- Open Access
Leveraging interactions for energy-efficient swarm-based Brownian computing
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.
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References (101)
- 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).
- M. G. Hinchey, R. Sterritt, and C. Rouff, Swarms and swarm intelligence, Computer 40, 111 (2007).
- M. Rubenstein, A. Cornejo, and R. Nagpal, Programmable self-assembly in a thousand-robot swarm, Science 345, 795 (2014).
- 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.
- 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).
- L. V. Nguyen, Swarm intelligence-based multi-robotics: A comprehensive review, AppliedMath 4, 1192 (2024).
- 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).
- 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.
- 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).
- 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).
- 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.
- 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).
- 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).
- 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).
- 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).
- D. Geiß and K. Kroy, Brownian thermometry beyond equilibrium, ChemSystemsChem 2, e1900041 (2020).
- K. Everschor-Sitte, A. Pignedoli, and B. Dörschel, Messverfahren mit Quasiteilchen, German Patent No. DE102023131171B3, German Patent and Trade Mark Office, Munich, Germany, 2025.
- J. D. Norton, Brownian computation is thermodynamically irreversible, Found. Phys. 43, 1384 (2013).
- F. Peper, J. Lee, J. Carmona, J. Cortadella, and K. Morita, Brownian circuits: Fundamentals, J. Emerg. Technol. Comput. Syst. 9, 1 (2013).
- M. Goto, H. Nomura, and Y. Suzuki, Stochastic skyrmion dynamics under alternating magnetic fields, J. Magn. Magn. Mater. 536, 167974 (2021).
- 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).
- K. Everschor-Sitte, A. Pignedoli, and B. Dörschel, Bestimmen einer optimalen Bewegungsbahn, German Patent Application No. DE102023131706A1, 2023.
- 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).
- 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).
- N. Mohseni, P. L. McMahon, and T. Byrnes, Ising machines as hardware solvers of combinatorial optimization problems, Nat. Rev. Phys. 4, 363 (2022).
- 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).
- K. Everschor-Sitte, A. Majumdar, K. Wolk, and D. Meier, Topological magnetic and ferroelectric systems for reservoir computing, Nat. Rev. Phys. 6, 455 (2024).
- D. Marković, A. Mizrahi, D. Querlioz, and J. Grollier, Physics for neuromorphic computing, Nat. Rev. Phys. 2, 499 (2020).
- 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).
- G. Finocchio et al., Roadmap for unconventional computing with nanotechnology, Nano Futures 8, 012001 (2024).
- 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).
- 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).
- A. Nitti, M. D. de Tullio, I. Federico, and G. Carbone, A collective intelligence model for swarm robotics applications, Nat. Commun. 16, 6572 (2025).
- Artificial Intelligence and Intelligent Matter: Nanoscience, Soft Matter, Philosophy, in Machine Intelligence for Materials Science, edited by M. te Vrugt (Springer, Cham, Switzerland, 2026).
- 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).
- We use the term “swarm” in the sense of the swarm-intelligence and unconventional-computing literature [100, 101], referring to a collection of 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.
- M. Widder and U. Titulaer, Brownian motion in a medium with inhomogeneous temperature, Physica A 154, 452 (1989).
- D. S. Dean, Langevin equation for the density of a system of interacting Langevin processes, J. Phys. A: Math. Gen. 29, L613 (1996).
- A. W. C. Lau and T. C. Lubensky, State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E 76, 011123 (2007).
- N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (North-Holland, Amsterdam, 2007).
- M. Yang and M. Ripoll, Brownian motion in inhomogeneous suspensions, Phys. Rev. E 87, 062110 (2013).
- R. J. Glauber, Time-dependent statistics of the Ising model, J. Math. Phys. 4, 294 (1963).
- U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
- 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).
- D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22, 403 (1976).
- D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).
- In this work, we restrict our analysis to nondegenerate landscapes, where the global minimum is unique. Extending the framework to degenerate global minima would require a generalized metric.
- The Manhattan distance between two lattice points is evaluated as the sum of their horizontal and vertical separation.
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/k8wj-xsl7 for the dynamics of the single-particle probability distribution as a function of the averaging window for selected values of and in a static temperature landscape.
- The value was chosen so that, for every combination of and , the steady-state distribution 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.
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/k8wj-xsl7 for the dynamics of the single-particle probability distribution as a function of using a sliding averaging window for selected values of and , tracking the relocation of the global minimum following an instantaneous switch of the temperature landscape.
- The nonidentical parameter dependence of the success ratio and the accuracy measure reflects the different averaging procedures used to evaluate these quantities.
- C. Schütte, J. Iwasaki, A. Rosch, and N. Nagaosa, Inertia, diffusion, and dynamics of a driven skyrmion, Phys. Rev. B 90, 174434 (2014).
- 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).
- 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).
- 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).
- 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).
- 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).
- M. Weißenhofer, L. Rózsa, and U. Nowak, Skyrmion dynamics at finite temperatures: Beyond Thiele's equation, Phys. Rev. Lett. 127, 047203 (2021).
- S. Koraltan et al., The 2026 skyrmionics roadmap, arXiv:2601.16575.
- 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).
- X. B. Xu, H. Fangohr, Z. H. Wang, M. Gu, S. L. Liu, D. Q. Shi, and S. X. Dou, Vortex dynamics for low- type-II superconductors, Phys. Rev. B 84, 014515 (2011).
- 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).
- 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).
- A. T. Liu et al., Colloidal robotics, Nat. Mater. 22, 1453 (2023).
- R. Fortulan, N. Raeisi Kheirabadi, A. Chiolerio, and A. Adamatzky, Thermal colloid programming, Sci. Rep. 15, 12646 (2025).
- N. Grønbech-jensen and S. Doniach, Long-time overdamped Langevin dynamics of molecular chains, J. Comput. Chem. 15, 997 (1994).
- F. Jülicher, A. Ajdari, and J. Prost, Modeling molecular motors, Rev. Mod. Phys. 69, 1269 (1997).
- 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).
- P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
- J. C. Crocker and D. G. Grier, Methods of digital video microscopy for colloidal studies, J. Colloid Interface Sci. 179, 298 (1996).
- J. Wu and M. Gu, Microfluidic sensing: State of the art fabrication and detection techniques, J. Biomed. Opt. 16, 080901 (2011).
- Y. Guang et al., Electrical detection of magnetic skyrmions in a magnetic tunnel junction, Adv. Electron. Mater. 9, 2370001 (2023).
- 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).
- C. J. Geyer, Practical Markov chain Monte Carlo, Stat. Sci. 7, 473 (1992).
- 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).
- J. K. G. Dhont, An Introduction to Dynamics of Colloids, Studies in Interface Science (Elsevier, Amsterdam, 1996), Vol. 2.
- A. Scacchi, M. Vuorte, and M. Sammalkorpi, Multiscale modelling of biopolymers, Adv. Phys.: X 9, 2358196 (2024).
- A. Pignedoli, Supplementary code to “Leveraging interactions for energy-efficient swarm-based Brownian computing”, Zenodo, 2026, https://doi.org/10.5281/zenodo.21915737.
- N. Van Kampen, Diffusion in inhomogeneous media, J. Phys. Chem. Solids 49, 673 (1988).
- N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotechnol. 8, 899 (2013).
- 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).
- 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).
- 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).
- 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).
- S. Eley, A. Glatz, and R. Willa, Challenges and transformative opportunities in superconductor vortex physics, J. Appl. Phys. 130, 050901 (2021).
- M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996).
- I. S. Veshchunov et al., Optical manipulation of single flux quanta, Nat. Commun. 7, 12801 (2016).
- 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).
- S. Hu, J. Qiao, G. Gu, Q.-K. Xue, and D. Zhang, Vortex entropy and superconducting fluctuations in ultrathin underdoped superconductor, Nat. Commun. 15, 4818 (2024).
- 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).
- C. Jooss, J. Albrecht, H. Kuhn, S. Leonhardt, and H. Kronmüller, Magneto-optical studies of current distributions in high- superconductors, Rep. Prog. Phys. 65, 651 (2002).
- J. R. Kirtley, Fundamental studies of superconductors using scanning magnetic imaging, Rep. Prog. Phys. 73, 126501 (2010).
- 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).
- E. Persky, I. Sochnikov, and B. Kalisky, Studying quantum materials with scanning SQUID microscopy, Annu. Rev. Condens. Matter Phys. 13, 385 (2022).
- J. Chen et al., Thermal gradient induced tweezers for the manipulation of particles and cells, Sci. Rep. 6, 35814 (2016).
- L. Lin et al., Opto-thermophoretic assembly of colloidal matter, Sci. Adv. 3, e1700458 (2017).
- X. Peng et al., Optothermophoretic manipulation of colloidal particles in nonionic liquids, J. Phys. Chem. C 122, 24226 (2018).
- 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).
- J. Kennedy, R. C. Eberhart, and Y. Shi, Swarm Intelligence (Morgan Kaufmann, San Francisco, CA, 2001).
- Unconventional Computing: A Volume in the Encyclopedia of Complexity and Systems Science, edited by A. Adamatzky (Springer, New York, 2018), 2nd ed.