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  • Letter
  • Open Access

Paramagnetism in spherically confined charged active matter

Balázs Németh* and Ronojoy Adhikari†

  • *Contact author: bn273@cam.ac.uk
  • †Contact author: ra413@cam.ac.uk
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Phys. Rev. Research 7, L022068 – Published 17 June, 2025

DOI: https://doi.org/10.1103/ccv9-c322

Abstract

The celebrated theorem of Bohr and van Leeuwen guarantees that a classical charged system cannot have a magnetization in thermal equilibrium. Quantum mechanically, however, a diamagnetic response is obtained. In contrast, we show here that a classical charged active system, consisting of a motile particle confined to the surface of a sphere, has a nonzero magnetization and a paramagnetic response. We numerically sample Langevin trajectories of this system and compare with limiting analytical solutions of the Fokker-Planck equation, at small and large temperatures, to find excellent agreement in the magnetic response. Our Letter suggests experimental routes to controlling and extracting work from charged active matter.

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References (52)

  1. N. Bohr, Studies on the electron theory of metals, in Niels Bohr Collected Works—Volume 1—Early Work (1905–1911), edited by L. Rosenfeld and J. Rud Nielsen (North-Holland, Amsterdam, 1972), pp. 163–393, translation of Ph.D. thesis, University of Copenhagen, 1911.
  2. H.-J. Van Leeuwen, Problèmes de la théorie électronique du magnétisme, J. Phys. Radium 2, 361 (1921).
  3. R. E. Peierls, Surprises in Theoretical Physics, Princeton Series in Physics (Princeton University, Princeton, NJ, 1979).
  4. L. Landau, Diamagnetismus der Metalle, Z. Phys. 64, 629 (1930).
  5. W. Fei, M. M. Driscoll, P. M. Chaikin, and K. J. M. Bishop, Magneto-capillary dynamics of amphiphilic Janus particles at curved liquid interfaces, Soft Matter 14, 4661 (2018).
  6. N. Kumar and K. Vijay Kumar, Classical Langevin dynamics of a charged particle moving on a sphere and diamagnetism: A surprise, Europhys. Lett. 86, 17001 (2009).
  7. P. Pradhan and U. Seifert, Nonexistence of classical diamagnetism and nonequilibrium fluctuation theorems for charged particles on a curved surface, Europhys. Lett. 89, 37001 (2010).
  8. H.-M. Chun, X. Durang, and J. D. Noh, Emergence of nonwhite noise in Langevin dynamics with magnetic Lorentz force, Phys. Rev. E 97, 032117 (2018).
  9. H. D. Vuijk, J. M. Brader, and A. Sharma, Anomalous fluxes in overdamped Brownian dynamics with Lorentz force, J. Stat. Mech. (2019) 063203.
  10. H. Poincaré, Sur une forme nouvelle des équations de la Mécanique, C. R. Acad. Sci. 132, 369 (1901).
  11. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ccv9-c322 for derivations of theoretical results, details of numerical simulations, and a movie of the dynamics, which includes Refs. [48, 49, 50, 51, 52].
  12. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd ed., Texts in Applied Mathematics Vol. 17 (Springer, New York, 2003).
  13. R. Sknepnek and S. Henkes, Active swarms on a sphere, Phys. Rev. E 91, 022306 (2015).
  14. S. Henkes, M. C. Marchetti, and R. Sknepnek, Dynamical patterns in nematic active matter on a sphere, Phys. Rev. E 97, 042605 (2018).
  15. M. Muhsin, M. Sahoo, and A. Saha, Orbital magnetism of an active particle in viscoelastic suspension, Phys. Rev. E 104, 034613 (2021).
  16. C. W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, 4th ed., Springer Series in Synergetics Vol. 13 (Springer-Verlag, Berlin, 2009).
  17. E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics Vol. 31 (Springer-Verlag, Berlin, 2006).
  18. S. A. Chin, Symplectic and energy-conserving algorithms for solving magnetic field trajectories, Phys. Rev. E 77, 066401 (2008).
  19. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed., North-Holland Personal Library (Elsevier, Amsterdam, 2007).
  20. H. Brinkman, Brownian motion in a field of force and the diffusion theory of chemical reactions, Physica 22, 29 (1956).
  21. P. Langevin, Sur la théorie du magnétisme, J. Phys. Theor. Appl. 4, 678 (1905).
  22. J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities (Oxford University, New York, 1965).
  23. S. Lee and C. Kwon, Nonequilibrium driven by an external torque in the presence of a magnetic field, Phys. Rev. E 99, 052142 (2019).
  24. H. D. Vuijk, J. U. Sommer, H. Merlitz, J. M. Brader, and A. Sharma, Lorentz forces induce inhomogeneity and flux in active systems, Phys. Rev. Res. 2, 013320 (2020).
  25. I. Abdoli, H. D. Vuijk, R. Wittmann, J. U. Sommer, J. M. Brader, and A. Sharma, Stationary state in Brownian systems with Lorentz force, Phys. Rev. Res. 2, 023381 (2020).
  26. I. Abdoli, E. Kalz, H. D. Vuijk, R. Wittmann, J.-U. Sommer, J. M. Brader, and A. Sharma, Correlations in multithermostat Brownian systems with Lorentz force, New J. Phys. 22, 093057 (2020).
  27. A. Matevosyan and A. E. Allahverdyan, Nonequilibrium, weak-field-induced cyclotron motion: A mechanism for magnetobiology, Phys. Rev. E 104, 064407 (2021).
  28. M. Muhsin and M. Sahoo, Inertial active Ornstein-Uhlenbeck particle in the presence of a magnetic field, Phys. Rev. E 106, 014605 (2022).
  29. M. Muhsin, F. Adersh, and M. Sahoo, Active magneto gyrator: Memory-induced trapped diamagnetism, Phys. Rev. E 111, 015411 (2025).
  30. W. Wang, L. A. Castro, M. Hoyos, and T. E. Mallouk, Autonomous motion of metallic microrods propelled by ultrasound, ACS Nano 6, 6122 (2012).
  31. L. K. Davis, K. Proesmans, and É. Fodor, Active matter under control: Insights from response theory, Phys. Rev. X 14, 011012 (2024).
  32. P. Tierno, R. Golestanian, I. Pagonabarraga, and F. Sagués, Controlled swimming in confined fluids of magnetically actuated colloidal rotors, Phys. Rev. Lett. 101, 218304 (2008).
  33. P. Guillamat, J. Ignés-Mullol, and F. Sagués, Control of active liquid crystals with a magnetic field, Proc. Natl. Acad. Sci. USA 113, 5498 (2016).
  34. P. Mandal, G. Patil, H. Kakoty, and A. Ghosh, Magnetic active matter based on helical propulsion, Acc. Chem. Res. 51, 2689 (2018).
  35. A. F. Demirörs, M. T. Akan, E. Poloni, and A. R. Studart, Active cargo transport with Janus colloidal shuttles using electric and magnetic fields, Soft Matter 14, 4741 (2018).
  36. K. Bente, A. Codutti, F. Bachmann, and D. Faivre, Biohybrid and bioinspired magnetic microswimmers, Small 14, 1704374 (2018).
  37. D. Matsunaga, J. K. Hamilton, F. Meng, N. Bukin, E. L. Martin, F. Y. Ogrin, J. M. Yeomans, and R. Golestanian, Controlling collective rotational patterns of magnetic rotors, Nat. Commun. 10, 4696 (2019).
  38. A. Théry, A. Chamolly, and E. Lauga, Controlling confined collective organization with taxis, Phys. Rev. Lett. 132, 108301 (2024).
  39. M. Baiesi, C. Maes, and B. Wynants, Fluctuations and response of nonequilibrium states, Phys. Rev. Lett. 103, 010602 (2009).
  40. É. Fodor and M. E. Cates, Active engines: Thermodynamics moves forward, Europhys. Lett. 134, 10003 (2021).
  41. I. Abdoli, R. Wittmann, J. M. Brader, J.-U. Sommer, H. Löwen, and A. Sharma, Tunable Brownian magneto heat pump, Sci. Rep. 12, 13405 (2022).
  42. A. M. Jayannavar and M. Sahoo, Charged particle in a magnetic field: Jarzynski equality, Phys. Rev. E 75, 032102 (2007).
  43. A. Saha and A. M. Jayannavar, Nonequilibrium work distributions for a trapped Brownian particle in a time-dependent magnetic field, Phys. Rev. E 77, 022105 (2008).
  44. J. I. Jiménez-Aquino, Work-fluctuation theorem for a charged harmonic oscillator, J. Phys. A 44, 295002 (2011).
  45. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  46. E. Lauga, W. R. DiLuzio, G. M. Whitesides, and H. A. Stone, Swimming in circles: Motion of bacteria near solid boundaries, Biophys. J. 90, 400 (2006).
  47. B. Liebchen and D. Levis, Chiral active matter, Europhys. Lett. 139, 67001 (2022).
  48. G. A. Pavliotis, Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, Texts in Applied Mathematics Vol. 60 (Springer, New York, 2014).
  49. B. Németh and R. Adhikari, Intrinsic Langevin dynamics of rigid inclusions on curved surfaces, Phys. Rev. E 111, 045418 (2025).
  50. J. O'Byrne and M. E. Cates, Geometric theory of (extended) time-reversal symmetries in stochastic processes: I. Finite dimension, J. Stat. Mech. (2024) 113207.
  51. M. Gopal, Control Systems: Principles and Design, 2nd ed. (McGraw-Hill, New York, 2003).
  52. R. Czopnik and P. Garbaczewski, Brownian motion in a magnetic field, Phys. Rev. E 63, 021105 (2001).

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Phys. Rev. Research 7, L022068– Published 17 June, 2025
  • Received 1 February 2025
  • Accepted 13 May 2025
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