- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Two-dimensional anomalous diffusion of coupled active-passive particles in viscoelastic media

Pankaj Jangid and Srabanti Chaudhury*

  • *Contact author: srabanti@iiserpune.ac.in
PDF

Phys. Rev. E 114, 044126 – Published 16 October, 2026

DOI: https://doi.org/10.1103/vhs8-5gc7

Abstract

Anomalous diffusion is well studied in complex environments where viscoelasticity, crowding, and nonequilibrium activity strongly influence particle motion. Several biological systems are not independent; rather, they are coupled to another system, which shows distinct viscoelastic responses and heterogeneous transport behavior. We have developed a theoretical model of an active and a passive particle, coupled via a harmonic interaction, whose motion is restricted to a plane within a viscoelastic medium described by a generalized Langevin framework with a long-tailed power-law memory kernel. We obtained analytical expressions for the mean-squared displacement (MSD) and interparticle separation. We demonstrate that the coupled system shows multiple dynamical crossovers such as subdiffusive, ballistic, and diffusive transport regimes, whose emergence is controlled by the relative viscoelastic exponents and coupling strength. Specifically, we show that long-term dynamics is mainly controlled by the subdiffusive component, where the passive particle can suppress activity and the active particle can dominate and drive enhanced transport. Our results also highlight non-Gaussian displacement statistics at intermediate lag times. These findings provide insight into coupled transport in active matter systems, including tethered biomolecules, molecular motors, cytoskeletal networks, and synthetic microswimmers in complex media.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (135)

  1. T. Skóra, F. Vaghefikia, J. Fitter, and S. Kondrat, Macromolecular crowding: How shape and interactions affect diffusion, J. Phys. Chem. B 124, 7537 (2020).
  2. M. Feig, I. Yu, P.-H. Wang, G. Nawrocki, and Y. Sugita, Crowding in cellular environments at an atomistic level from computer simulations, J. Phys. Chem. B 121, 8009 (2017).
  3. V. P. S. Kompella, I. Stansfield, M. C. Romano, and R. L. Mancera, Definition of the minimal contents for the molecular simulation of the yeast cytoplasm, Front. Mol. Biosci. 6, 97 (2019).
  4. V. P. S. Kompella, M. C. Romano, I. Stansfield, and R. L. Mancera, What determines sub-diffusive behavior in crowded protein solutions? Biophys. J. 123, 134 (2024).
  5. S. R. McGuffee and A. H. Elcock, Diffusion, crowding & protein stability in a dynamic molecular model of the bacterial cytoplasm, PLoS Comput. Biol. 6, e1000694 (2010).
  6. T. A. Waigh and N. Korabel, Heterogeneous anomalous transport in cellular and molecular biology, Rep. Prog. Phys. 86, 126601 (2023).
  7. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  8. I. M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043 (2012).
  9. I. Eliazar and J. Klafter, Anomalous is ubiquitous, Ann. Phys. 326, 2517 (2011).
  10. J. M. Sancho, A. M. Lacasta, K. Lindenberg, I. M. Sokolov, and A. H. Romero, Diffusion on a solid surface: Anomalous is normal, Phys. Rev. Lett. 92, 250601 (2004).
  11. J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
  12. A. Godec, A. V. Chechkin, E. Barkai, H. Kantz, and R. Metzler, Localisation and universal fluctuations in ultraslow diffusion processes, J. Phys. A: Math. Theor. 47, 492002 (2014).
  13. I. Goychuk, V. O. Kharchenko, and R. Metzler, Persistent Sinai-type diffusion in Gaussian random potentials with decaying spatial correlations, Phys. Rev. E 96, 052134 (2017).
  14. V. Adrien, G. Rayan, K. Astafyeva, I. Broutin, M. Picard, P. Fuchs, W. Urbach, and N. Taulier, How to best estimate the viscosity of lipid bilayers, Biophys. Chem. 281, 106732 (2022).
  15. E. Flenner, J. Das, M. C. Rheinstädter, and I. Kosztin, Subdiffusion and lateral diffusion coefficient of lipid atoms and molecules in phospholipid bilayers, Phys. Rev. E 79, 011907 (2009).
  16. M. A. Dichtl and E. Sackmann, Colloidal probe study of short time local and long time reptational motion of semiflexible macromolecules in entangled networks, New J. Phys. 1, 18 (1999).
  17. A. Caspi, R. Granek, and M. Elbaum, Enhanced diffusion in active intracellular transport, Phys. Rev. Lett. 85, 5655 (2000).
  18. O. Chaudhuri, J. Cooper-White, P. A. Janmey, D. J. Mooney, and V. B. Shenoy, Effects of extracellular matrix viscoelasticity on cellular behaviour, Nature (London) 584, 535 (2020).
  19. Y. Wu, Y. Song, J. Soto, T. Hoffman, X. Lin, A. Zhang, S. Chen, R. N. Massad, X. Han, D. Qi, K.-W. Yeh, Z. Fang, J. Eoh, L. Gu, A. C. Rowat, Z. Gu, and S. Li, Viscoelastic extracellular matrix enhances epigenetic remodeling and cellular plasticity, Nat. Commun. 16, 4054 (2025).
  20. Z. Zhang, E. Bouchbinder, and E. Lerner, The strain-stiffening critical exponents in polymer networks and their universality, J. Chem. Phys. 163, 111102 (2025).
  21. F. J. Vernerey, Transient response of nonlinear polymer networks: A kinetic theory, J. Mech. Phys. Solids 115, 230 (2018).
  22. S. Joo, X. Durang, O.-C. Lee, and J.-H. Jeon, Anomalous diffusion of active Brownian particles cross-linked to a networked polymer: Langevin dynamics simulation and theory, Soft Matter 16, 9188 (2020).
  23. S. P. O. Danielsen, H. K. Beech, S. Wang, B. M. El-Zaatari, X. Wang, L. Sapir, T. Ouchi, Z. Wang, P. N. Johnson, Y. Hu, D. J. Lundberg, G. Stoychev, S. L. Craig, J. A. Johnson, J. A. Kalow, B. D. Olsen, and M. Rubinstein, Molecular characterization of polymer networks, Chem. Rev. 121, 5042 (2021).
  24. G. A. Parada and X. Zhao, Ideal reversible polymer networks, Soft Matter 14, 5186 (2018).
  25. S. Aney, P. Pandit, L. Ratke, B. Milow, and A. Rege, On the origin of power-scaling exponents in silica aerogels, J. Solgel Sci. Technol. 114, 98 (2025).
  26. T. S. K. Ng and G. H. McKinley, Power law gels at finite strains: The nonlinear rheology of gluten gels, J. Rheol. 52, 417 (2008).
  27. S. Aime, L. Cipelletti, and L. Ramos, Power law viscoelasticity of a fractal colloidal gel, J. Rheol. 62, 1429 (2018).
  28. M. Souzy, I. Zuriguel, and A. Marin, Transition from clogging to continuous flow in constricted particle suspensions, Phys. Rev. E 101, 060901(R) (2020).
  29. A. Maiti, Y. Koyano, H. Kitahata, and K. K. Dey, Activity-induced diffusion recovery in crowded colloidal suspensions, Phys. Rev. E 109, 054607 (2024).
  30. S. G. Leyva and I. Pagonabarraga, Clogging transition and anomalous transport in driven suspensions in a disordered medium, Phys. Rev. E 109, 014618 (2024).
  31. S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm, Phys. Rev. Lett. 104, 238102 (2010).
  32. G. G. Cabal, A. Genovesio, S. Rodriguez-Navarro, C. Zimmer, O. Gadal, A. Lesne, H. Buc, F. Feuerbach-Fournier, J.-C. Olivo-Marin, E. C. Hurt, and U. Nehrbass, SAGA interacting factors confine sub-diffusion of transcribed genes to the nuclear envelope, Nature (London) 441, 770 (2006).
  33. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, International Series of Monographs on Physics (Clarendon Press / Oxford University Press, Oxford, 1986).
  34. K. Speckner, L. Stadler, and M. Weiss, Anomalous dynamics of the endoplasmic reticulum network, Phys. Rev. E 98, 012406 (2018).
  35. A. Rajyaguru, R. Metzler, I. Dror, D. Grolimund, and B. Berkowitz, Diffusion in porous rock is anomalous, Environ. Sci. Technol. 58, 8946 (2024).
  36. A. Rajyaguru, R. Metzler, A. G. Cherstvy, and B. Berkowitz, Quantifying anomalous chemical diffusion through disordered porous rock materials, Phys. Chem. Chem. Phys. 27, 9056 (2025).
  37. V. Plerou, P. Gopikrishnan, L. A. Nunes Amaral, X. Gabaix, and H. Eugene Stanley, Economic fluctuations and anomalous diffusion, Phys. Rev. E 62, R3023(R) (2000).
  38. X. Gabaix, Power laws in economics and finance, Annu. Rev. Econ. 1, 255 (2009).
  39. A. Zöttl and H. Stark, Modeling active colloids: From active Brownian particles to hydrodynamic and chemical fields, Annu. Rev. Condens. Matter Phys. 14, 109 (2023).
  40. P. Romanczuk and L. Schimansky-Geier, Brownian motion with active fluctuations, Phys. Rev. Lett. 106, 230601 (2011).
  41. I. S. Aranson, Bacterial active matter, Rep. Prog. Phys. 85, 076601 (2022).
  42. H. C. Berg, Random Walks in Biology (Princeton University Press, Princeton, 2025).
  43. A. J. T. M. Mathijssen, N. Figueroa-Morales, G. Junot, É. Clément, A. Lindner, and A. Zöttl, Oscillatory surface rheotaxis of swimming E. coli bacteria, Nat. Commun. 10, 3434 (2019).
  44. L. Xie, T. Altindal, S. Chattopadhyay, and X.-L. Wu, From the cover: Bacterial flagellum as a propeller and as a rudder for efficient chemotaxis, Proc. Natl. Acad. Sci. USA 108, 2246 (2011).
  45. A. Ghosh, W. Xu, N. Gupta, and D. H. Gracias, Active matter therapeutics, Nano Today 31, 100836 (2020).
  46. P. Chaudhary, L. Ahamad, A. Chaudhary, G. Kumar, W.-J. Chen, and S. Chen, Nanoparticle-mediated bioremediation as a powerful weapon in the removal of environmental pollutants, J. Environ. Chem. Eng. 11, 109591 (2023).
  47. X. Xin, X. Pei, X. Yang, Y. Lv, L. Zhang, W. He, and L. Yin, Rod-shaped active drug particles enable efficient and safe gene delivery, Adv. Sci. 4, 1700324 (2017).
  48. R. S. Klessen and D. N. C. Lin, Diffusion in supersonic turbulent compressible flows, Phys. Rev. E 67, 046311 (2003).
  49. G. L. Eyink and K. R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phys. 78, 87 (2006).
  50. O. Vilk, M. Charter, S. Toledo, E. Barkai, and R. Nathan, Strong anomalous diffusion for free-ranging birds, PRX Life 3, 033020 (2025).
  51. G. M. Viswanathan, E. P. Raposo, and M. G. E. da Luz, Lévy flights and superdiffusion in the context of biological encounters and random searches, Phys. Life Rev. 5, 133 (2008).
  52. N. Narinder, C. Bechinger, and J. R. Gomez-Solano, Memory-induced transition from a persistent random walk to circular motion for achiral microswimmers, Phys. Rev. Lett. 121, 078003 (2018).
  53. C. Jin, Y. Chen, C. C. Maass, and A. J. T. M. Mathijssen, Collective entrainment and confinement amplify transport by schooling microswimmers, Phys. Rev. Lett. 127, 088006 (2021).
  54. C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett. 93, 098103 (2004).
  55. H. P. Zhang, A. Be'er, E.-L. Florin, and H. L. Swinney, Collective motion and density fluctuations in bacterial colonies, Proc. Natl. Acad. Sci. USA 107, 13626 (2010).
  56. B. Sun, J. Guo, B. Hao, Y. Cao, T. K. F. Chan, M. Sun, J. J. Y. Sung, and L. Zhang, Liquid-bodied antibiofilm robot with switchable viscoelastic response for biofilm eradication on complex surface topographies, Sci. Adv. 11, eadt8213 (2025).
  57. D. Schamel, A. G. Mark, J. G. Gibbs, C. Miksch, K. I. Morozov, A. M. Leshansky, and P. Fischer, Nanopropellers and their actuation in complex viscoelastic media, ACS Nano 8, 8794 (2014).
  58. F. Guzmán-Lastra, A. Kaiser, and H. Löwen, Fission and fusion scenarios for magnetic microswimmer clusters, Nat. Commun. 7, 13519 (2016).
  59. J. K. Hamilton, P. G. Petrov, C. P. Winlove, A. D. Gilbert, M. T. Bryan, and F. Y. Ogrin, Magnetically controlled ferromagnetic swimmers, Sci. Rep. 7, 44142 (2017).
  60. C. M. Buness, A. Rana, C. C. Maass, and R. Dey, Electrotaxis of self-propelling artificial swimmers in microchannels, Phys. Rev. Lett. 133, 158301 (2024).
  61. H.-W. Huang, F. E. Uslu, P. Katsamba, E. Lauga, M. S. Sakar, and B. J. Nelson, Adaptive locomotion of artificial microswimmers, Sci. Adv. 5, eaau1532 (2019).
  62. P. Witzel, M. Götz, Y. Lanoiselée, T. Franosch, D. S. Grebenkov, and D. Heinrich, Heterogeneities shape passive intracellular transport, Biophys. J. 117, 203 (2019).
  63. L. Bruno, V. Levi, M. Brunstein, and M. A. Despósito, Transition to superdiffusive behavior in intracellular actin-based transport mediated by molecular motors, Phys. Rev. E 80, 011912 (2009).
  64. J. F. Reverey, J.-H. Jeon, H. Bao, M. Leippe, R. Metzler, and C. Selhuber-Unkel, Superdiffusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic Acanthamoeba castellanii, Sci. Rep. 5, 11690 (2015).
  65. S. Park, O.-C. Lee, X. Durang, and J.-H. Jeon, A mini-review of the diffusion dynamics of DNA-binding proteins: Experiments and models, J. Korean Phys. Soc. 78, 408 (2021).
  66. Y. M. Wang, R. H. Austin, and E. C. Cox, Single molecule measurements of repressor protein 1D diffusion on DNA, Phys. Rev. Lett. 97, 048302 (2006).
  67. E. Yamamoto, T. Akimoto, A. C. Kalli, K. Yasuoka, and M. S. P. Sansom, Dynamic interactions between a membrane binding protein and lipids induce fluctuating diffusivity, Sci. Adv. 3, e1601871 (2017).
  68. M. Javanainen, H. Hammaren, L. Monticelli, J.-H. Jeon, M. S. Miettinen, H. Martinez-Seara, R. Metzler, and I. Vattulainen, Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes, Faraday Discuss. 161, 397 (2013).
  69. C. Lim and J.-H. Jeon, Anomalous diffusion in coupled viscoelastic media: A fractional Langevin equation approach, Phys. Rev. Res. 7, 043356 (2025).
  70. J. Balbo, P. Mereghetti, D.-P. Herten, and R. C. Wade, The shape of protein crowders is a major determinant of protein diffusion, Biophys. J. 104, 1576 (2013).
  71. D. Dey, A. Nunes-Alves, R. C. Wade, and G. Schreiber, Diffusion of small molecule drugs is affected by surface interactions and crowder proteins, iScience 25, 105088 (2022).
  72. F. Jülicher, K. Kruse, J. Prost, and J.-F. Joanny, Active behavior of the cytoskeleton, Phys. Rep. 449, 3 (2007).
  73. A. Taloni, Diffusion of an active particle bound to a generalized elastic model: Fractional Langevin equation, Fractal Fract. 8, 76 (2024).
  74. H. R. Vutukuri, B. Bet, R. van Roij, M. Dijkstra, and W. T. S. Huck, Rational design and dynamics of self-propelled colloidal bead chains: From rotators to flagella, Sci. Rep. 7, 16758 (2017).
  75. A. V. Singh, V. Kishore, G. Santomauro, O. Yasa, J. Bill, and M. Sitti, Mechanical coupling of puller and pusher active microswimmers influences motility, Langmuir 36, 5435 (2020).
  76. W. Gao, A. Pei, X. Feng, C. Hennessy, and J. Wang, Organized self-assembly of Janus micromotors with hydrophobic hemispheres, J. Am. Chem. Soc. 135, 998 (2013).
  77. H. Massana-Cid, J. Codina, I. Pagonabarraga, and P. Tierno, Active apolar doping determines routes to colloidal clusters and gels, Proc. Natl. Acad. Sci. USA 115, 10618 (2018).
  78. H. Ishikawa, Y. Koyano, H. Kitahata, and Y. Sumino, Pairing-induced motion of source and inert particles driven by surface tension, Phys. Rev. E 106, 024604 (2022).
  79. H. Ishikawa, Y. Koyano, H. Ito, Y. Sumino, and H. Kitahata, Simple mathematical model for a pairing-induced motion of active and passive particles, J. Phys. Soc. Jpn. 94, 044006 (2025).
  80. I. Santra, Dynamical fluctuations of a tracer coupled to active and passive particles, J. Phys. Complex. 4, 015013 (2023).
  81. M. Ann, K.-J.-B. Lee, and P. J. Park, Stochastic dynamics of coupled active particles in an overdamped limit, Physica A 436, 126 (2015).
  82. I. Goychuk, V. O. Kharchenko, and R. Metzler, Molecular motors pulling cargos in the viscoelastic cytosol: How power strokes beat subdiffusion, Phys. Chem. Chem. Phys. 16, 16524 (2014).
  83. D. Bhat and M. Gopalakrishnan, Transport of organelles by elastically coupled motor proteins, Eur. Phys. J. E 39, 71 (2016).
  84. A. V. Singh, Z. Hosseinidoust, B.-W. Park, O. Yasa, and M. Sitti, Microemulsion-based soft bacteria-driven microswimmers for active cargo delivery, ACS Nano 11, 9759 (2017).
  85. V. M. S. G. Tanuku, I. Malhotra, L. Caprini, H. Löwen, T. Palberg, and I. Buttinoni, Active particles in tunable compressible environments, Small Science 6, e202500599 (2026).
  86. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  87. B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. Soc. Ind. Appl. Math. 10, 422 (1968).
  88. H. Mori, Transport, collective motion, and Brownian motion, Prog. Theor. Phys. 33, 423 (1965).
  89. A. D. Viñales and M. A. Despósito, Anomalous diffusion: Exact solution of the generalized Langevin equation for harmonically bounded particle, Phys. Rev. E 73, 016111 (2006).
  90. M. A. Despósito and A. D. Viñales, Subdiffusive behavior in a trapping potential: Mean square displacement and velocity autocorrelation function, Phys. Rev. E 80, 021111 (2009).
  91. D. Panja, Generalized Langevin equation formulation for anomalous polymer dynamics, J. Stat. Mech. (2010) L02001.
  92. M. A. Despósito, Superdiffusion induced by a long-correlated external random force, Phys. Rev. E 84, 061114 (2011).
  93. M. A. Despósito, C. Pallavicini, V. Levi, and L. Bruno, Active transport in complex media: Relationship between persistence and superdiffusion, Physica A 390, 1026 (2011).
  94. D. S. Grebenkov, M. Vahabi, E. Bertseva, L. Forró, and S. Jeney, Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium, Phys. Rev. E 88, 040701(R) (2013).
  95. D. S. Grebenkov and M. Vahabi, Analytical solution of the generalized Langevin equation with hydrodynamic interactions: Subdiffusion of heavy tracers, Phys. Rev. E 89, 012130 (2014).
  96. A. A. Tateishi, E. K. Lenzi, L. R. da Silva, H. V. Ribeiro, S. Picoli, and R. S. Mendes, Different diffusive regimes, generalized Langevin and diffusion equations, Phys. Rev. E 85, 011147 (2012).
  97. D. S. Quevedo, M. Conte, M. Dijkstra, and C. M. Smith, Active Brownian particles in power-law viscoelastic media, Phys. Rev. E 114, L013401 (2026).
  98. G. Strobl, The Physics of Polymers: Concepts for Understanding their Structures and Behavior (Springer, Berlin, Heidelberg, 2007).
  99. W. Min and X. S. Xie, Kramers model with a power-law friction kernel: Dispersed kinetics and dynamic disorder of biochemical reactions, Phys. Rev. E 73, 010902(R) (2006).
  100. T. Akimoto, E. Yamamoto, K. Yasuoka, Y. Hirano, and M. Yasui, Non-Gaussian fluctuations resulting from power-law trapping in a lipid bilayer, Phys. Rev. Lett. 107, 178103 (2011).
  101. H. Li, S.-X. Dou, Y.-R. Liu, W. Li, P. Xie, W.-C. Wang, and P.-Y. Wang, Mapping intracellular diffusion distribution using single quantum dot tracking: Compartmentalized diffusion defined by endoplasmic reticulum, J. Am. Chem. Soc. 137, 436 (2015).
  102. R. Zwanzig, Nonlinear generalized Langevin equations, J. Stat. Phys. 9, 215 (1973).
  103. S. C. Kou and X. S. Xie, Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule, Phys. Rev. Lett. 93, 180603 (2004).
  104. E. Lutz, Fractional Langevin equation, Phys. Rev. E 64, 051106 (2001).
  105. I. Goychuk, Viscoelastic subdiffusion: Generalized Langevin equation approach, Adv. Chem. Phys. 150, 187 (2012).
  106. T. G. Mason and D. A. Weitz, Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids, Phys. Rev. Lett. 74, 1250 (1995).
  107. F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations, Phys. Rev. Lett. 79, 3286 (1997).
  108. P. Sollich, Rheological constitutive equation for a model of soft glassy materials, Phys. Rev. E 58, 738 (1998).
  109. B. Fabry, G. N. Maksym, J. P. Butler, M. Glogauer, D. Navajas, and J. J. Fredberg, Scaling the microrheology of living cells, Phys. Rev. Lett. 87, 148102 (2001).
  110. É. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland, How far from equilibrium is active matter? Phys. Rev. Lett. 117, 038103 (2016).
  111. D. Martin, J. O'Byrne, M. E. Cates, É. Fodor, C. Nardini, J. Tailleur, and F. van Wijland, Statistical mechanics of active Ornstein-Uhlenbeck particles, Phys. Rev. E 103, 032607 (2021).
  112. M. Javanainen, H. Martinez-Seara, C. V. Kelly, P. Jungwirth, and B. Fábián, Anisotropic diffusion of membrane proteins at experimental timescales, J. Chem. Phys. 155, 015102 (2021).
  113. F. R. de Hoog, J. H. Knight, and A. N. Stokes, An improved method for numerical inversion of Laplace transforms, SIAM J. Sci. Stat. Comput. 3, 357 (1982).
  114. The mpmath development team, Mpmath: A Python library for arbitrary-precision floating-point arithmetic (version 1.4.0) (2026), http://mpmath.org/.
  115. I. M. Tolić-Nørrelykke, E.-L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, Anomalous diffusion in living yeast cells, Phys. Rev. Lett. 93, 078102 (2004).
  116. B. Wang, S. M. Anthony, S. C. Bae, and S. Granick, Anomalous yet Brownian, Proc. Natl. Acad. Sci. USA 106, 15160 (2009).
  117. X. Zhang, H. Wang, and W. Deng, Brownian non-Gaussian polymer diffusion in non-static media, Chaos 34, 123144 (2024).
  118. H. Xu, X. Zheng, and X. Shi, Deciphering non-Gaussianity of diffusion based on the evolution of diffusivity, Phys. Rev. Res. 6, 023014 (2024).
  119. E. Lemaitre, I. M. Sokolov, R. Metzler, and A. V. Chechkin, Non-Gaussian displacement distributions in models of heterogeneous active particle dynamics, New J. Phys. 25, 013010 (2023).
  120. F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76, 046602 (2013).
  121. T. Kanaya, I. Tsukushi, and K. Kaji, Non-Gaussian parameter and heterogeneity of amorphous polymers, Prog. Theor. Phys. Suppl. 126, 133 (1997).
  122. A. Godec, M. Bauer, and R. Metzler, Collective dynamics effect transient subdiffusion of inert tracers in flexible gel networks, New J. Phys. 16, 092002 (2014).
  123. D. S. Grebenkov, Time-averaged quadratic functionals of a Gaussian process, Phys. Rev. E 83, 061117 (2011).
  124. X. Zheng, B. ten Hagen, A. Kaiser, M. Wu, H. Cui, Z. Silber-Li, and H. Löwen, Non-Gaussian statistics for the motion of self-propelled Janus particles: Experiment versus theory, Phys. Rev. E 88, 032304 (2013).
  125. Q. Yin, Y. Li, F. Marchesoni, S. Nayak, and P. K. Ghosh, Non-Gaussian normal diffusion in low dimensional systems, Front. Phys. 16, 33203 (2021).
  126. A. A. Anashkina, Protein-DNA recognition mechanisms and specificity, Biophys. Rev. 15, 1007 (2023).
  127. A. Akhmanova and J. A. Hammer, Linking molecular motors to membrane cargo, Curr. Opin. Cell Biol. 22, 479 (2010).
  128. L. L. Gutierrez-Martinez and M. Sandoval, Time-dependent propulsion of fully inertial active stochastic particles: Theory and simulations, J. Phys.: Condens. Matter 37, 135102 (2025).
  129. C.-H. Li, X. Yin, S.-Q. He, and G.-K. Xu, Time-dependent active force drives periodic reversal in collective cell migration, Biophys. J. 124, 3542 (2025).
  130. H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math. 2011, 298628 (2011).
  131. F. Mainardi, A. Mura, R. Gorenflo, and M. Stojanović, The two forms of fractional relaxation of distributed order, J. Vib. Control 13, 1249 (2007).
  132. Z. Odibat, Approximations of fractional integrals and Caputo fractional derivatives, Appl. Math. Comput. 178, 527 (2006).
  133. H. Brunner, 1896–1996: One hundred years of Volterra integral equations of the first kind, Appl. Numer. Math. 24, 83 (1997).
  134. M. Stynes, A survey of the L1 scheme in the discretisation of time-fractional problems, Numer. Math. Theor. Methods Appl. 15, 1173 (2022).
  135. J. R. M. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resour. Res. 20, 1898 (1984).

Outline

Information

Phys. Rev. E 114, 044126– Published 16 October, 2026
  • Received 24 June 2026
  • Revised 21 August 2026
  • Accepted 29 September 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation