and J. F. Marko, How do site‐specific DNA‐binding proteins find their targets? Nucleic Acids Res. 32, 3040 (2004).
  5. K. Kamagata, R. Kusano, S. Kanbayashi, T. Banerjee, and H. Takahashi, Single-molecule characterization of target search dynamics of DNA-binding proteins in DNA-condensed droplets, Nucleic Acids Res. 51, 6654 (2023).
  6. O. G. Berg, R. B. Winter, and P. H. Von Hippel, Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. Models and theory, Biochemistry 20, 6929 (1981).
  7. W. J. O'Brien, H. I. Browman, and B. I. Evans, Search strategies of foraging animals, Am. Sci. 78, 152 (1990).
  8. P. Louâpre, X. Fauvergue, J. van Baaren, and V. Martel, The male mate search: An optimal foraging issue? Curr. Opin. Insect Sci. 9, 91 (2015).
  9. A. J. King and H. H. Marshall, Optimal foraging, Curr. Biol. 32, R680 (2022).
  10. R. J. Cowie, Optimal foraging in great tits (Parus major), Nature (London) 268, 137 (1977).
  11. E. L. Charnov, Optimal foraging, the marginal value theorem, Theor. Popul Biol. 9, 129 (1976).
  12. F. Bartumeus, Lévy processes in animal movement: an evolutionary hypothesis, Fractals 15, 151 (2007).
  13. R. P. D. Atkinson, C. J. Rhodes, D. W. Macdonald, and R. M. Anderson, Scale-free dynamics in the movement patterns of jackals, Oikos 98, 134 (2002).
  14. 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).
  15. G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. da Luz, E. P. Raposo, and H. Eugene Stanley, Optimizing the success of random searches, Nature (London) 401, 911 (1999).
  16. J. O. Ares and M. Bertiller, Modeling high-frequency position data of large herbivores with a phase-state model, Ecol. Modell. 221, 2323 (2010).
  17. G. M. Viswanathan, M. G. E. 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).
  18. J. Ma, M. Do, M. A. Le Gros, C. S. Peskin, C. A. Larabell, Y. Mori, S. A. Isaacson, and J. R. Faeder, Strong intracellular signal inactivation produces sharper and more robust signaling from cell membrane to nucleus, PLoS Comput. Biol. 16, e1008356 (2020).
  19. S. D. Lawley, The effects of fast inactivation on conditional first passage times of mortal diffusive searchers, SIAM J. Appl. Math. 81, 1 (2021).
  20. B. Meerson and S. Redner, Mortality, redundancy, and diversity in stochastic search, Phys. Rev. Lett. 114, 198101 (2015).
  21. M. R. Shaebani, M. Piel, and F. Lautenschläger, Distinct speed and direction memories of migrating dendritic cells diversify their search strategies, Biophys. J. 121, 4099 (2022).
  22. M. F. Krummel, F. Bartumeus, and A. Gérard, T cell migration, search strategies and mechanisms, Nat. Rev. Immunol. 16, 193 (2016).
  23. J.-F. Rupprecht, O. Bénichou, and R. Voituriez, Optimal search strategies of run-and-tumble walks, Phys. Rev. E 94, 012117 (2016).
  24. N. Hirokawa, Y. Noda, Y. Tanaka, and S. Niwa, Kinesin superfamily motor proteins and intracellular transport, Nat. Rev. Mol. Cell Biol. 10, 682 (2009).
  25. S. Condamin, V. Tejedor, R. Voituriez, O. Bénichou, and J. Klafter, Probing microscopic origins of confined subdiffusion by first-passage observables, Proc. Natl. Acad. Sci. USA 105, 5675 (2008).
  26. O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. 83, 81 (2011).
  27. D. L. Kramer and R. L. McLaughlin, The behavioral ecology of intermittent locomotion, Am. Zool. 41, 137 (2001).
  28. M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601 (2011).
  29. S. N. Majumdar, S. Sabhapandit, and G. Schehr, Dynamical transition in the temporal relaxation of stochastic processes under resetting, Phys. Rev. E 91, 052131 (2015).
  30. S. Eule and J. J. Metzger, Non-equilibrium steady states of stochastic processes with intermittent resetting, New J. Phys. 18, 033006 (2016).
  31. A. Pal, Diffusion in a potential landscape with stochastic resetting, Phys. Rev. E 91, 012113 (2015).
  32. M. R. Evans and S. N. Majumdar, Diffusion with resetting in arbitrary spatial dimension, J. Phys. A: Math. Theor. 47, 285001 (2014).
  33. A. Pal, A. Kundu, and M. R. Evans, Diffusion under time-dependent resetting, J. Phys. A: Math. Theor. 49, 225001 (2016).
  34. M. R. Evans and S. N. Majumdar, Diffusion with optimal resetting, J. Phys. A: Math. Theor. 44, 435001 (2011).
  35. A. Pal and V. V. Prasad, First passage under stochastic resetting in an interval, Phys. Rev. E 99, 032123 (2019).
  36. G. Mercado-Vásquez, D. Boyer, S. N. Majumdar, and G. Schehr, Intermittent resetting potentials, J. Stat. Mech. (2020) 113203.
  37. A. Chechkin and I. M. Sokolov, Random search with resetting: A unified renewal approach, Phys. Rev. Lett. 121, 050601 (2018).
  38. M. R. Evans and S. N. Majumdar, Effects of refractory period on stochastic resetting, J. Phys. A: Math. Theor. 52, 01LT01 (2019).
  39. M. Radice, Diffusion processes with gamma-distributed resetting and non-instantaneous returns, J. Phys. A: Math. Theor. 55, 224002 (2022).
  40. A. P. Riascos, D. Boyer, P. Herringer, and J. L. Mateos, Random walks on networks with stochastic resetting, Phys. Rev. E 101, 062147 (2020).
  41. J. Fuchs, S. Goldt, and U. Seifert, Stochastic thermodynamics of resetting, Europhys. Lett. 113, 60009 (2016).
  42. B. De Bruyne, J. Randon-Furling, and S. Redner, Optimization in first-passage resetting, Phys. Rev. Lett. 125, 050602 (2020).
  43. A. Biswas, S. N. Majumdar, and A. Pal, Target search optimization by threshold resetting, Phys. Rev. Lett. 135, 227101 (2025).
  44. B. Besga, A. Bovon, A. Petrosyan, S. N. Majumdar, and S. Ciliberto, Optimal mean first-passage time for a Brownian searcher subjected to resetting: Experimental and theoretical results, Phys. Rev. Res. 2, 032029(R) (2020).
  45. F. Faisant, B. Besga, A. Petrosyan, S. Ciliberto, and S. N. Majumdar, Optimal mean first-passage time of a Brownian searcher with resetting in one and two dimensions: Experiments, theory and numerical tests, J. Stat. Mech. (2021) 113203.
  46. O. Tal-Friedman, A. Pal, A. Sekhon, S. Reuveni, and Y. Roichman, Experimental realization of diffusion with stochastic resetting, J. Phys. Chem. Lett. 11, 7350 (2020).
  47. C. A. Plata, D. Gupta, and S. Azaele, Asymmetric stochastic resetting: Modeling catastrophic events, Phys. Rev. E 102, 052116 (2020).
  48. M. Biroli, H. Larralde, S. N. Majumdar, and G. Schehr, Extreme statistics and spacing distribution in a Brownian gas correlated by resetting, Phys. Rev. Lett. 130, 207101 (2023).
  49. S. Y. Ali, N. Choudhury, and D. Mondal, Asymmetric restart in a stochastic climate model: A theoretical perspective to prevent the abnormal precipitation accumulation caused by global warming, J. Phys. A: Math. Theor. 55, 301001 (2022).
  50. M. Montero, J. Perelló, and J. Masoliver, Valuing the distant future under stochastic resettings: The effect on discounting, J. Phys. A: Math. Theor. 55, 464001 (2022).
  51. G. Mercado-Vásquez and D. Boyer, Lotka–Volterra systems with stochastic resetting, J. Phys. A: Math. Theor. 51, 405601 (2018).
  52. M. R. Evans, S. N. Majumdar, and G. Schehr, An exactly solvable predator prey model with resetting, J. Phys. A: Math. Theor. 55, 274005 (2022).
  53. A. Genthon, R. García-García, and D. Lacoste, Branching processes with resetting as a model for cell division, J. Phys. A: Math. Theor. 55, 074001 (2022).
  54. F. H. González, A. P. Riascos, and D. Boyer, Diffusive transport on networks with stochastic resetting to multiple nodes, Phys. Rev. E 103, 062126 (2021).
  55. P. Julián-Salgado, L. Dagdug, and D. Boyer, Diffusion with two resetting points, Phys. Rev. E 109, 024134 (2024).
  56. P. Julián-Salgado, L. Dagdug, and D. Boyer, Abrupt transitions in the optimization of diffusion with distributed resetting, Phys. Rev. E 112, 044110 (2025).
  57. A. Pal and V. V. Prasad, Landau-like expansion for phase transitions in stochastic resetting, Phys. Rev. Res. 1, 032001(R) (2019).
  58. S. Ahmad, K. Rijal, and D. Das, First passage in the presence of stochastic resetting and a potential barrier, Phys. Rev. E 105, 044134 (2022).
  59. S. Ray and S. Reuveni, Resetting transition is governed by an interplay between thermal and potential energy, J. Chem. Phys. 154, 171103 (2021).
  60. S. Jain, D. Boyer, A. Pal, and L. Dagdug, Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting, J. Chem. Phys. 158, 054113 (2023).
  61. J. C. Sunil, R. A. Blythe, M. R. Evans, and S. N. Majumdar, Minimizing the profligacy of searches with reset, Phys. Rev. E 110, 054122 (2024).
  62. L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and G. Schehr, First order transition for the optimal search time of Lévy flights with resetting, Phys. Rev. Lett. 113, 220602 (2014).
  63. R. Yin and E. Barkai, Restart expedites quantum walk hitting times, Phys. Rev. Lett. 130, 050802 (2023).
  64. T. Rotbart, S. Reuveni, and M. Urbakh, Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem, Phys. Rev. E 92, 060101(R) (2015).
  65. A. Chatterjee, C. Christou, and A. Schadschneider, Diffusion with resetting inside a circle, Phys. Rev. E 97, 062106 (2018).
  66. A. Kolmogoroff, Zur Umkehrbarkeit der statistischen Naturgesetze, Math. Ann. 113, 766 (1937).
  67. R. L. Stratonovich, Auxiliary information concerning probability theory and equilibrium thermodynamics, in Nonlinear Nonequilibrium Thermodynamics I: Linear and Nonlinear Fluctuation-Dissipation Theorems, edited by R. L. Stratonovich (Springer, Berlin, 1992), pp. 25–58.
  68. M. Polettini, Generally covariant state-dependent diffusion, J. Stat. Mech. (2013) P07005.
  69. P. Castro-Villarreal, A. Villada-Balbuena, J. M. Méndez-Alcaraz, R. Castañeda-Priego, and S. Estrada-Jiménez, A Brownian dynamics algorithm for colloids in curved manifolds, J. Chem. Phys. 140, 214115 (2014).
  70. S. Giordano, Stochastic thermodynamics of holonomic systems, Eur. Phys. J. B 92, 174 (2019).
  71. P. Castro-Villarreal, C. O. Solano-Cabrera, and R. Castañeda-Priego, Covariant description of the colloidal dynamics on curved manifolds, Front. Phys. 11, 1204751 (2023).
  72. C. O. Solano-Cabrera, P. Castro-Villarreal, R. E. Moctezuma, F. Donado, J. C. Conrad, and R. Castañeda-Priego, Self-assembly and transport phenomena of colloids: Confinement and geometrical effects, Annu. Rev. Condens. Matter Phys. 16, 41 (2025).
  73. M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, J. Phys. A: Math. Theor. 53, 193001 (2020).
  74. L. Dagdug, J. Peña, and I. Pompa-Gracía, Diffusion under Confinement: A Journey through Counterintuition (Springer, Cham, 2024).
  75. S. Reuveni, Optimal stochastic restart renders fluctuations in first passage times universal, Phys. Rev. Lett. 116, 170601 (2016).
  76. A. Pal and S. Reuveni, First passage under restart, Phys. Rev. Lett. 118, 030603 (2017).
  77. D. Boyer, G. Mercado-Vásquez, S. N. Majumdar, and G. Schehr, Optimizing the random search of a finite-lived target by a lévy flight, Phys. Rev. E 109, L022103 (2024).
  78. I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Series, and Products, 7th ed., edited by A. Jeffrey and D. Zwillinger (Academic Press, San Diego, CA, 2007).

Outline

Information

Phys. Rev. E 114, 044142– Published 30 October, 2026
  • Received 8 May 2026
  • Accepted 29 September 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation