- 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

Macroscopic localization and collective memory in Poisson renewal resetting

Ohad Vilk*

  • *Contact author: ohad.vilk@mail.huji.ac.il
PDF

Phys. Rev. Research 8, 023305 – Published 16 June, 2026

DOI: https://doi.org/10.1103/2rz6-8n5v

Abstract

Stochastic renewal processes are ubiquitous across physics, biology, and the social sciences. Here, we show that continuous-time renewal dynamics can naturally produce a mixed discrete-continuous structure, with a macroscopic fraction of particles occupying a discrete state. For ensembles of continuous-time random walkers subject to Poissonian renewal resets, we develop an age-structured framework showing that this discrete component corresponds to localization at the reset configuration. We next show that collective interactions can retain memory although all reset events are memoryless. Remarkably, the transition to collective memory is discontinuous, and we identify a discontinuous dynamical phase transition from weak collective bias, where the dynamics are stationary, to strong collective bias, where the dynamics are nonstationary and display aging up to finite-size effects. We explicitly discuss ecological implications of our work, illustrating how continuous-time renewal dynamics shape macroscopic structure and collective organization with long-term memory.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. 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).
  2. S. Bianco, M. Ignaccolo, M. S. Rider, M. J. Ross, P. Winsor, and P. Grigolini, Brain, music, and non-Poisson renewal processes, Phys. Rev. E 75, 061911 (2007).
  3. M. C. Gonzalez, C. A. Hidalgo, and A.-L. Barabasi, Understanding individual human mobility patterns, Nature (London) 453, 779 (2008).
  4. V. Zaburdaev, S. Denisov, and J. Klafter, Lévy walks, Rev. Mod. Phys. 87, 483 (2015).
  5. J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, and R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules, Phys. Rev. Lett. 106, 048103 (2011).
  6. 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).
  7. E. Barkai and S. Burov, Packets of diffusing particles exhibit universal exponential tails, Phys. Rev. Lett. 124, 060603 (2020).
  8. O. Vilk, Y. Orchan, M. Charter, N. Ganot, S. Toledo, R. Nathan, and M. Assaf, Ergodicity breaking in area-restricted search of avian predators, Phys. Rev. X 12, 031005 (2022).
  9. O. Vilk, E. Aghion, T. Avgar, C. Beta, O. Nagel, A. Sabri, R. Sarfati, D. K. Schwartz, M. Weiss, D. Krapf, et al., Unravelling the origins of anomalous diffusion: From molecules to migrating storks, Phys. Rev. Res. 4, 033055 (2022).
  10. O. Vilk, E. Aghion, R. Nathan, S. Toledo, R. Metzler, and M. Assaf, Classification of anomalous diffusion in animal movement data using power spectral analysis, J. Phys. A: Math. Theor. 55, 334004 (2022).
  11. C. Godrèche and J. Luck, Statistics of the occupation time of renewal processes, J. Stat. Phys. 104, 489 (2001).
  12. J. H. Schulz, E. Barkai, and R. Metzler, Aging renewal theory and application to random walks, Phys. Rev. X 4, 011028 (2014).
  13. R. Cakir, P. Grigolini, and A. A. Krokhin, Dynamical origin of memory and renewal, Phys. Rev. E 74, 021108 (2006).
  14. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  15. G. Bel and E. Barkai, Weak ergodicity breaking in the continuous-time random walk, Phys. Rev. Lett. 94, 240602 (2005).
  16. R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys. 16, 24128 (2014).
  17. M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601 (2011).
  18. S. Reuveni, M. Urbakh, and J. Klafter, The role of substrate unbinding in Michaelis-Menten enzymatic reactions, Biophys. J. 106, 677a (2014).
  19. A. Kundu and S. Reuveni, Preface: Stochastic resetting—Theory and applications, J. Phys. A: Math. Theor. 57, 060301 (2024).
  20. J. R. Koenig, O. Padon, and A. Aiken, Adaptive restarts for stochastic synthesis, in Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation (Association for Computing Machinery, New York, NY, 2021), pp. 696–709.
  21. S. Picardi, B. J. Smith, M. E. Boone, P. C. Frederick, J. G. Cecere, D. Rubolini, L. Serra, S. Pirrello, R. R. Borkhataria, and M. Basille, Analysis of movement recursions to detect reproductive events and estimate their fate in central place foragers, Mov. Ecol. 8, 24 (2020).
  22. O. Blumer, S. Reuveni, and B. Hirshberg, Stochastic resetting for enhanced sampling, J. Phys. Chem. Lett. 13, 11230 (2022).
  23. M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, J. Phys. A: Math. Theor. 53, 193001 (2020).
  24. S. Gupta and A. M. Jayannavar, Stochastic resetting: A (very) brief review, Front. Phys. 10, 789097 (2022).
  25. S. Meir, T. D. Keidar, S. Reuveni, and B. Hirshberg, First-passage approach to optimizing perturbations for improved training of machine learning models, Mach. Learn.: Sci. Technol. 6, 025053 (2025).
  26. J. R. Church, O. Blumer, T. D. Keidar, L. Ploutno, S. Reuveni, and B. Hirshberg, Accelerating molecular dynamics through informed resetting, J. Chem. Theory Comput. 21, 605 (2025).
  27. Ł. Kuśmierz and E. Gudowska-Nowak, Subdiffusive continuous-time random walks with stochastic resetting, Phys. Rev. E 99, 052116 (2019).
  28. A. Masó-Puigdellosas, D. Campos, and V. Méndez, Transport properties and first-arrival statistics of random motion with stochastic reset times, Phys. Rev. E 99, 012141 (2019).
  29. A. Nagar and S. Gupta, Stochastic resetting in interacting particle systems: A review, J. Phys. A: Math. Theor. 56, 283001 (2023).
  30. M. Siboni, P. Sasorov, and B. Meerson, Fluctuations of a swarm of Brownian bees, Phys. Rev. E 104, 054131 (2021).
  31. 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).
  32. 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).
  33. M. Biroli, S. N. Majumdar, and G. Schehr, Resetting Dyson Brownian motion, Phys. Rev. E 112, 014101 (2025).
  34. O. Vilk and B. Meerson, Collective behavior of independent scaled Brownian particles with renewal resetting, Physica A 692, 131542 (2026).
  35. O. Vilk, M. Assaf, and B. Meerson, Fluctuations and first-passage properties of systems of Brownian particles with reset, Phys. Rev. E 106, 024117 (2022).
  36. J. Berestycki, É. Brunet, J. Nolen, and S. Penington, Brownian bees in the infinite swarm limit, Ann. Probab. 50, 2133 (2022).
  37. J. Berestycki, É. Brunet, J. Nolen, and S. Penington, A free boundary problem arising from branching Brownian motion with selection, Trans. Am. Math. Soc. 374, 6269 (2021).
  38. B. Meerson and O. Vilk, Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting, Phys. Rev. Res. 8, 023103 (2026).
  39. Aging is defined here as the breakdown of time-translation invariance in macroscopic observables, manifested by their explicit dependence on the time since system preparation. In renewal processes, this is typically associated with a time-dependent renewal rate.
  40. J. P. Garrahan, R. L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, and F. van Wijland, Dynamical first-order phase transition in kinetically constrained models of glasses, Phys. Rev. Lett. 98, 195702 (2007).
  41. G. Gradenigo and S. N. Majumdar, A first-order dynamical transition in the displacement distribution of a driven run-and-tumble particle, J. Stat. Mech. (2019) 053206.
  42. J. Meibohm and M. Esposito, Finite-time dynamical phase transition in nonequilibrium relaxation, Phys. Rev. Lett. 128, 110603 (2022).
  43. H. Berry, T. Lepoutre, and Á. M. González, Quantitative convergence towards a self-similar profile in an age-structured renewal equation for subdiffusion, Acta Appl. Math. 145, 15 (2016).
  44. By “diffusion limit,” we mean the continuum limit ℓ→0, τ0→0 with the effective transport coefficient Deff held fixed; this reduces the discrete CTRW to a (fractional) diffusion equation regardless of whether transport is normal or anomalous.
  45. M. R. Evans and S. N. Majumdar, Effects of refractory period on stochastic resetting, J. Phys. A: Math. Theor. 52, 01LT01 (2019).
  46. M. M. Meerschaert and H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab. 41, 623 (2004).
  47. J. C. Alonso, J. A. Alonso, and L. M. Carrascal, Habitat selection by foraging White Storks, Ciconia ciconia, during the breeding season, Can. J. Zool. 69, 1957 (1991).
  48. D. Zurell, H. Von Wehrden, S. Rotics, M. Kaatz, H. Groß, L. Schlag, M. Schäfer, N. Sapir, S. Turjeman, M. Wikelski, et al., Home range size and resource use of breeding and non-breeding white storks along a land use gradient, Front. Ecol. Evol. 6, 79 (2018).
  49. G. Blanco, Population dynamics and communal roosting of white storks foraging at a Spanish refuse dump, Colonial Waterbirds 19, 273 (1996).
  50. A. Flack, M. Nagy, W. Fiedler, I. D. Couzin, and M. Wikelski, From local collective behavior to global migratory patterns in white storks, Science 360, 911 (2018).
  51. R. Lande, S. Engen, and B.-E. Sæther, Extinction times in finite metapopulation models with stochastic local dynamics, Oikos 83, 383 (1998).
  52. A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biol. Cybern. 95, 1 (2006).
  53. H. Kautz, E. Horvitz, Y. Ruan, C. Gomes, and B. Selman, Dynamic restart policies, AAAI/IAAI 97, 674 (2002).
  54. https://gitlab.com/ohad.vilk/macroscopic_localization_and_collective_memory_in_poisson_renewal_resetting.
  55. D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22, 403 (1976).
  56. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).

Outline

Information

Phys. Rev. Research 8, 023305– Published 16 June, 2026
  • Received 20 January 2026
  • Accepted 19 May 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