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

Leveraging interactions for energy-efficient swarm-based Brownian computing

Alessandro Pignedoli, Atreya Majumdar*, and Karin Everschor-Sitte†

  • *Contact author: atreya.majumdar@uni-due.de
  • †Contact author: karin.everschor-sitte@uni-due.de

Phys. Rev. Research 8, 033381 – Published 30 September, 2026

DOI: https://doi.org/10.1103/k8wj-xsl7

Abstract

Drawing inspiration from swarm intelligence, we show that short-range attractive interactions between thermally driven Brownian quasiparticles enable energy-efficient optimization. As quasiparticles can be generated directly within a material, the swarm size can be adjusted with minimal energy overhead. Using an optimization task defined by a spatially varying temperature landscape, we quantitatively show that interacting swarms reliably identify global optima and significantly outperform noninteracting searchers within a well-defined regime of interaction strength and swarm size. This improvement arises from emergent cooperative behavior, where local interactions guide the swarm toward high-quality solutions without central coordination. To link our physical model to experimental realizations, we coarse grain the quasiparticle dynamics onto a sensor lattice and generate trajectories emulating particle-tracking measurements. We further show that the interacting swarm adapts robustly to landscapes that evolve over time. These findings establish interacting Brownian quasiparticles as a physical platform for scalable and energy-efficient unconventional computing.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (101)

  1. E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Santa Fe Institute Studies in the Sciences of Complexity (Oxford University Press, New York, NY, 1999).
  2. M. G. Hinchey, R. Sterritt, and C. Rouff, Swarms and swarm intelligence, Computer 40, 111 (2007).
  3. M. Rubenstein, A. Cornejo, and R. Nagpal, Programmable self-assembly in a thousand-robot swarm, Science 345, 795 (2014).
  4. Nature-Inspired Computing and Optimization: Theory and Applications, Modeling and Optimization in Science and Technologies, edited by S. Patnaik, X.-S. Yang, and K. Nakamatsu (Springer International Publishing, Cham, Switzerland, 2017), Vol. 10.
  5. C. Kaspar, B. J. Ravoo, W. G. van der Wiel, S. V. Wegner, and W. H. Pernice, The rise of intelligent matter, Nature (London) 594, 345 (2021).
  6. L. V. Nguyen, Swarm intelligence-based multi-robotics: A comprehensive review, AppliedMath 4, 1192 (2024).
  7. S. Kernbach, R. Thenius, O. Kernbach, and T. Schmickl, Re-embodiment of honeybee aggregation behavior in an artificial micro-robotic system, Adapt. Behav. 17, 237 (2009).
  8. T. Schmickl and H. Hamann, Beeclust: A swarm algorithm derived from honeybees, in Bio-Inspired Computing and Communication Networks, edited by Y. Xiao (CRC Press, Boca Raton, FL, 2011), pp. 95–137.
  9. M. Bodi, R. Thenius, M. Szopek, T. Schmickl, and K. Crailsheim, Interaction of robot swarms using the honeybee-inspired control algorithm Beeclust, Math. Comput. Modell. Dyn. Syst. 18, 87 (2012).
  10. S. Kernbach, D. Häbe, O. Kernbach, R. Thenius, G. Radspieler, T. Kimura, and T. Schmickl, Adaptive collective decision-making in limited robot swarms without communication, Int. J. Rob. Res. 32, 35 (2013).
  11. M. K. Heinrich, M. Wahby, M. Dorigo, and H. Hamann, Swarm robotics, in Cognitive Robotics, edited by A. Cangelosi and M. Asada (MIT Press, Cambridge, MA, 2022), pp. 77–98.
  12. Y. Y. Rios, O. Acevedo, and L. L. García, Robot swarm aggregation using an improved Beeclust method, Int. J. Intell. Rob. Appl. 9, 804 (2025).
  13. R. Brown, A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies, Philos. Mag. 4, 161 (1828).
  14. J. S. Park, C. K. Choi, and K. D. Kihm, Temperature measurement for a nanoparticle suspension by detecting the Brownian motion using optical serial sectioning microscopy (OSSM), Meas. Sci. Technol. 16, 1418 (2005).
  15. K. Chung, J. K. Cho, E. S. Park, V. Breedveld, and H. Lu, Three-dimensional in situ temperature measurement in microsystems using Brownian motion of nanoparticles, Anal. Chem. 81, 991 (2009).
  16. D. Geiß and K. Kroy, Brownian thermometry beyond equilibrium, ChemSystemsChem 2, e1900041 (2020).
  17. K. Everschor-Sitte, A. Pignedoli, and B. Dörschel, Messverfahren mit Quasiteilchen, German Patent No. DE102023131171B3, German Patent and Trade Mark Office, Munich, Germany, 2025.
  18. J. D. Norton, Brownian computation is thermodynamically irreversible, Found. Phys. 43, 1384 (2013).
  19. F. Peper, J. Lee, J. Carmona, J. Cortadella, and K. Morita, Brownian circuits: Fundamentals, J. Emerg. Technol. Comput. Syst. 9, 1 (2013).
  20. M. Goto, H. Nomura, and Y. Suzuki, Stochastic skyrmion dynamics under alternating magnetic fields, J. Magn. Magn. Mater. 536, 167974 (2021).
  21. K. Raab, M. A. Brems, G. Beneke, T. Dohi, J. Rothörl, F. Kammerbauer, J. H. Mentink, and M. Kläui, Brownian reservoir computing realized using geometrically confined skyrmion dynamics, Nat. Commun. 13, 6982 (2022).
  22. K. Everschor-Sitte, A. Pignedoli, and B. Dörschel, Bestimmen einer optimalen Bewegungsbahn, German Patent Application No. DE102023131706A1, 2023.
  23. G. Beneke, T. B. Winkler, K. Raab, M. A. Brems, F. Kammerbauer, P. Gerhards, K. Knobloch, S. Krishnia, J. H. Mentink, and M. Kläui, Gesture recognition with Brownian reservoir computing using geometrically confined skyrmion dynamics, Nat. Commun. 15, 8103 (2024).
  24. P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, H. Mabuchi, and Y. Yamamoto, A fully programmable 100-spin coherent Ising machine with all-to-all connections, Science 354, 614 (2016).
  25. N. Mohseni, P. L. McMahon, and T. Byrnes, Ising machines as hardware solvers of combinatorial optimization problems, Nat. Rev. Phys. 4, 363 (2022).
  26. G. Tanaka, T. Yamane, J. B. Héroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano, and A. Hirose, Recent advances in physical reservoir computing: A review, Neural Netw. 115, 100 (2019).
  27. K. Everschor-Sitte, A. Majumdar, K. Wolk, and D. Meier, Topological magnetic and ferroelectric systems for reservoir computing, Nat. Rev. Phys. 6, 455 (2024).
  28. D. Marković, A. Mizrahi, D. Querlioz, and J. Grollier, Physics for neuromorphic computing, Nat. Rev. Phys. 2, 499 (2020).
  29. H. Jaeger, B. Noheda, and W. G. van der Wiel, Toward a formal theory for computing machines made out of whatever physics offers, Nat. Commun. 14, 4911 (2023).
  30. G. Finocchio et al., Roadmap for unconventional computing with nanotechnology, Nano Futures 8, 012001 (2024).
  31. H. Kurebayashi, G. Finocchio, K. Everschor-Sitte, J. C. Gartside, T. Taniguchi, A. Litvinenko, A. Kumar, J. Åkerman, E. Vasilaki, K. Selçuk, K. Y. Çamsarý, A. Madhavan, and S. Fukami, Metrics for spin-based computing, Nat. Rev. Phys. 8, 208 (2026).
  32. W. Zhu, S. Oğuz, M. K. Heinrich, M. Allwright, M. Wahby, A. L. Christensen, E. Garone, and M. Dorigo, Self-organizing nervous systems for robot swarms, Sci. Rob. 9, eadl5161 (2024).
  33. A. Nitti, M. D. de Tullio, I. Federico, and G. Carbone, A collective intelligence model for swarm robotics applications, Nat. Commun. 16, 6572 (2025).
  34. Artificial Intelligence and Intelligent Matter: Nanoscience, Soft Matter, Philosophy, in Machine Intelligence for Materials Science, edited by M. te Vrugt (Springer, Cham, Switzerland, 2026).
  35. M. Alhafnawi, J. Bendarkawi, Y. Tafesse, L. Stein-Montalvo, A. Jones, V. Chow, S. Adriaenssens, and R. Nagpal, Architectural swarms for responsive façades and creative expression, Sci. Rob. 11, eady7233 (2026).
  36. We use the term “swarm” in the sense of the swarm-intelligence and unconventional-computing literature [100, 101], referring to a collection of N interacting agents performing a collective task, characterized by (1) decentralization, with no central coordinator and behavior emerging from local interactions; (2) emergence, where simple individual actions produce nontrivial global outcomes; (3) self-organization, through continuous adaptation to local conditions; and (4) robustness, maintaining functionality despite the loss or malfunction of individual agents. In the present Brownian-computing realization, thermal motion provides stochastic exploration, while local interactions couple the individual searches and produce collective localization.
  37. M. Widder and U. Titulaer, Brownian motion in a medium with inhomogeneous temperature, Physica A 154, 452 (1989).
  38. D. S. Dean, Langevin equation for the density of a system of interacting Langevin processes, J. Phys. A: Math. Gen. 29, L613 (1996).
  39. A. W. C. Lau and T. C. Lubensky, State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E 76, 011123 (2007).
  40. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (North-Holland, Amsterdam, 2007).
  41. M. Yang and M. Ripoll, Brownian motion in inhomogeneous suspensions, Phys. Rev. E 87, 062110 (2013).
  42. R. J. Glauber, Time-dependent statistics of the Ising model, J. Math. Phys. 4, 294 (1963).
  43. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  44. L. T. Stutzer, C. Dieball, and A. c. v. Godec, Stochastic calculus for pathwise observables of Markov-jump processes: Unification of diffusion and jump dynamics, Phys. Rev. X 16, 021038 (2026).
  45. D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22, 403 (1976).
  46. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).
  47. In this work, we restrict our analysis to nondegenerate landscapes, where the global minimum s0 is unique. Extending the framework to degenerate global minima would require a generalized metric.
  48. The Manhattan distance between two lattice points is evaluated as the sum of their horizontal and vertical separation.
  49. See Supplemental Material at http://link.aps.org/supplemental/10.1103/k8wj-xsl7 for the dynamics of the single-particle probability distribution p̂ as a function of the averaging window K for selected values of ε/kBT0 and ν in a static temperature landscape.
  50. The value K=12500 was chosen so that, for every combination of ν and ε/kBT0, the steady-state distribution p̂(J,K) is statistically converged. This value is held fixed to ensure that the bias introduced by the finite window averaging is identical and the dynamical performance can be compared.
  51. See Supplemental Material at http://link.aps.org/supplemental/10.1103/k8wj-xsl7 for the dynamics of the single-particle probability distribution p̂ as a function of J using a sliding averaging window for selected values of ε/kBT0 and ν, tracking the relocation of the global minimum following an instantaneous switch of the temperature landscape.
  52. The nonidentical parameter dependence of the success ratio R and the accuracy measure A reflects the different averaging procedures used to evaluate these quantities.
  53. C. Schütte, J. Iwasaki, A. Rosch, and N. Nagaosa, Inertia, diffusion, and dynamics of a driven skyrmion, Phys. Rev. B 90, 174434 (2014).
  54. L. Rózsa, A. Deák, E. Simon, R. Yanes, L. Udvardi, L. Szunyogh, and U. Nowak, Skyrmions with attractive interactions in an ultrathin magnetic film, Phys. Rev. Lett. 117, 157205 (2016).
  55. J. Zázvorka, F. Jakobs, D. Heinze, N. Keil, S. Kromin, S. Jaiswal, K. Litzius, G. Jakob, P. Virnau, D. Pinna, K. Everschor-Sitte, L. Rózsa, A. Donges, U. Nowak, and M. Kläui, Thermal skyrmion diffusion used in a reshuffler device, Nat. Nanotechnol. 14, 658 (2019).
  56. K. Litzius, J. Leliaert, P. Bassirian, D. Rodrigues, S. Kromin, I. Lemesh, J. Zázvorka, K.-J. Lee, J. Mulkers, N. Kerber, D. Heinze, N. Keil, R. M. Reeve, M. Weigand, B. Van Waeyenberge, G. Schütz, K. Everschor-Sitte, G. S. D. Beach, and M. Kläui, The role of temperature and drive current in skyrmion dynamics, Nat. Electron. 3, 30 (2020).
  57. O. Lee, R. Msiska, M. A. Brems, M. Kläui, H. Kurebayashi, and K. Everschor-Sitte, Perspective on unconventional computing using magnetic skyrmions, Appl. Phys. Lett. 122, 260501 (2023).
  58. L. Zhao, Z. Wang, X. Zhang, X. Liang, J. Xia, K. Wu, H.-A. Zhou, Y. Dong, G. Yu, K. L. Wang, X. Liu, Y. Zhou, and W. Jiang, Topology-dependent Brownian gyromotion of a single skyrmion, Phys. Rev. Lett. 125, 027206 (2020).
  59. M. Weißenhofer, L. Rózsa, and U. Nowak, Skyrmion dynamics at finite temperatures: Beyond Thiele's equation, Phys. Rev. Lett. 127, 047203 (2021).
  60. S. Koraltan et al., The 2026 skyrmionics roadmap, arXiv:2601.16575.
  61. G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Vortices in high-temperature superconductors, Rev. Mod. Phys. 66, 1125 (1994).
  62. X. B. Xu, H. Fangohr, Z. H. Wang, M. Gu, S. L. Liu, D. Q. Shi, and S. X. Dou, Vortex dynamics for low-κ type-II superconductors, Phys. Rev. B 84, 014515 (2011).
  63. J. L. Garcia-Palacios and F. J. Lazaro, Langevin-dynamics study of the dynamical properties of small magnetic particles, Phys. Rev. B 58, 14937 (1998).
  64. S. Xin, J. Sun, Z. Shi, R. Li, X. Liu, N. Wang, J. B. Weaver, and K. Wu, Study and optimization on hyperthermia performance of magnetic fluids modeled by coupled Brownian-Néel rotations, J. Appl. Phys. 137, 054702 (2025).
  65. A. T. Liu et al., Colloidal robotics, Nat. Mater. 22, 1453 (2023).
  66. R. Fortulan, N. Raeisi Kheirabadi, A. Chiolerio, and A. Adamatzky, Thermal colloid programming, Sci. Rep. 15, 12646 (2025).
  67. N. Grønbech-jensen and S. Doniach, Long-time overdamped Langevin dynamics of molecular chains, J. Comput. Chem. 15, 997 (1994).
  68. F. Jülicher, A. Ajdari, and J. Prost, Modeling molecular motors, Rev. Mod. Phys. 69, 1269 (1997).
  69. Z. Liu and M. Dijkstra, Collective dynamics of intelligent active Brownian particles with visual perception and velocity alignment in 3D: Spheres, rods, and worms, Soft Matter 21, 1529 (2025).
  70. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  71. J. C. Crocker and D. G. Grier, Methods of digital video microscopy for colloidal studies, J. Colloid Interface Sci. 179, 298 (1996).
  72. J. Wu and M. Gu, Microfluidic sensing: State of the art fabrication and detection techniques, J. Biomed. Opt. 16, 080901 (2011).
  73. Y. Guang et al., Electrical detection of magnetic skyrmions in a magnetic tunnel junction, Adv. Electron. Mater. 9, 2370001 (2023).
  74. M. Zhao, A. Chen, P.-Y. Huang, C. Liu, L. Shen, J. Liu, L. Zhao, B. Fang, W.-C. Yue, D. Zheng, L. Wang, H. Bai, K. Shen, Y. Zhou, S. Wang, E. Liu, S. He, Y.-L. Wang, X. Zhang, and W. Jiang, Electrical detection of mobile skyrmions with 100% tunneling magnetoresistance in a racetrack-like device, npj Quantum Mater. 9, 50 (2024).
  75. C. J. Geyer, Practical Markov chain Monte Carlo, Stat. Sci. 7, 473 (1992).
  76. S. Liu, S. P. Chepuri, M. Fardad, E. Maşazade, G. Leus, and P. K. Varshney, Sensor selection for estimation with correlated measurement noise, IEEE Trans. Signal Process. 64, 3509 (2016).
  77. J. K. G. Dhont, An Introduction to Dynamics of Colloids, Studies in Interface Science (Elsevier, Amsterdam, 1996), Vol. 2.
  78. A. Scacchi, M. Vuorte, and M. Sammalkorpi, Multiscale modelling of biopolymers, Adv. Phys.: X 9, 2358196 (2024).
  79. A. Pignedoli, Supplementary code to “Leveraging interactions for energy-efficient swarm-based Brownian computing”, Zenodo, 2026, https://doi.org/10.5281/zenodo.21915737.
  80. N. Van Kampen, Diffusion in inhomogeneous media, J. Phys. Chem. Solids 49, 673 (1988).
  81. N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotechnol. 8, 899 (2013).
  82. K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kläui, Perspective: Magnetic skyrmions—Overview of recent progress in an active research field, J. Appl. Phys. 124, 240901 (2018).
  83. S. Chen, J. Lourembam, P. Ho, A. K. J. Toh, J. Huang, X. Chen, H. K. Tan, S. L. K. Yap, R. J. J. Lim, H. R. Tan, T. S. Suraj, M. I. Sim, Y. T. Toh, I. Lim, N. C. B. Lim, J. Zhou, H. J. Chung, S. T. Lim, and A. Soumyanarayanan, All-electrical skyrmionic magnetic tunnel junction, Nature (London) 627, 522 (2024).
  84. J. Kim, S. Yang, D. Kim, K.-W. Moon, C. Kim, C. Hwang, and M.-K. Seo, Photothermal skyrmion tweezer: Programmable optical manipulation of magnetic topological quasiparticles, Nat. Commun. 16, 11375 (2025).
  85. S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena, Particle model for skyrmions in metallic chiral magnets: Dynamics, pinning, and creep, Phys. Rev. B 87, 214419 (2013).
  86. S. Eley, A. Glatz, and R. Willa, Challenges and transformative opportunities in superconductor vortex physics, J. Appl. Phys. 130, 050901 (2021).
  87. M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996).
  88. I. S. Veshchunov et al., Optical manipulation of single flux quanta, Nat. Commun. 7, 12801 (2016).
  89. H. Heo, W. B. Choi, S. Ha, H. Park, and J. Jang, Magneto-optical measurements of mesoscopic Nb superconducting structures using a ferromagnetic metal indicator layer, J. Appl. Phys. 131, 233901 (2022).
  90. S. Hu, J. Qiao, G. Gu, Q.-K. Xue, and D. Zhang, Vortex entropy and superconducting fluctuations in ultrathin underdoped Bi2Sr2CaCu2O8+x superconductor, Nat. Commun. 15, 4818 (2024).
  91. S. Ooi, M. Tachiki, T. Mochiku, H. Ito, T. Kubo, A. Kikuchi, S. Arisawa, and K. Umemori, Dynamical visualization of attractively interacting single vortices in type-II/1 superconducting Nb by magneto-optical imaging, Phys. Rev. B 111, 094519 (2025).
  92. C. Jooss, J. Albrecht, H. Kuhn, S. Leonhardt, and H. Kronmüller, Magneto-optical studies of current distributions in high-Tc superconductors, Rep. Prog. Phys. 65, 651 (2002).
  93. J. R. Kirtley, Fundamental studies of superconductors using scanning magnetic imaging, Rep. Prog. Phys. 73, 126501 (2010).
  94. S. C. Scholten, A. J. Healey, I. O. Robertson, G. J. Abrahams, D. A. Broadway, and J.-P. Tetienne, Widefield quantum microscopy with nitrogen-vacancy centers in diamond: Strengths, limitations, and prospects, J. Appl. Phys. 130, 150902 (2021).
  95. E. Persky, I. Sochnikov, and B. Kalisky, Studying quantum materials with scanning SQUID microscopy, Annu. Rev. Condens. Matter Phys. 13, 385 (2022).
  96. J. Chen et al., Thermal gradient induced tweezers for the manipulation of particles and cells, Sci. Rep. 6, 35814 (2016).
  97. L. Lin et al., Opto-thermophoretic assembly of colloidal matter, Sci. Adv. 3, e1700458 (2017).
  98. X. Peng et al., Optothermophoretic manipulation of colloidal particles in nonionic liquids, J. Phys. Chem. C 122, 24226 (2018).
  99. W. B. Rogers and J. C. Crocker, Direct measurements of DNA-mediated colloidal interactions and their quantitative modeling, Proc. Natl. Acad. Sci. USA 108, 15687 (2011).
  100. J. Kennedy, R. C. Eberhart, and Y. Shi, Swarm Intelligence (Morgan Kaufmann, San Francisco, CA, 2001).
  101. Unconventional Computing: A Volume in the Encyclopedia of Complexity and Systems Science, edited by A. Adamatzky (Springer, New York, 2018), 2nd ed.

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation