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Gigantic dynamical spreading and anomalous diffusion of jerky active particles

Hartmut Löwen

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Phys. Rev. E 112, 045412 – Published 17 October, 2025

DOI: https://doi.org/10.1103/976t-qry7

Abstract

Jerky active particles are Brownian self-propelled particles which are dominated by “jerk,” the change in acceleration. They represent a generalization of inertial active particles. In order to describe jerky active particles, a linear jerk equation of motion which involves a third-order derivative in time, Stokes friction, and a spring force is combined with activity modeled by an active Ornstein-Uhlenbeck process. This equation of motion is solved analytically and the associated mean-square displacement (MSD) is extracted as a function of time. For small damping and small spring constants, the MSD shows an enormous superballistic spreading with different scaling regimes characterized by anomalous high dynamical exponents 6, 5, 4, or 3 arising from a competition among jerk, inertia, and activity. When exposed to a harmonic potential, the gigantic spreading tendency induced by jerk gives rise to an enormous increase of the kinetic temperature and even to a sharp localization-delocalization transition, i.e., a jerky particle can escape from harmonic confinement. The transition can be either first or second order as a function of jerkiness. Finally it is shown that self-propelled jerky particles governed by the basic equation of motion can be realized experimentally both in feedback-controlled macroscopic particles and in active colloids governed by friction with memory.

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References (134)

  1. I. Newton, Philosophiae Naturalis Principia Mathematica (Royal Society Press, London, 1687).
  2. E. Rebhan, Theoretische Physik: Mechanik (Spektrum Akademischer Verlag, Heidelberg, 2006), Vol. 1.
  3. M. W. McCall, Classical Mechanics: From Newton to Einstein: A Modern Introduction (Wiley, Chichester, UK, 2011), Vol. 2.
  4. A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. Löwen, Phys. Rev. X 5, 011035 (2015).
  5. A. Dinelli, J. O'Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Nat. Commun. 14, 7035 (2023).
  6. M. Knežević, T. Welker, and H. Stark, Sci. Rep. 12, 19437 (2022).
  7. S. A. M. Loos and S. H. L. Klapp, New J. Phys. 22, 123051 (2020).
  8. S. Osat and R. Golestanian, Nat. Nanotechnol. 18, 79 (2023).
  9. S. Osat, J. Metson, M. Kardar, and R. Golestanian, Phys. Rev. Lett. 133, 028301 (2024).
  10. G. Gompper et al., J. Phys.: Condens. Matter 37, 143501 (2025).
  11. J. Bechhoefer, Rev. Mod. Phys. 77, 783 (2005).
  12. B. van Heerden, N. A. Vickers, T. P. J. Krüger, and S. B. Andersson, Small 18, 2107024 (2022).
  13. D. Saha, S. Tarama, H. Löwen, and S. U. Egelhaaf, Soft Matter 20, 8112 (2024).
  14. Z. Hou, Z. Zhang, J. Li, K. Yasuda, and S. Komura, Europhys. Lett. 150, 27001 (2025).
  15. J. Li, Z. Zhang, Z. Hou, Y. Hosaka, K. Yasuda, L. He, and S. Komura, J. Phys. Soc. Jpn. 94, 064802 (2025).
  16. S. H. Schot, Am. J. Phys. 46, 1090 (1978).
  17. S. J. Linz, Am. J. Phys. 66, 1109 (1998).
  18. S. J. Linz, Am. J. Phys. 65, 523 (1997).
  19. R. Eichhorn, S. J. Linz, and P. Hänggi, Phys. Rev. E 58, 7151 (1998).
  20. R. Eichhorn, S. J. Linz, and P. Hänggi, Chaos Solit. Fractals 13, 1 (2002).
  21. O. Umut and S. Yaşar, Int. J. Mod. Nonlin. Theory Appl. 2, 60 (2013).
  22. K. Rajagopal, S. Takougang Kingni, G. Fautso, V. Kamdoum Tamba, and V.-T. Pham, Adv. Theor. Math. Phys. 2018, 273531 (2018).
  23. B. Wu, C. Lim, and W. Sun, Phys. Lett. A 354, 95 (2006).
  24. H. Gottlieb, J. Sound Vib. 297, 243 (2006).
  25. H. Gottlieb, J. Sound Vib. 271, 671 (2004).
  26. H. Gottlieb, Am. J. Phys. 66, 903 (1998).
  27. S. J. Linz, Phys. Lett. A 275, 204 (2000).
  28. V. Patidar and K. K. Sud, Pramana 64, 75 (2005).
  29. Z. Njitacke, J. Kengne, and L. Kengne, Chaos. Solitons. Fract. 105, 77 (2017).
  30. H. von Baeyer, Sciences 38, 12 (1998).
  31. F. X. Liu, R. J. Cheng, and H. X. Ge, Nonlin. Dyn. 83, 793 (2016).
  32. C. Zhai and W. T. Wu, Nonlin. Dyn. 93, 2185 (2018).
  33. R. K. Tiwari, D. Sofuoglu, and A. Beesham, Gravit. Cosmol. 28, 196 (2022).
  34. N. J. Poplawski, Phys. Lett. B 640, 135 (2006).
  35. S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  36. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013).
  37. J. Elgeti, R. Winkler, and G. Gompper, Rep. Prog. Phys. 78, 056601 (2015).
  38. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Rev. Mod. Phys. 88, 045006 (2016).
  39. G. Gompper et al., J. Phys.: Condens. Matter 32, 193001 (2020).
  40. M. te Vrugt and R. Wittkowski, Eur. Phys. J. E 48, 12 (2025).
  41. R. Fürth, Z. Phys. 2, 244 (1920).
  42. G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930).
  43. P. Hänggi and P. Jung, Colored Noise in Dynamical Systems (John Wiley & Sons, New York, 1994).
  44. G. Szamel, Phys. Rev. E 90, 012111 (2014).
  45. L. L. Bonilla, Phys. Rev. E 100, 022601 (2019).
  46. D. Martin, J. O'Byrne, M. E. Cates, E. Fodor, C. Nardini, J. Tailleur, and F. van Wijland, Phys. Rev. E 103, 032607 (2021).
  47. L. Dabelow and R. Eichhorn, Front. Phys. 8, 582992 (2021).
  48. R. Wittmann, J. M. Brader, A. Sharma, and U. M. B. Marconi, Phys. Rev. E 97, 012601 (2018).
  49. T. J. H. Fritz and U. Seifert, J. Stat. Mech. (2023) 093204.
  50. D. Gupta, S. H. L. Klapp, and D. A. Sivak, Phys. Rev. E 108, 024117 (2023).
  51. A. Crisanti and M. Paoluzzi, Phys. Rev. E 107, 034110 (2023).
  52. A. Pacheco-Pozo, I. M. Sokolov, R. Metzler, and D. Krapf, arXiv:2505.13363.
  53. M. te Vrugt, J. Jeggle, and R. Wittkowski, New J. Phys. 23, 063023 (2021).
  54. L. Caprini and U. M. B. Marconi, J. Chem. Phys. 154, 024902 (2021).
  55. G. H. P. Nguyen, R. Wittmann, and H. Löwen, J. Phys.: Condens. Matter 34, 035101 (2022).
  56. H. Löwen, J. Chem. Phys. 152, 040901 (2020).
  57. M. Sandoval, Phys. Rev. E 101, 012606 (2020).
  58. L. L. Gutierrez-Martinez and M. Sandoval, J. Chem. Phys. 153, 044906 (2020).
  59. H. Li, G. He, Y. Peng, and H. Cheng, Chin. J. Phys. 77, 1997 (2022).
  60. J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Phys. Rev. Lett. 99, 048102 (2007).
  61. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
  62. R. Metzler, J. Jeon, A. Cherstvy, and E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014).
  63. S. Babel, B. ten Hagen, and H. Löwen, J. Stat. Mech. (2014) P02011.
  64. J. Masoliver and J. M. Porrà, Phys. Rev. E 48, 4309 (1993).
  65. S. Ye, P. Liu, F. Ye, K. Chen, and M. Yang, Soft Matter 16, 4655 (2020).
  66. A. Cavagna, L. D. Castello, I. Giardina, T. S. Grigera, A. Jelic, S. Melillo, T. Mora, L. Parisi, E. Silvestri, M. Viale, and A. M. Walczak, J. Stat. Phys. 158, 601 (2015).
  67. L. Caprini and H. Löwen, Phys. Rev. Lett. 130, 148202 (2023).
  68. M. Pavšič, Phys. Rev. D 87, 107502 (2013).
  69. A. Argun, J. Soni, L. Dabelow, S. Bo, G. Pesce, R. Eichhorn, and G. Volpe, Phys. Rev. E 96, 052106 (2017).
  70. M. A. Fernandez-Rodriguez, F. Grillo, L. Alvarez, M. Rathlef, I. Buttinoni, G. Volpe, and L. Isa, Nat. Commun. 11, 4223 (2020).
  71. A. R. Sprenger, M. A. Fernandez-Rodriguez, L. Alvarez, L. Isa, R. Wittkowski, and H. Löwen, Langmuir 36, 7066 (2020).
  72. C. Scholz, S. Jahanshahi, A. Ldov, and H. Löwen, Nat. Commun. 9, 5156 (2018).
  73. L. Caprini, D. Breoni, A. Ldov, C. Scholz, and H. Löwen, Commun. Phys. 7, 343 (2024).
  74. V. Nosenko, F. Luoni, A. Kaouk, M. Rubin-Zuzic, and H. Thomas, Phys. Rev. Res. 2, 033226 (2020).
  75. V. Holubec, G. Volpe, and F. Cichos, arXiv:2505.11042.
  76. N. Narinder, C. Bechinger, and J. R. Gomez-Solano, Phys. Rev. Lett. 121, 078003 (2018).
  77. F. J. Sevilla, R. F. Rodríguez, and J. R. Gomez-Solano, Phys. Rev. E 100, 032123 (2019).
  78. A. R. Sprenger, C. Bair, and H. Löwen, Phys. Rev. E 105, 044610 (2022).
  79. S. Milster, F. Koch, C. Widder, T. Schilling, and J. Dzubiella, J. Chem. Phys. 160, 094901 (2024).
  80. M. Wiśniewski, J. Łuczka, and J. Spiechowicz, Phys. Rev. E 109, 044116 (2024).
  81. A. Ruiz-Silva, B. B. Cassal-Quiroga, R. d. J. Escalante-Gonzalez, J. A. Del-Puerto-Flores, H. E. Gilardi-Velazquez, and E. Campos, Mathematics 13, 804 (2025).
  82. G. Huerta-Cuellar, E. Jiménez-López, E. Campos-Cantón, and A. N. Pisarchik, Commun. Nonlin. Sci. Numer. Simul. 19, 2740 (2014).
  83. M. R. Spiegel, Mathematical Handbook of Formulas and Tables (Schaum, New York, 1968).
  84. H. Risken, The Fokker-Planck Equation, Methods of Solution and Applications (Springer, Berlin, 1996), Vol. 2.
  85. J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, 1996), Vol. 2.
  86. M. Doi and S. Edwards, Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986).
  87. H. Löwen, J. Phys.: Condens. Matter 21, 474203 (2009).
  88. J. Berner, B. Müller, J. R. Gomez-Solano, M. Krüger, and C. Bechinger, Nat. Commun. 9, 999 (2018).
  89. D. S. Dean, S. N. Majumdar, and H. Schawe, Phys. Rev. E 103, 012130 (2021).
  90. B. ten Hagen, R. Wittkowski, and H. Löwen, Phys. Rev. E 84, 031105 (2011).
  91. B. ten Hagen, S. van Teeffelen, and H. Löwen, J. Phys.: Condens. Matter 23, 194119 (2011).
  92. A. P. Antonov, L. Caprini, A. Ldov, C. Scholz, and H. Löwen, Phys. Rev. Lett. 133, 198301 (2024).
  93. D. Breoni, M. Schmiedeberg, and H. Löwen, Phys. Rev. E 102, 062604 (2020).
  94. A. P. Antonov, Y. Zheng, B. Liebchen, and H. Löwen, Phys. Rev. Res. 7, 033008 (2025).
  95. A. Attanasi, A. Cavagna, L. D. Castello, I. Giardina, T. S. Grigera, A. Jelic, S. Melillo, L. Parisi, O. Pohl, E. Shen, and M. Viale, Nat. Phys. 10, 691 (2014).
  96. M. V. Ostrogradski, Mem. Acad. Sci. St. Petersb. VI 4, 385 (1850).
  97. V. V. Nesterenko, Phys. Rev. D 75, 087703 (2007).
  98. I. Buttinoni, L. Caprini, L. Alvarez, F. Schwarzendahl, and H. Löwen, Europhys. Lett. 140, 27001 (2022).
  99. G. Szamel, Phys. Rev. E 107, 054602 (2023).
  100. L. Hecht, L. Caprini, H. Löwen, and B. Liebchen, J. Chem. Phys. 161, 224904 (2024).
  101. F. A. Lavergne, H. Wendehenne, T. Bäuerle, and C. Bechinger, Science 364, 70 (2019).
  102. M. Nasiri, H. Löwen, and B. Liebchen, Europhys. Lett. 142, 17001 (2023).
  103. S. Goh, R. G. Winkler, and G. Gompper, Commun. Phys. 6, 310 (2023).
  104. S. Goh, R. G. Winkler, and G. Gompper, New J. Phys. 24, 093039 (2022).
  105. S. M. J. Khadem and S. H. L. Klapp, Phys. Chem. Chem. Phys. 21, 13776 (2019).
  106. P.-C. Chen, K. Kroy, F. Cichos, X. Wang, and V. Holubec, Europhys. Lett. 142, 67003 (2023).
  107. U. Töpfer, M. R. Bailey, S. Schreiber, F. Paratore, and L. Isa, arXiv:2505.15396.
  108. C. R. Packard and D. M. Sussman, arXiv:2505.10657.
  109. R. A. Kopp and S. H. L. Klapp, Phys. Rev. E 110, 054126 (2024).
  110. S. Dago, J. Pereda, S. Ciliberto, and L. Bellon, J. Stat. Mech. (2022) 053209.
  111. S. Chatterjee, J. Sound Vib. 330, 1860 (2011).
  112. R. van Leeuwen, D. M. Karabacak, H. S. J. van der Zant, and W. J. Venstra, Phys. Lett. B 88, 214301 (2013).
  113. E. Agoritsas and P. K. Morse, arXiv:2403.11701.
  114. R. Kürsten, V. Sushkov, and T. Ihle, Phys. Rev. Lett. 119, 188001 (2017).
  115. B. A. Dalton, A. Klimek, H. Kiefer, F. N. Brünig, H. Colinet, L. Tepper, A. Abbasi, and R. R. Netz, Annu. Rev. Phys. Chem. 76, 431 (2025).
  116. C. Sandford and A. Y. Grosberg, Phys. Rev. E 97, 012602 (2018).
  117. M. Mungan, E. Clement, D. Vandembroucq, and S. Sastry (unpublished).
  118. U. Khadka, V. Holubec, H. Yang, and F. Cichos, Nat. Commun. 9, 3864 (2018).
  119. C. Huang, M. Ding, and X. Xing, Phys. Rev. Res. 2, 043222 (2020).
  120. A. Fischer, F. Schmid, and T. Speck, Phys. Rev. E 101, 012601 (2020).
  121. L. Caprini, A. R. Sprenger, H. Löwen, and R. Wittmann, J. Chem. Phys. 156, 071102 (2022).
  122. S. Jose and H. Löwen, arXiv:2508.18180.
  123. P.-G. de Gennes, J. Stat. Phys. 119, 953 (2005).
  124. H. Hayakawa, Physica D 205, 48 (2005).
  125. S. Daniel, M. K. Chaudhury, and P.-G. de Gennes, Langmuir 21, 4240 (2005).
  126. Y. Chen and W. Just, Phys. Rev. E 89, 022103 (2014).
  127. T. Lequy and A. M. Menzel, Phys. Rev. E 108, 064606 (2023).
  128. M. Cates and J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  129. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995).
  130. J. Toner, The Physics of Flocking: Birth, Death, and Flight in Active Matter (Cambridge University Press, Cambridge, UK, 2024).
  131. H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. Löwen, and J. M. Yeomans, Proc. Natl. Acad. Sci. USA 109, 14308 (2012).
  132. D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009).
  133. P. Forgács, A. Libál, C. Reichhardt, N. Hengartner, and C. J. O. Reichhardt, Sci. Rep. 12, 11229 (2022).
  134. R. Kailasham and A. S. Khair, Soft Matter 20, 9193 (2024).

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Phys. Rev. E 112, 045412– Published 17 October, 2025
  • Received 12 July 2025
  • Accepted 10 September 2025
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