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  • Letter
  • Open Access

Memory-multi-fractional Brownian motion with continuous correlations

Wei Wang1, Michał Balcerek2, Krzysztof Burnecki2, Aleksei V. Chechkin1,2,3, Skirmantas Janušonis4, Jakub Ślęzak2, Thomas Vojta5, Agnieszka Wyłomańska2, and Ralf Metzler1,6

  • 1Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
  • 2Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
  • 3Akhiezer Institute for Theoretical Physics, National Science Center “Kharkov Institute of Physics and Technology,” Kharkov 61108, Ukraine
  • 4Department of Psychological and Brain Sciences, University of California, Santa Barbara, Santa Barbara, California 93106, USA
  • 5Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA
  • 6Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea

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Phys. Rev. Research 5, L032025 – Published 23 August, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L032025

Abstract

We propose a generalization of the widely used fractional Brownian motion (FBM), memory-multi-FBM (MMFBM), to describe viscoelastic or persistent anomalous diffusion with time-dependent memory exponent α(t) in a changing environment. In MMFBM the built-in, long-range memory is continuously modulated by α(t). We derive the essential statistical properties of MMFBM such as its response function, mean-squared displacement (MSD), autocovariance function, and Gaussian distribution. In contrast to existing forms of FBM with time-varying memory exponents but a reset memory structure, the instantaneous dynamic of MMFBM is influenced by the process history, e.g., we show that after a steplike change of α(t) the scaling exponent of the MSD after the α step may be determined by the value of α(t) before the change. MMFBM is a versatile and useful process for correlated physical systems with nonequilibrium initial conditions in a changing environment.

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References (97)

  1. 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).
  2. E. Barkai, Y. Garini, and R. Metzler, Strange kinetics of single molecules in living cells, Phys. Today 65(8), 29 (2012).
  3. M. J. Saxton, Anomalous subdiffusion in fluorescence photobleaching recovery: A Monte Carlo study, Biophys. J. 81, 2226 (2001).
  4. F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76, 046602 (2013).
  5. J. Szymanski and M. Weiss, Elucidating the Origin of Anomalous Diffusion in Crowded Fluids, Phys. Rev. Lett. 103, 038102 (2009).
  6. J.-H. Jeon, N. Leijnse, L. B. Oddershede, and R. Metzler, Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions, New J. Phys. 15, 045011 (2013).
  7. 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).
  8. 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).
  9. J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, and R. Metzler, In Vivo Anomalous Diffusion and Weak Ergodicity Breaking of Lipid Granules, Phys. Rev. Lett. 106, 048103 (2011).
  10. M. Magdziarz, A. Weron, K. Burnecki, and J. Klafter, Fractional Brownian Motion Versus the Continuous-Time Random Walk: A Simple Test for Subdiffusive Dynamics, Phys. Rev. Lett. 103, 180602 (2009).
  11. I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Shav-Tal, E. Barkai, and Y. Garini, Transient Anomalous Diffusion of Telomeres in the Nucleus of Mammalian Cells, Phys. Rev. Lett. 103, 018102 (2009).
  12. A. Caspi, R. Granek, and M. Elbaum, Enhanced Diffusion in Active Intracellular Transport, Phys. Rev. Lett. 85, 5655 (2000).
  13. G. Seisenberger, M. U. Ried, T. Endress, H. Büning, M. Hallek, and C. Bräuchle, Real-time single-molecule imaging of the infection pathway of an adeno-associated virus, Science 294, 1929 (2001).
  14. K. J. Chen, B. Wang, and S. Granick, Memoryless self-reinforcing directionality in endosomal active transport within living cells, Nat. Mater. 14, 589 (2015).
  15. A. V. Weigel, B. Simon, M. M. Tamkun, and D. Krapf, Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking, Proc. Natl. Acad. Sci. U.S.A. 108, 6438 (2011).
  16. W. He, H. Song, Y. Su, L. Geng, B. J. Ackerson, H. B. Peng, and P. Tong, Dynamic heterogeneity and non-Gaussian statistics for acetylcholine receptors on live cell membrane, Nat. Commun. 7, 11701 (2016).
  17. C. Manzo, J. A. Torreno-Pina, P. Massignan, G. J. Lapeyre Jr., M. Lewenstein, and M. F. Garcia Parajo, Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity, Phys. Rev. X 5, 011021 (2015).
  18. M. J. Skaug, L. Wang, Y. Ding, and D. K. Schwartz, Hindered nanoparticle diffusion and void accessibility in a three-dimensional porous medium, ACS Nano 9, 2148 (2015).
  19. J.-H. Jeon, H. Martinez-Seara Monne, M. Javanainen, and R. Metzler, Anomalous Diffusion of Phospholipids and Cholesterols in a Lipid Bilayer and its Origins, Phys. Rev. Lett. 109, 188103 (2012).
  20. J.-H. Jeon, M. Javanainen, H. Martinez-Seara, R. Metzler, and I. Vattulainen, Protein Crowding in Lipid Bilayers Gives Rise to Non-Gaussian Anomalous Lateral Diffusion of Phospholipids and Proteins, Phys. Rev. X 6, 021006 (2016).
  21. 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).
  22. X. Hu, L. Hong, M. Dean Smith, T. Neusius, and J. C. Smith, The dynamics of single protein molecules is non-equilibrium and self-similar over thirteen decades in time, Nat. Phys. 12, 171 (2016).
  23. S. Hapca, J. W. Crawford, and I. M. Young, Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level, J. R. Soc. Interface 6, 111 (2009).
  24. 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).
  25. 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).
  26. N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. Brunnschweiler, T. K. Doyle, J. D. R. Houghton, G. C. Hays, C. S. Jones, L. R. Noble, V. J. Wearmouth, E. J. Southall, and D.W. Sims, Environmental context explains Lévy and Brownian movement patterns of marine predators, Nature (London) 465, 1066 (2010).
  27. D. W. Sims, N. E. Humphries, N. Hu, V. Medan, and J. Berni, Optimal searching behaviour generated intrinsically by the central pattern generator for locomotion, eLife 8, e50316 (2019).
  28. R. Nathan, C. T. Monk, R. Arlinghaus, T. Adam, J. Alós, M. Assaf, H. Baktoft, C. E. Beardsworth, M. G. Bertram, A. I. Bijleveld et al., Big-data approaches lead to an increased understanding of the ecology of animal movement, Science 375, eabg1780 (2022).
  29. O. Vilk, Y. Orchan, M. Charter, N. Ganot, S. Toledo, R. Nathan, and M. Assaf, Ergodicity Breaking in Area-Restricted Search of Avian Predators, Phys. Rev. X 12, 031005 (2022).
  30. O. Vilk, E. Aghion, T. Avgar, C. Beta, O. Nagel, A. Sabri, R. Sarfati, D. K. Schwartz, M. Weiss, D. Krapf , Unravelling the origins of anomalous diffusion: From molecules to migrating storks, Phys. Rev. Res. 4, 033055 (2022).
  31. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, (North Holland, Amsterdam, 1981).
  32. J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
  33. 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).
  34. I. M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043 (2012).
  35. G. Muñoz-Gil, G. Volpe, M. A.Garcia-March, E. Aghion, A. Argun, C. B. Hong, T. Bland, S. Bo, J. A. Conejero, N. Firbas et al., Objective comparison of methods to decode anomalous diffusion, Nat. Commun. 12, 6253 (2021).
  36. H. Seckler and R. Metzler, Bayesian deep learning for error estimation in the analysis of anomalous diffusion, Nat. Commun. 13, 6717 (2022).
  37. M. Levin, G. Bel, and Y. Roichman, Measurements and characterization of the dynamics of tracer particles in an actin network, J. Chem. Phys. 154, 144901 (2021).
  38. I. Y. Wong, M. L. Gardel, D. R. Reichman, E. R. Weeks, M. T. Valentine, A. R. Bausch, and D. A. Weitz, Anomalous Diffusion Probes Microstructure Dynamics of Entangled F-Actin Networks, Phys. Rev. Lett. 92, 178101 (2004).
  39. A. Díez Fernández, P. Charchar, A. G. Cherstvy, R. Metzler, and M. W. Finnis, The diffusion of doxorubicin drug molecules in silica nanoslits is non-Gaussian, intermittent and anticorrelated, Phys. Chem. Chem. Phys. 22, 27955 (2020).
  40. M. S. Song, H. C. Moon, J.-H. Jeon, and H. Y. Park, Neuronal messenger ribonucleoprotein transport follows an aging Lévy walk, Nat. Commun. 9, 344 (2018).
  41. T. H. Solomon, E. R. Weeks, and H. L. Swinney, Observation of Anomalous Diffusion and Lévy Flights in a Two-Dimensional Rotating Flow, Phys. Rev. Lett. 71, 3975 (1993).
  42. T. Geisel and S. Thomae, Anomalous Diffusion in Intermittent Chaotic Systems, Phys. Rev. Lett. 52, 1936 (1984).
  43. A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (doklady), Acad. Sci. URSS (NS) 26, 115 (1940).
  44. B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10, 422 (1968).
  45. H. Qian, in Processes with Long-Range Correlations: Theory and Applications, edited by G. Rangarajan and M. Z. Ding, Lecture Notes in Physics Vol. 621 (Springer, New York, 2003).
  46. W. H. Deng and E. Barkai, Ergodic properties of fractional Brownian-Langevin motion, Phys. Rev. E 79, 011112 (2009).
  47. 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).
  48. D. Krapf, N. Lukat, E. Marinari, R. Metzler, G. Oshanin, C. Selhuber-Unkel, A. Squarcini, L. Stadler, M. Weiss, X. Xu et al., Spectral Content of a Single Non-Brownian Trajectory, Phys. Rev. X 9, 011019 (2019).
  49. 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).
  50. Z. R. Fox, E. Barkai, and D. Krapf, Aging power spectrum of membrane protein transport and other subordinated random walks, Nat. Commun. 12, 6162 (2021).
  51. B. J. Alder and T. E. Wainwright, Decay of the velocity autocorrelation function, Phys. Rev. A 1, 18 (1970).
  52. A. McDonough, S. P. Russo, and I. K. Snook, Long-time behavior of the velocity autocorrelation function for moderately dense, soft-repulsive, and Lennard-Jones fluids, Phys. Rev. E 63, 026109 (2001).
  53. G. L. Paul and P. N. Pusey, Observation of a long-time tail in Brownian motion, J. Phys. A 14, 3301 (1981).
  54. T. Franosch, M. Grimm, M. Belushkin, F. M. Mor, G. Foffi, L. Forro, and S. Jeney, Resonances arising from hydrodynamic memory in Brownian motion, Nature (London) 478, 85 (2011).
  55. F. Comte and E. Renault, Long memory in continuous-time stochastic volatility models, Math. Fin. 8, 291 (1998).
  56. J. Gatheral, T. Jaisson, and M. Rosenbaum, Volatility is rough, Quant. Fin. 18, 933 (2018).
  57. B. Wang, J. Kuo, S. C. Bae, and S. Granick, When Brownian diffusion is not Gaussian, Nat. Mater. 11, 481 (2012).
  58. Y. Lanoiselée, N. Moutal, and D. S. Grebenkov, Diffusion-limited reactions in dynamic heterogeneous media, Nat. Commun. 9, 4398 (2018).
  59. J. Ślȩzak, R. Metzler, and M. Magdziarz, Superstatistical generalised Langevin equation: Non-Gaussian viscoelastic anomalous diffusion, New J. Phys. 20, 023026 (2018).
  60. A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov, Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities, Phys. Rev. X 7, 021002 (2017).
  61. M. V. Chubynsky and G. W. Slater, Diffusing Diffusivity: A Model for Anomalous, yet Brownian, Diffusion, Phys. Rev. Lett. 113, 098302 (2014).
  62. W. Wang, A. G. Cherstvy, A. V. Chechkin, S. Thapa, F. Seno, X. Liu, and R. Metzler, Fractional Brownian motion with random diffusivity: Emerging residual nonergodicity below the correlation time, J. Phys. A 53, 474001 (2020).
  63. W. Wang, F. Seno, I. M. Sokolov, A. V. Chechkin, and R. Metzler, Unexpected crossovers in correlated random-diffusivity processes, New J. Phys. 22, 083041 (2020).
  64. D. Han, N. Korabel, R. Chen, M. Johnston, A.Gavrilova, V. J. Allan, S. Fedotov, and T. A. Waigh, Deciphering anomalous heterogeneous intracellular transport with neural networks, elife 9, e52224 (2020).
  65. M. Balcerek, K. Burnecki, S. Thapa, A. Wyłomańska, and A. Chechkin, Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions, Chaos 32, 093114 (2022).
  66. P. D. Odermatt, T. P. Miettinen, J. Lemiére, J. H. Kang, E. Bostan, S. R. Manalis, K. C. Huang, and F. Chang, Variations of intracellular density during the cell cycle arise from tip-growth regulation in fission yeast, eLife 10, e64901 (2021).
  67. A. J. Barlow, G. Harrison, J. B. Irving, M. G. Kim, J. Lamb, and W. C. Pursley, The effect of pressure on the viscoelastic properties of liquids, P. Roy. Soc. A-Math. Phy. 327, 403 (1972).
  68. J. Caspers, N. Ditz, K. K. Kumar, F. Ginot, C. Bechinger, M. Fuchs, and M. Krüger, How are mobility and friction related in viscoelastic fluids?, J. Chem. Phys. 158, 024901 (2023).
  69. F. Etoc, E. Balloul, C. Vicario, D. Normanno, D. Liße, A. Sittner, J. Piehler, M. Dahan, and M. Coppey, Non-specific interactions govern cytosolic diffusion of nanosized objects in mammalian cells, Nat. Mater. 17, 740 (2018).
  70. I. Heller, G. Sitters, O. D. Broekmans, G. Farge, C. Menges, W. Wende, S. W. Hell, E. J. G. Peterman, and G. J. L. Wuite, STED nanoscopy combined with optical tweezers reveals protein dynamics on densely covered DNA, Nat. Methods 10, 910 (2013).
  71. D. Robert, T. Nguyen, F. Gallet, and C. Wilhelm, In vivo determination of fluctuating forces during endosome trafficking using a combination of active and passive microrheology, PLoS One 5, e10046 (2010).
  72. I. Goychuk, V. O. Kharchenko, and R. Metzler, Molecular motors pulling cargos in the viscoelastic cytosol: Power strokes beat subdiffusion, Phys. Chem. Chem. Phys. 16, 16524 (2014).
  73. P. Lévy, Random Functions: General Theory with Special Reference to Laplacian Random Functions (University of California Press, Berkeley, 1953).
  74. We here use a dimensionless notation. To restore physical units the associated prefactor α(s) in Eq. (1) can be replaced by the form [Kα(s)]=α(s)ℓ02/t0α(s). Dimensionality can then be removed by setting the length scale ℓ0 and the timescale t0 to unity. The specific choice of the dimensionless prefactor α(s) in this expression for Kα(s) allows a compact notation and to illustrate the most essential features of the MMFBM model.
  75. A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66, 046129 (2002).
  76. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.5.L032025 for details on the definitions and properties of MFBM and MMFBM.
  77. A. Ayache, C. Esser, and J. Hamonier, A new multifractional process with random exponent, Risk Decis. Anal. 7, 5 (2018).
  78. A. Philippe, D. Surgailis, and M.-C. Viano, Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl. 52, 651 (2008).
  79. D. Surgailis, Nonhomogeneous fractional integration and multifractional processes, Stochastic Processes Appl. 118, 171 (2008).
  80. A. Ayache and F. Bouly, Moving average Multifractional processes with random exponent: Lower bounds for local oscillations, Stochastic Processes Appl. 146, 143 (2022).
  81. S. A. Stoev and M. S. Taqqu, How rich is the class of multifractional Brownian motions?, Stochastic Processes Appl. 116, 200 (2006).
  82. A. Ayache and J. Lévy-Vehel, The generalized multifractional Brownian motion, Stat. Infererence Stochastic Processes 3, 7 (2000).
  83. M. Balcerek and K. Burnecki, Testing of multifractional Brownian motion, Entropy 22, 1403 (2020).
  84. D. Szarek, I. Jabłoński, D. Krapf, and A. Wyłomańska, Multifractional Brownian motion characterization based on Hurst exponent estimation and statistical learning, Chaos 32, 083148 (2022).
  85. M. Li, S. C. Lim, B.-J. Hu, and H. Feng, in Lecture Notes in Computer Science, edited by Y. Shi, G. Dick van Albada, J. Dongarra, and P. M. A. Sloot (Springer, Berlin, 1987), Vol. 4488.
  86. J. Lévy-Vehel and R. Riedi, in Fractals in Engineering, edited by J. Lévy Véhel, E. Lutton, and C. Tricot (Springer, Berlin, 1997).
  87. S. Bianchi, A. Pantanella, and A. Pianese, Modeling stock prices by multifractional Brownian motion: An improved estimation of the pointwise regularity, Quant. Fin. 13, 1317 (2011).
  88. K. C. Lee, Characterization of turbulence stability through the identification of multifractional Brownian motions, Nonlin. Proc. Geophys. 20, 97 (2013).
  89. C. Box, Multifractional Brownian motion and its applications to factor analysis on consumer confidence index, Senior thesis, Claremont College, 2021.
  90. We note that we choose the additional prefactor α(t) in the definition of MFBM; compare [77] in agreement with RL-FBM [44]. This choice does not change the fundamental behavior.
  91. K. Falconer, Fractal Geometry (Wiley, Chichester, UK, 1990).
  92. J. Picard, in Séminaire de Probabilités XLIII, edited by C. Donati-Martin, A. Lejay, and A. Rouault, Lecture Notes in Mathematics Vol. 2006 (Springer, New York, 2011).
  93. W. Wang et al. (unpublished).
  94. D. Molina-Garcia, T. Sandev, H. Safdari, G. Pagnini, A. Chechkin, and R. Metzler, Crossover from anomalous to normal diffusion: Truncated power-law noise correlations and applications to dynamics in lipid bilayers, New J. Phys. 20, 103027 (2018).
  95. T. Vojta, S. Halladay, S. Skinner, S. Janušonis, T. Guggenberger, and R. Metzler, Reflected fractional Brownian motion in one and higher dimensions, Phys. Rev. E 102, 032108 (2020).
  96. T. Guggenberger, A. Chechkin, and R. Metzler, Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions, J. Phys. A 54, 29LT01 (2021).
  97. S. Janušonis, N. Detering, R. Metzler, and T. Vojta, Serotonergic axons as fractional Brownian motion paths: Insights into the self-organization of regional densities, Front. Comput. Neurosci. 14, 56 (2020); S. Janušonis, J. H. Haiman, R. Metzler, and T. Vojta, Predicting the distribution of serotonergic axons: A supercomputing simulation of reflected fractional Brownian motion in a 3D-mouse brain model, ibid. 17, 1189853 (2023).

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Phys. Rev. Research 5, L032025– Published 23 August, 2023
  • Received 9 October 2022
  • Accepted 14 July 2023
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