- Physical Review Journals
All JournalsPhysics Magazine

APS Open Science

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

Resetting optimized competitive first-passage outcomes in non-Markovian systems

Suvam Pal1,*, Rahul Das2,†, and Arnab Pal2,‡

  • *Contact author: suvamjoy256@gmail.com
  • †Contact author: rahuldas@imsc.res.in
  • ‡Contact author: arnabpal@imsc.res.in
PDF

APS Open Sci. 1, 000155 – Published 30 September, 2026

DOI: https://doi.org/10.1103/tk5f-y65r

Abstract

We investigate the role of stochastic resetting in non-Markovian systems, where memory effects arise due to slow relaxation, rugged energy landscapes, disordered environments, and molecular crowding. Using the celebrated continuous-time random walk framework, we analyze first-passage processes with multiple competing outcomes and examine how resetting can selectively enhance desired outcomes. We characterize the efficiency of resetting through conditional mean first-passage times and demonstrate that its impact is highly sensitive to the underlying waiting-time statistics. Furthermore, we derive an inequality that quantifies how resetting controls fluctuations in conditional first-passage times, revealing regimes where variability is significantly suppressed. Our results provide a systematic understanding of how long-term memory influences competitive first-passage outcomes and establish resetting as a powerful control mechanism beyond the conventional Markovian setting.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (93)

  1. I. M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043 (2012).
  2. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  3. 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).
  4. J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
  5. R. Metzler, Brownian motion and beyond: First-passage, power spectrum, non-Gaussianity, and anomalous diffusion, J. Stat. Mech. (2019) 114003.
  6. K. Pearson, The problem of the random walk, Nature (London) 72, 342 (1905).
  7. T. Guérin, O. Bénichou, and R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem. 4, 568 (2012).
  8. O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Geometry-controlled kinetics, Nat. Chem. 2, 472 (2010).
  9. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, First-passage times in complex scale-invariant media, Nature (London) 450, 77 (2007).
  10. 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).
  11. K. Lindenberg, R. Metzler, and G. Oshanin, Chemical Kinetics: Beyond the Textbook (World Scientific, 2019).
  12. O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. 83, 81 (2011).
  13. B. M. Regner, D. Vučinić, C. Domnisoru, T. M. Bartol, M. W. Hetzer, D. M. Tartakovsky, and T. J. Sejnowski, Anomalous diffusion of single particles in cytoplasm, Biophys. J. 104, 1652 (2013).
  14. I. M. Sokolov, Lévy flights from a continuous-time process, Phys. Rev. E 63, 011104 (2000).
  15. I. M. Sokolov, Solutions of a class of non-Markovian Fokker-Planck equations, Phys. Rev. E 66, 041101 (2002).
  16. 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).
  17. A. Vezzani, E. Barkai, and R. Burioni, Single-big-jump principle in physical modeling, Phys. Rev. E 100, 012108 (2019).
  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. M. Höll, A. Nissan, B. Berkowitz, and E. Barkai, Controls that expedite first-passage times in disordered systems, Phys. Rev. E 108, 034124 (2023).
  20. J.-P. Bouchaud, Weak ergodicity breaking and aging in disordered systems, J. Phys. I 2, 1705 (1992).
  21. J. Rajoria and A. Pal, Broad distributions of sliding times are fingerprints of efficient target search on DNA, bioRxiv (2026), https://doi.org/10.64898/2026.03.21.713314.
  22. H. Kesten, Percolation theory and first-passage percolation, Ann. Probab. 15, 1231 (1987).
  23. M. Dai, Y. Sun, Y. Sun, L. Xi, and S. Shao, The entire mean weighted first-passage time on a family of weighted treelike networks, Sci. Rep. 6, 28733 (2016).
  24. M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601 (2011).
  25. M. R. Evans and S. N. Majumdar, Diffusion with optimal resetting, J. Phys. A: Math. Theor. 44, 435001 (2011).
  26. M. R. Evans, S. N. Majumdar, and K. Mallick, Optimal diffusive search: Nonequilibrium resetting versus equilibrium dynamics, J. Phys. A: Math. Theor. 46, 185001 (2013).
  27. A. Pal, Diffusion in a potential landscape with stochastic resetting, Phys. Rev. E 91, 012113 (2015).
  28. S. Reuveni, Optimal stochastic restart renders fluctuations in first passage times universal, Phys. Rev. Lett. 116, 170601 (2016).
  29. A. Pal and S. Reuveni, First passage under restart, Phys. Rev. Lett. 118, 030603 (2017).
  30. A. Chechkin and I. M. Sokolov, Random search with resetting: A unified renewal approach, Phys. Rev. Lett. 121, 050601 (2018).
  31. J. Masoliver and M. Montero, Anomalous diffusion under stochastic resettings: A general approach, Phys. Rev. E 100, 042103 (2019).
  32. W. Wang, A. G. Cherstvy, R. Metzler, and I. M. Sokolov, Restoring ergodicity of stochastically reset anomalous-diffusion processes, Phys. Rev. Res. 4, 013161 (2022).
  33. M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, J. Phys. A: Math. Theor. 53, 193001 (2020).
  34. 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).
  35. S. Gupta and A. M. Jayannavar, Stochastic resetting: A (very) brief review, Front. Phys. 10, 789097 (2022).
  36. A. Kumar and A. Pal, Universal framework for record ages under restart, Phys. Rev. Lett. 130, 157101 (2023).
  37. S. Pal, D. Boyer, L. Dagdug, and A. Pal, Channel-facilitated transport under resetting dynamics, J. Chem. Phys. 161, 144114 (2024).
  38. A. Biswas, S. N. Majumdar, and A. Pal, Target search optimization by threshold resetting, Phys. Rev. Lett. 135, 227101 (2025).
  39. S. Pal and A. Pal, Optimal resetting mediated universal fluctuations in conditional first-passage times: Application to diffusive transport processes, Phys. Fluids 37, 077126 (2025).
  40. T. Sandev, A. Iomin, J. Kurths, and L. Kocarev, Shear-driven anomalous diffusion: Memory effects and stochastic resetting, Phys. Fluids 37, 067101 (2025).
  41. Y. Liang, Q. Wei, W. Wang, and A. G. Cherstvy, Ultraslow diffusion processes under stochastic resetting, Phys. Fluids 37, 032014 (2025).
  42. P. Trajanovski, I. Petreska, K. Górska, L. Kocarev, and T. Sandev, Generalized diffusion process with nonlocal interactions: Continuous time random walk model and stochastic resetting, Rep. Math. Phys. 96, 385 (2025).
  43. A. Pal and V. Prasad, First passage under stochastic resetting in an interval, Phys. Rev. E 99, 032123 (2019).
  44. A. Pal, Ł. Kuśmierz, and S. Reuveni, Search with home returns provides advantage under high uncertainty, Phys. Rev. Res. 2, 043174 (2020).
  45. A. Pal and V. Prasad, Landau-like expansion for phase transitions in stochastic resetting, Phys. Rev. Res. 1, 032001(R) (2019).
  46. É. Roldán and S. Gupta, Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement, Phys. Rev. E 96, 022130 (2017).
  47. U. Basu, A. Kundu, and A. Pal, Symmetric exclusion process under stochastic resetting, Phys. Rev. E 100, 032136 (2019).
  48. S. Gupta and A. Nagar, Resetting of fluctuating interfaces at power-law times, J. Phys. A: Math. Theor. 49, 445001 (2016).
  49. A. Nagar and S. Gupta, Diffusion with stochastic resetting at power-law times, Phys. Rev. E 93, 060102(R) (2016).
  50. A. S. Bodrova and I. M. Sokolov, Continuous-time random walks under power-law resetting, Phys. Rev. E 101, 062117 (2020).
  51. D. Boyer and S. N. Majumdar, Power-law relaxation of a confined diffusing particle subject to resetting with memory, J. Stat. Mech. (2024) 073206.
  52. A. Pal, V. Stojkoski, and T. Sandev, Random resetting in search problems, in Target Search Problems (Springer, 2024), pp. 323–355.
  53. A. Biswas, J. L. Dubbeldam, T. Sandev, and A. Pal, A resetting particle embedded in a viscoelastic bath, Chaos 35, 031102 (2025).
  54. A. Biswas, S. N. Majumdar, and A. Pal, Optimal threshold resetting in collective diffusive search, J. Phys. A: Math. Theor. 59, 385002 (2026).
  55. 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).
  56. 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.
  57. 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, Phys. Rev. Res. 6, 023255 (2024).
  58. R. Vatash and Y. Roichman, Many-body colloidal dynamics under stochastic resetting: Competing effects of particle interactions on the steady-state distribution, Phys. Rev. Res. 7, L032020 (2025).
  59. S. Paramanick, A. Biswas, H. Soni, A. Pal, and N. Kumar, Uncovering universal characteristics of homing paths using foraging robots, PRX Life 2, 033007 (2024).
  60. R. Goerlich, M. Li, L. B. Pires, P.-A. Hervieux, G. Manfredi, and C. Genet, Taming a Maxwell's demon for experimental stochastic resetting, Phys. Rev. E 112, 064116 (2025).
  61. S. Kundu, D. Mondal, A. Biswas, A. Pal, and M. Khan, Emulating microbial run-and-tumble and tactic motion by stochastically reorienting synthetic active Brownian particles, arXiv:2509.21903.
  62. A. Pal, S. Kostinski, and S. Reuveni, The inspection paradox in stochastic resetting, J. Phys. A: Math. Theor. 55, 021001 (2022).
  63. T. D. Keidar, S. Meir, N. Sherf, R. Goerlich, S. Reuveni, Y. Roichman, and B. Hirshberg, Stochastic resetting: A non-equilibrium framework for prediction, inference and design, arXiv:2607.16474.
  64. E. W. Montroll and G. H. Weiss, Random walks on lattices. II, J. Math. Phys. 6, 167 (1965).
  65. V. Méndez, A. Masó-Puigdellosas, and D. Campos, Nonstandard diffusion under Markovian resetting in bounded domains, Phys. Rev. E 105, 054118 (2022).
  66. V. Méndez, R. Flaquer-Galmés, and A. Pal, Occupation-time statistics for non-Markovian random walks, Phys. Rev. E 111, 044119 (2025).
  67. O. Bénichou and R. Voituriez, From first-passage times of random walks in confinement to geometry-controlled kinetics, Phys. Rep. 539, 225 (2014).
  68. R. Metzler, S. Redner, and G. Oshanin, First-Passage Phenomena and Their Applications (World Scientific, 2014), Vol. 35.
  69. S. Pal, L. Dagdug, D. Ghosh, D. Boyer, and A. Pal, Universal criterion for selective outcomes under stochastic resetting, Phys. Rev. E 112, 034116 (2025).
  70. A. Szabo, K. Schulten, and Z. Schulten, First passage time approach to diffusion controlled reactions, J. Chem. Phys. 72, 4350 (1980).
  71. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, 2001).
  72. S. Condamin, O. Bénichou, and M. Moreau, First-passage times for random walks in bounded domains, Phys. Rev. Lett. 95, 260601 (2005).
  73. J. Elf, G.-W. Li, and X. S. Xie, Probing transcription factor dynamics at the single-molecule level in a living cell, Science 316, 1191 (2007).
  74. D. Holcman and Z. Schuss, The narrow escape problem, SIAM Rev. 56, 213 (2014).
  75. M. Dolgushev, T. V. Mendes, B. Gorin, K. Xie, N. Levernier, O. Bénichou, H. Kellay, R. Voituriez, and T. Guérin, Evidence and quantification of memory effects in competitive first-passage events, Sci. Adv. 11, eadp2386 (2025).
  76. H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12, 2455 (1975).
  77. B. Berkowitz and H. Scher, Theory of anomalous chemical transport in random fracture networks, Phys. Rev. E 57, 5858 (1998).
  78. R. D. Hanes, C. Dalle-Ferrier, M. Schmiedeberg, M. C. Jenkins, and S. U. Egelhaaf, Colloids in one dimensional random energy landscapes, Soft Matter 8, 2714 (2012).
  79. A. M. Berezhkovskii, M. A. Pustovoit, and S. M. Bezrukov, Channel-facilitated membrane transport: Transit probability and interaction with the channel, J. Chem. Phys. 116, 9952 (2002).
  80. R. Satija, A. M. Berezhkovskii, and D. E. Makarov, Broad distributions of transition-path times are fingerprints of multidimensionality of the underlying free energy landscapes, Proc. Natl. Acad. Sci. USA 117, 27116 (2020).
  81. A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, Exact solutions for distributions of first-passage, direct-transit, and looping times in symmetric cusp potential barriers and wells, J. Phys. Chem. B 123, 3786 (2019).
  82. L. Dagdug, J. Peña, and I. Pompa-García, Diffusion Under Confinement: A Journey Through Counterintuition (Springer, 2024), Vol. 1.
  83. A. Pal, A. Kundu, and M. R. Evans, Diffusion under time-dependent resetting, J. Phys. A: Math. Theor. 49, 225001 (2016).
  84. 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).
  85. 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, 2005RG000178 (2006).
  86. X. Brokmann, J.-P. Hermier, G. Messin, P. Desbiolles, J.-P. Bouchaud, and M. Dahan, Statistical aging and nonergodicity in the fluorescence of single nanocrystals, Phys. Rev. Lett. 90, 120601 (2003).
  87. A. V. Weigel, B. Simon, M. M. Tamkun, and D. Krapf, Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking, Proc. Natl. Acad. Sci. USA 108, 6438 (2011).
  88. S. C. Kou and X. S. Xie, Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule, Phys. Rev. Lett. 93, 180603 (2004).
  89. F. Santamaria, S. Wils, E. De Schutter, and G. J. Augustine, Anomalous diffusion in Purkinje cell dendrites caused by spines, Neuron 52, 635 (2006).
  90. S. Fedotov and V. Méndez, Non-Markovian model for transport and reactions of particles in spiny dendrites, Phys. Rev. Lett. 101, 218102 (2008).
  91. F. Santamaria, S. Wils, E. De Schutter, and G. J. Augustine, The diffusional properties of dendrites depend on the density of dendritic spines, Eur. J. Neurosci. 34, 561 (2011).
  92. G. Margolin, V. Protasenko, M. Kuno, and E. Barkai, Power law blinking quantum dots: Stochastic and physical models, Adv. Chem. Phys. 133, 327 (2006).
  93. M. Biroli, S. Ciliberto, M. Kulkarni, S. N. Majumdar, A. Petrosyan, and G. Schehr, Experimental evidence for strong emergent correlations between particles in a switching trap, Phys. Rev. Lett. 137, 037102 (2026).

Outline

Information

APS Open Sci. 1, 000155– Published 30 September, 2026
Vol. 1
  • Received 27 May 2026
  • Revised 20 August 2026
  • Accepted 15 September 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