- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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

Active wave-particle clusters

Rahil N. Valani1,* and David M. Paganin2,†

  • *Contact author: rahil.valani@physics.ox.ac.uk
  • †Contact author: david.paganin@monash.edu
PDF

Phys. Rev. E 112, 065103 – Published 8 December, 2025

DOI: https://doi.org/10.1103/4cgg-hnyh

Abstract

Active particles are nonequilibrium entities that uptake energy and convert it into self-propulsion. A dynamically rich class of inertial active particles having features of wave-particle coupling and wave memory are walking/superwalking droplets. Such classical, active wave-particle entities (WPEs) have previously been shown to exhibit hydrodynamic analogs of many single-particle quantum systems. Inspired by the rich dynamics of strongly interacting superwalking droplets in experiments, we numerically investigate the dynamics of WPE clusters using a stroboscopic model. We find that several interacting WPEs self-organize into a stable bound cluster, reminiscent of an atomic nucleus. This active cluster exhibits a rich spectrum of collective excitations, including shape oscillations and chiral rotating modes, akin to vibrational and rotational modes of nuclear excitations, as the spatial extent of the waves and their temporal decay rate (memory) are varied. Dynamically distinct excitation modes create a common time-averaged collective wave field potential, bearing qualitative similarities with the nuclear shell model and the bag model of hadrons. For high memory and rapid spatial decay of waves, the active cluster becomes unstable and disintegrates; however, within a narrow regime of the parameter space, the cluster ejects single particles whose decay statistics follow exponential laws, reminiscent of radioactive nuclear decay. Our study uncovers a rich spectrum of dynamical behaviors in clusters of active particles, opening new avenues for exploring hydrodynamic quantum analogs in active matter systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (93)

  1. V. I. Strazhev and P. L. Shkol'nikov, Electrodynamics without potentials, Sov. Phys. J. 32, 378 (1989).
  2. B. S. DeWitt, Quantum theory without electromagnetic potentials, Phys. Rev. 125, 2189 (1962).
  3. R. M. Wald, Advanced Classical Electromagnetism (Princeton University Press, Princeton, NJ, 2022), pp. 8–9.
  4. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, Hoboken, NJ, 1999).
  5. P. A. M. Dirac, Classical theory of radiating electrons, Proc. R. Soc. Lond. A 167, 148 (1938).
  6. E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, NY, 1998), pp. 560–564.
  7. M. F. de la Ripelle and H. Fiedeldey, An integro-differential equation approach to the many-body problem, Phys. Lett. B 171, 325 (1986).
  8. M. F. de la Ripelle, H. Fiedeldey, and S. A. Sofianos, Integrodifferential equation for few- and many-body systems, Phys. Rev. C 38, 449 (1988).
  9. W. Oehm, S. A. Sofianos, H. Fiedeldey, and M. F. de la Ripelle, Integrodifferential equation approach. I. Triton and α-particle binding energies, Phys. Rev. C 42, 2322 (1990).
  10. W. Oehm, S. A. Sofianos, H. Fiedeldey, and M. F. de la Ripelle, Integro-differential equation approach. II. Triton and α-particle wave functions, graphical plots, Phys. Rev. C 43, 25 (1991).
  11. W. Oehm, H. Fiedeldey, S. A. Sofianos, and M. F. de la Ripelle, Integro-differential equation approach. III. Triton and α-particle bound states. Realistic forces and two-body correlations, Phys. Rev. C 44, 81 (1991).
  12. R. M. Adam, S. A. Sofianos, H. Fiedeldey, and M. F. de la Ripelle, Integro-differential equation approach extended to larger nuclei, J. Phys. G: Nucl. Part. Phys. 18, 1365 (1992).
  13. S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  14. L. Pismen, Active colloids, in Active Matter Within and Around Us: From Self-Propelled Particles to Flocks and Living Forms (Springer International Publishing, Cham, 2021), pp. 43–64.
  15. L. Pismen, Motion of microorganisms, in Active Matter Within and Around Us: From Self-Propelled Particles to Flocks and Living Forms (Springer International Publishing, Cham, 2021), pp. 65–90.
  16. R. N. Valani, A. C. Slim, and T. Simula, Superwalking droplets, Phys. Rev. Lett. 123, 024503 (2019).
  17. Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Walking and orbiting droplets, Nature (London) 437, 208 (2005).
  18. S. Protière, A. Boudaoud, and Y. Couder, Particle-wave association on a fluid interface, J. Fluid Mech. 554, 85 (2006).
  19. J. W. M. Bush, Pilot-wave hydrodynamics, Annu. Rev. Fluid Mech. 47, 269 (2015).
  20. J. W. M. Bush and A. U. Oza, Hydrodynamic quantum analogs, Rep. Prog. Phys. 84, 017001 (2021).
  21. J. W. M. Bush, K. Papatryfonos, and V. Frumkin, The state of play in hydrodynamic quantum analogs, in Advances in Pilot Wave Theory: From Experiments to Foundations, edited by P. Castro, J. W. M. Bush, and J. Croca (Springer International Publishing, Cham, 2024), pp. 7–34.
  22. J. W. M. Bush, V. Frumkin, and P. J. Sáenz, Perspectives on pilot-wave hydrodynamics, Appl. Phys. Lett. 125, 030503 (2024).
  23. E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Y. Couder, Path-memory induced quantization of classical orbits, Proc. Natl. Acad. Sci. USA 107, 17515 (2010).
  24. D. M. Harris and J. W. M. Bush, Droplets walking in a rotating frame: From quantized orbits to multimodal statistics, J. Fluid Mech. 739, 444 (2014).
  25. A. U. Oza, D. M. Harris, R. R. Rosales, and J. W. M. Bush, Pilot-wave dynamics in a rotating frame: On the emergence of orbital quantization, J. Fluid Mech. 744, 404 (2014).
  26. S. Perrard, M. Labousse, E. Fort, and Y. Couder, Chaos driven by interfering memory, Phys. Rev. Lett. 113, 104101 (2014).
  27. S. Perrard, M. Labousse, M. Miskin, E. Fort, and Y. Couder, Self-organization into quantized eigenstates of a classical wave-driven particle, Nat. Commun. 5, 3219 (2014).
  28. M. Labousse, S. Perrard, Y. Couder, and E. Fort, Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction, Phys. Rev. E 94, 042224 (2016).
  29. A. Eddi, E. Fort, F. Moisy, and Y. Couder, Unpredictable tunneling of a classical wave-particle association, Phys. Rev. Lett. 102, 240401 (2009).
  30. A. Nachbin, P. A. Milewski, and J. W. M. Bush, Tunneling with a hydrodynamic pilot-wave model, Phys. Rev. Fluids 2, 034801 (2017).
  31. L. Tadrist, T. Gilet, P. Schlagheck, and J. W. M. Bush, Predictability in a hydrodynamic pilot-wave system: Resolution of walker tunneling, Phys. Rev. E 102, 013104 (2020).
  32. D. M. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. W. M. Bush, Wavelike statistics from pilot-wave dynamics in a circular corral, Phys. Rev. E 88, 011001(R) (2013).
  33. T. Gilet, Quantumlike statistics of deterministic wave-particle interactions in a circular cavity, Phys. Rev. E 93, 042202 (2016).
  34. P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, Statistical projection effects in a hydrodynamic pilot-wave system, Nat. Phys. 14, 315 (2018).
  35. T. Cristea-Platon, P. J. Sáenz, and J. W. M. Bush, Walking droplets in a circular corral: Quantisation and chaos, Chaos 28, 096116 (2018).
  36. M. Durey, P. A. Milewski, and Z. Wang, Faraday pilot-wave dynamics in a circular corral, J. Fluid Mech. 891, A3 (2020).
  37. P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, A hydrodynamic analog of Friedel oscillations, Sci. Adv. 6, eaay9234 (2020).
  38. K. Papatryfonos, M. Ruelle, C. Bourdiol, A. Nachbin, J. W. M. Bush, and M. Labousse, Hydrodynamic superradiance in wave-mediated cooperative tunneling, Commun. Phys. 5, 142 (2022).
  39. V. Frumkin, J. W. M. Bush, and K. Papatryfonos, Superradiant droplet emission from parametrically excited cavities, Phys. Rev. Lett. 130, 064002 (2023).
  40. A. Hélias and M. Labousse, Statistical self-organization of an assembly of interacting walking drops in a confining potential, Eur. Phys. J. E 46, 29 (2023).
  41. T. P. Simula, Droplet time crystals, Phys. Scr. 98, 035004 (2023).
  42. M. M. P. Couchman and J. W. M. Bush, Free rings of bouncing droplets: Stability and dynamics, J. Fluid Mech. 903, A49 (2020).
  43. S. Protière, Y. Couder, E. Fort, and A. Boudaoud, The self-organization of capillary wave sources, J. Phys.: Condens. Matter 17, S3529 (2005).
  44. A. Eddi, A. Boudaoud, and Y. Couder, Oscillating instability in bouncing droplet crystals, Europhys. Lett. 94, 20004 (2011).
  45. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  46. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016).
  47. M. E. Cates and J. Tailleur, Motility-induced phase separation, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  48. A. Mauleon-Amieva, M. P. Allen, and C. P. Royall, Dynamics and interactions of Quincke roller clusters: From orbits and flips to excited states, Sci. Adv. 9, eabf5144 (2021).
  49. I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert, and L. Bocquet, Dynamic clustering in active colloidal suspensions with chemical signaling, Phys. Rev. Lett. 108, 268303 (2012).
  50. A. U. Oza, R. R. Rosales, and J. W. M. Bush, A trajectory equation for walking droplets: Hydrodynamic pilot-wave theory, J. Fluid Mech. 737, 552 (2013).
  51. A. P. Damiano, P.-T. Brun, D. M. Harris, C. A. Galeano-Rios, and J. W. M. Bush, Surface topography measurements of the bouncing droplet experiment, Exp. Fluids 57, 163 (2016).
  52. R. N. Valani, A. C. Slim, D. M. Paganin, T. P. Simula, and T. Vo, Unsteady dynamics of a classical particle-wave entity, Phys. Rev. E 104, 015106 (2021).
  53. J. Moláček and J. W. M. Bush, Drops walking on a vibrating bath: Towards a hydrodynamic pilot-wave theory, J. Fluid Mech. 727, 612 (2013).
  54. R. N. Valani, J. Dring, T. P. Simula, and A. C. Slim, Emergence of superwalking droplets, J. Fluid Mech. 906, A3 (2021).
  55. P. N. Bogolioubov, Sur un modèle à quarks quasi-indépendants, Annales de l'institut Henri Poincaré. Section A, Physique Théorique 8, 163 (1968).
  56. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, New extended model of hadrons, Phys. Rev. D 9, 3471 (1974).
  57. A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn, Baryon structure in the bag theory, Phys. Rev. D 10, 2599 (1974).
  58. B. L. Cohen, Concepts of Nuclear Physics (McGraw–Hill, New York, NY, 1971), p. 67.
  59. K. S. Krane, Introductory Nuclear Physics (Wiley, New York, NY, 1987).
  60. A. Eddi, A. Decelle, E. Fort, and Y. Couder, Archimedean lattices in the bound states of wave interacting particles, Europhys. Lett. 87, 56002 (2009).
  61. See Supplemental Material at http://link.aps.org/supplemental/10.1103/4cgg-hnyh for Videos S1–S15.
  62. J. P. Blaizot, Nuclear compressibilities, Phys. Rep. 64, 171 (1980).
  63. D. H. Youngblood, H. L. Clark, and Y.-W. Lui, Incompressibility of nuclear matter from the giant monopole resonance, Phys. Rev. Lett. 82, 691 (1999).
  64. D. Vretenar, N. Paar, P. Ring, and T. Nikšić, Toroidal dipole resonances in the relativistic random phase approximation, Phys. Rev. C 65, 021301(R) (2002).
  65. A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. II: Nuclear Deformations (W. A. Benjamin, Reading, MA, 1975).
  66. P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980).
  67. K. Heyde, Basic Ideas and Concepts in Nuclear Physics, 3rd ed. (CRC Press, Boca Raton, FL, 2017).
  68. B. V. Hokmabad, A. Nishide, P. Ramesh, C. Krüger, and C. C. Maass, Spontaneously rotating clusters of active droplets, Soft Matter 18, 2731 (2022).
  69. J. Yan, M. Bloom, S. C. Bae, E. Luijten, and S. Granick, Linking synchronization to self-assembly using magnetic Janus colloids, Nature (London) 491, 578 (2012).
  70. M. R. Nejad and J. M. Yeomans, Spontaneous rotation of active droplets in two and three dimensions, PRX Life 1, 023008 (2023).
  71. The finite size of the particle as a disk of diameter a is taken into account while calculating the probability density for particles.
  72. C. DeTar and J. F. Donoghue, Bag models of hadrons, Annu. Rev. Nucl. Part. Sci. 33, 235 (1983).
  73. N. B. Padhan and R. Pandit, Activity-induced droplet propulsion and multifractality, Phys. Rev. Res. 5, L032013 (2023).
  74. M. Durey, P. A. Milewski, and J. W. M. Bush, Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets, Chaos 28, 096108 (2018).
  75. Y. Ye, X. Yang, H. Sakurai, and B. Hu, Physics of exotic nuclei, Nat. Rev. Phys. 7, 21 (2025).
  76. C. Kittel, Introduction to Solid State Physics, 8th ed. (John Wiley & Sons, New York, NY, 2005).
  77. T. Tan, X. Ma, D. Ran, M. Dao, G. Li, D. A. Weitz, and D. Bi, Odd dynamics of living chiral crystals, Nature (London) 607, 287 (2022).
  78. A. F. Kay, M. Durey, and J. W. M. Bush, Classical pilot-wave dynamics in three dimensions, Proc. R. Soc. A 481, 20240986 (2025).
  79. H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer Verlag, Berlin, 1989).
  80. D. Pelliccia and D. M. Paganin, Synchrotron Light (Oxford University Press, Oxford, UK, 2025).
  81. J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, Boston, MA, 1967).
  82. D. Darrow and J. W. M. Bush, Single-particle Fraunhofer diffraction in a classical pilot-wave model, Phys. Rev. Res. 7, 033288 (2025).
  83. J. Moláček and J. W. M. Bush, Drops bouncing on a vibrating bath, J. Fluid Mech. 727, 582 (2013).
  84. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, CA, 2003).
  85. N. Akhmediev, J. M. Soto-Crespo, and G. Town, Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach, Phys. Rev. E 63, 056602 (2001).
  86. P. B. Umbanhowar, F. Melo, and H. L. Swinney, Localized excitations in a vertically vibrated granular layer, Nature (London) 382, 793 (1996).
  87. E. Clément, L. Vanel, J. Rajchenbach, and J. Duran, Pattern formation in a vibrated two-dimensional granular layer, Phys. Rev. E 53, 2972 (1996).
  88. L. S. Tsimring and I. S. Aranson, Localized and cellular patterns in a vibrated granular layer, Phys. Rev. Lett. 79, 213 (1997).
  89. I. S. Aranson and L. S. Tsimring, Patterns and collective behavior in granular media: Theoretical concepts, Rev. Mod. Phys. 78, 641 (2006).
  90. G. El and A. Tovbis, Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation, Phys. Rev. E 101, 052207 (2020).
  91. L. de Broglie, Non-Linear Wave Mechanics: A Causal Interpretation (Elsevier, Amsterdam, 1960).
  92. Page 289 of the book by de Broglie [91] suggests—in the context of the “double solution” model—that “a piling up of [a] great number of singular regions” which are “very close together or even overlapping might permit the correct forecasting of certain nuclear phenomena.” This conjecture bears some conceptual similarity to the core study of the present paper.
  93. G. 't Hooft, The Cellular Automaton Interpretation of Quantum Mechanics (Springer, Cham, 2016).

Outline

Information

Phys. Rev. E 112, 065103– Published 8 December, 2025
  • Received 19 March 2025
  • Revised 20 October 2025
  • Accepted 6 November 2025
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