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Identifying the interpretation in two-dimensional diffusion processes with power-law spatially dependent diffusivity

He-Chuan Liu1, Rui-Bin Zhong1, Jun-Feng Zhang1, Li-Ming Fan2, Peng-Cheng Li1, and Ming-Gen Li1,*

  • *Contact author: mgli@stu.edu.cn
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Phys. Rev. Research 8, 013143 – Published 9 February, 2026

DOI: https://doi.org/10.1103/xz7y-n8w2

Abstract

Heterogeneous diffusion processes (HDPs) are ubiquitous in macromolecules or submicron tracers in biological cells. The Fokker-Planck equation (FPE) provides a fundamental framework for analyzing these diffusion dynamics. However, its formulations for HDPs with spatially dependent diffusivity are not unique, which can lead to potentially divergent dynamical predictions. We investigate three conventional interpretations (Itô, Stratonovich, and Hänggi-Klimontovich) of FPEs describing two-dimensional diffusion processes with power-law spatially dependent diffusivity. Through analytical solutions of time-dependent probability density functions, which are validated by numerical simulations, we reveal significant interpretation-dependent discrepancies in several physical quantities. These include distinct temporal evolution of mean-square displacement (MSD), survival probabilities, and threshold times at which half of the particles reach a set boundary. Furthermore, we demonstrate that the mean of time-averaged MSDs varies across different interpretations, yielding distinct effective diffusion coefficients. This offers a chance to identify the applicable interpretation by comparing theoretical effective diffusion coefficients with those extracted from single-particle tracking experiments. Our findings reveal that the appropriate interpretation could depend on measurement timescales and system's parameters. We believe that these results are helpful for investigating diffusion dynamics in complex heterogeneous environments.

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References (62)

  1. Y. J. Liang, W. Wang, R. Metzler, and A. G. Cherstvy, Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks, Phys. Rev. E 108, 034113 (2023).
  2. W. Wang, A. G. Cherstvy, H. Kantz, R. Metzler, and I. M. Sokolov, Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes, Phys. Rev. E 104, 024105 (2021).
  3. V. Sposini, S. Nampoothiri, A. Chechkin, E. Orlandini, F. Seno, and F. Baldovin, Being heterogeneous is advantageous: Extreme Brownian non-Gaussian searches, Phys. Rev. Lett. 132, 117101 (2024).
  4. W. Wang, A. G. Cherstvy, X. Liu, and R. Metzler, Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise, Phys. Rev. E 102, 012146 (2020).
  5. A. G. Cherstvy, S. Thapa, C. E. Wagner, and R. Metzler, Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels, Soft Matter 15, 2526 (2019).
  6. Y. Xu, X. Liu, Y. Li, and R. Metzler, Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes, Phys. Rev. E 102, 062106 (2020).
  7. T. A. Waigh and N. Korabel, Heterogeneous anomalous transport in cellular and molecular biology, Rep. Prog. Phys. 86, 126601 (2023).
  8. K. Sakamoto, T. Akimoto, M. Muramatsu, M. S. Sansom, R. Metzler, and E. Yamamoto, Heterogeneous biological membranes regulate protein partitioning via fluctuating diffusivity, PNAS Nexus 2, pgad258 (2023).
  9. L. S. Ferreira, F. L. Metz, and P. Barucca, Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures, Phys. Rev. E 111, 014151 (2025).
  10. M. G. Li and J. D. Bao, Resonance behavior for the dynamical friction of a system in a trapping potential, Phys. Rev. Res. 4, 043058 (2022).
  11. L. M. Xiang, K. Chen, R. Yan, W. Li, and K. Xu, Single-molecule displacement mapping unveils nanoscale heterogeneities in intracellular diffusivity, Nat. Methods 17, 524 (2020).
  12. A. G. Cherstvy and R. Metzler, Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes, J. Chem. Phys. 142, 144105 (2015).
  13. Q. Xuan, F. Du, T. J. Wu, and G. R. Chen, Emergence of heterogeneous structures in chemical reaction-diffusion networks, Phys. Rev. E 82, 046116 (2010).
  14. L. Menon and C. Anteneodo, Random search with resetting in heterogeneous environments, Phys. Rev. E 110, 054111 (2024).
  15. M. G. Li, M. Hu, L. M. Fan, J. D. Bao, and P. C. Li, Quantifying the energy landscape in weakly and strongly disordered frictional media, J. Chem. Phys. 160, 024903 (2024).
  16. M. G. Li, L. M. Fan, and J. D. Bao, Power-law frictional landscapes induce anomalous diffusion, Physica A 597, 127249 (2022).
  17. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1989).
  18. K. S. Fa, Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients, Phys. Rev. E 84, 012102 (2011).
  19. O. Farago and N. Grønbech-Jensen, Langevin dynamics in inhomogeneous media: Re-examining the Itô-Stratonovich dilemma, Phys. Rev. E 89, 013301 (2014).
  20. G. Volpe and J. Wehr, Effective drifts in dynamical systems with multiplicative noise: A review of recent progress, Rep. Prog. Phys. 79, 053901 (2016).
  21. I. M. Sokolov, Itô, and Stratonovich, Hänggi and all the rest: The thermodynamics of interpretation, Chem. Phys. 375, 359 (2010).
  22. M. G. Li, R. Xing, L. M. Fan, M. Hu, J. D. Bao, and P. C. Li, On the interpretation of mean-square displacement in heterogeneous systems, New J. Phys. 26, 093024 (2024).
  23. J. M. Sancho, Brownian colloidal particles: Itô, Stratonovich, or a different stochastic interpretation, Phys. Rev. E 84, 062102 (2011).
  24. A. Bhattacharyay, Brownian motion near a wall: The dilemma of Itô or Stratonovich, J. Phys. A: Math. Theor. 58, 213001 (2025).
  25. G. Pesce, A. McDaniel, S. Hottovy, J. Wehr, and G. Volpe, Stratonovich-to-Itô transition in noisy systems with multiplicative feedback, Nat. Commun. 4, 2733 (2013).
  26. G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger, Influence of noise on force measurements, Phys. Rev. Lett. 104, 170602 (2010).
  27. T. Brettschneider, G. Volpe, L. Helden, J. Wehr, and C. Bechinger, Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method, Phys. Rev. E 83, 041113 (2011).
  28. A. Pacheco-Pozo, I. M. Sokolov, R. Metzler, and D. Krapf, Heterogeneous diffusion in an harmonic potential: The role of the interpretation, New J. Phys. 27, 064602 (2025).
  29. A. Pacheco-Pozo, M. Balcerek, A. Wyłomanska, K. Burnecki, I. M. Sokolov, and D. Krapf, Langevin equation in heterogeneous landscapes: How to choose the interpretation, Phys. Rev. Lett. 133, 067102 (2024).
  30. A. G. Cherstvy, A. V. Chechkin, and R. Metzler, Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity, Soft Matter 10, 1591 (2014).
  31. T. Kühn, T. O. Ihalainen, J. Hyväluoma, N. Dross, S. F. Willman, J. Langowski, M. Vihinen-Ranta, and J. Timonen, Protein diffusion in mammalian cell cytoplasm, PLoS One 6, e22962 (2011).
  32. Y. L. Klimontovich, Nonlinear Brownian motion, Phys. Usp. 37, 737 (1994).
  33. Y. L. Klimontovich, Statistical Physics (Nauka, Moscow, 1982).
  34. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications (Springer, Berlin, 2013).
  35. M. Heidernätsch, On the diffusion in inhomogeneous systems, Ph.D. thesis, 2015.
  36. Muñoz-Gil, et al., Quantitative evaluation of methods to analyze motion changes in single-particle experiments, Nat. Commun. 16, 6749 (2025).
  37. C. Manzo and M. F. Garcia-Parajo, A review of progress in single particle tracking: From methods to biophysical insights, Rep. Prog. Phys. 78, 124601 (2015).
  38. X. S. Xie, P. J. Choi, G.-W. Li, N. K. Lee, and G. Lia, Single-molecule approach to molecular biology in living bacterial cells, Annu. Rev. Biophys. 37, 417 (2008).
  39. H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, Single particle tracking: From theory to biophysical applications, Chem. Rev. 117, 7331 (2017).
  40. S. Giordano and R. Blossey, Effective diffusion constant of stochastic processes with spatially periodic noise, Phys. Rev. E 110, 044123 (2024).
  41. D. Breoni, H. Löwen, and R. Blossey, Active noise-driven particles under space-dependent friction in one dimension, Phys. Rev. E 103, 052602 (2021).
  42. N. Leibovich and E. Barkai, Infinite ergodic theory for heterogeneous diffusion processes, Phys. Rev. E 99, 042138 (2019).
  43. G. Vaccario, C. Antoine, and J. Talbot, First-passage times in d-dimensional heterogeneous media, Phys. Rev. Lett. 115, 240601 (2015).
  44. T. C. Gard, Introduction to Stochastic Differential Equations (Dekker, New York, 1988).
  45. P. Hänggi, Connection between deterministic and stochastic descriptions of nonlinear systems, Helv. Phys. Acta 53, 491 (1980).
  46. B. Lindner, The diffusion coefficient of nonlinear Brownian motion, New J. Phys. 9, 136 (2007).
  47. S. Regev, N. Grønbech-Jensen, and O. Farago, Isothermal Langevin dynamics in systems with power-law spatially dependent friction, Phys. Rev. E 94, 012116 (2016).
  48. R. Metzler, J. H. Jeon, and A. G. Cherstvy, Non-Brownian diffusion in lipid membranes: Experiments and simulations, Biochim. Biophys. Acta: Biomembranes 1858, 2451 (2016).
  49. I. M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043 (2012).
  50. T. G. Mattos, C. Mejía-Monasterio, R. Metzler, and G. Oshanin, First passages in bounded domains: When is the mean first passage time meaningful? Phys. Rev. E 86, 031143 (2012).
  51. W. Wang, A. G. Cherstvy, R. Metzler, and I. M. Sokolov, Restoring ergodicity of stochastically reset anomalous-diffusion processes Phys. Rev. Res. 4, 013161 (2022).
  52. A. Sabri, X. Xu, D. Krapf, and M. Weiss, Elucidating the origin of heterogeneous anomalous diffusion in the cytoplasm of mammalian cells, Phys. Rev. Lett. 125, 058101 (2020).
  53. E. Barkai, Y. Garini, and R. Metzler, Strange kinetics of single molecules in living cells, Phys. Today 65(8), 29 (2012).
  54. T. Neusius, I. M. Sokolov, and J. C. Smith, Subdiffusion in time-averaged, confined random walks, Phys. Rev. E 80, 011109 (2009).
  55. Y. He, S. Burov, R. Metzler, and E. Barkai, Random time-scale invariant diffusion and transport coefficients, Phys. Rev. Lett. 101, 058101 (2008).
  56. 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).
  57. A. Godec and R. Metzler, Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics, Phys. Rev. Lett. 110, 020603 (2013).
  58. 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).
  59. G. Seisenberger, M. U. Ried, T. Endreß, H. Buning, M. Hallek, and C. Brauchle, Real-time single-molecule imaging of the infection pathway of an adeno-associated virus, Science 294, 1929 (2001).
  60. T. Akimoto, J.-H. Jeon, R. Metzler, T. Miyaguchi, T. Uneyama, and E. Yamamoto, Anomalous statistics in the Langevin equation with fluctuating diffusivity: From Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking, arXiv:2509.12571.
  61. T. Akimoto and E. Yamamoto, Distributional behaviors of time-averaged observables in the Langevin equation with fluctuating diffusivity: Normal diffusion but anomalous fluctuations, Phys. Rev. E 93, 062109 (2016).
  62. M. Kimura and T. Akimoto, Non-Markovian effects of conformational fluctuations on the global diffusivity in Langevin equation with fluctuating diffusivity, J. Chem. Phys. 159, 055102 (2023).

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Phys. Rev. Research 8, 013143– Published 9 February, 2026
  • Received 29 October 2025
  • Accepted 26 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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