Journal of Statistical Physics

, Volume 146, Issue 5, pp 989–1000 | Cite as

A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes

  • Daniel O’MalleyEmail author
  • John H. Cushman


Anomalous diffusion processes are often classified by their mean square displacement. If the mean square displacement grows linearly in time, the process is considered classical. If it grows like t β with β<1 or β>1, the process is considered subdiffusive or superdiffusive, respectively. Processes with infinite mean square displacement are considered superdiffusive. We begin by examining the ways in which power-law mean square displacements can arise; namely via non-zero drift, nonstationary increments, and correlated increments. Subsequently, we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present. Finally, we introduce an alternative classification scheme based on renormalization groups. This scheme classifies processes with stationary increments such as Brownian motion and fractional Brownian motion in the same groups as the mean square displacement scheme, but does a better job of classifying processes with nonstationary increments and/or processes with infinite second moments such as α-stable Lévy motion. A numerical approach to analyzing data based on the renormalization group classification is also presented.


Anomalous diffusion Renormalization group Self-similarity Scaling Nonstationary increments Correlated increments 



The authors thank the National Science Foundation for supporting this work under contracts CMG-0934806 and EAR-0838224.


  1. 1.
    Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322, 549–560 (1905) CrossRefGoogle Scholar
  2. 2.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000) MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Cushman, J.H., O’Malley, D., Park, M.: Anomalous diffusion as modeled by a nonstationary extension of Brownian motion. Phys. Rev. E 79, 032101 (2009) ADSCrossRefGoogle Scholar
  4. 4.
    O’Malley, D., Cushman, J.H.: Fractional Brownian motion run with a non-linear clock. Phys. Rev. E 82, 032102 (2010) ADSCrossRefGoogle Scholar
  5. 5.
    Golding, I., Cox, E.C.: Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96(9), 098102 (2006) ADSCrossRefGoogle Scholar
  6. 6.
    Kantor, Y., Kardar, M.: Anomalous diffusion with absorbing boundary. Phys. Rev. E 76, 061121 (2007) ADSCrossRefGoogle Scholar
  7. 7.
    Bouchaud, J.-P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms and physical applications. Phys. Rep. 195, 127–293 (1990) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Richardson, L.F.: Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. Ser. A 110, 709–737 (1926) ADSCrossRefGoogle Scholar
  9. 9.
    Humphries, N.E., Queiroz, N., Dyer, J.R.M., Pade, N.G., Musyl, M.K., Schaefer, K.M., Fuller, D.W., Brunnschweiler, J.M., Doyle, T.K., Houghton, J.D.R., Hays, G.C., Jones, C.S., Noble, L.R., Wearmouth, V.J., Southall, E.J., Sims, D.W.: Environmental context explains Levy and Brownian movement patterns of marine predators. Nature 465, 1066–1069 (2010) ADSCrossRefGoogle Scholar
  10. 10.
    Viswanathan, G., Afanasyev, V., Buldyrev, S., Murphy, E.J., Prince, P.A., Stanley, H.E.: Levy flight search patterns of wandering albatrosses. Nature 381, 413–415 (1996) ADSCrossRefGoogle Scholar
  11. 11.
    Bassler, K.E., McCauley, J.L., Gunaratne, G.H.: Nonstationary increments, scaling distributions and variable diffusion processes in financial markets. Proc. Natl. Acad. Sci. 104, 17287–17290 (2007) ADSCrossRefGoogle Scholar
  12. 12.
    Seemann, L., McCauley, J.L., Gunaratne, G.H.: Intraday volatility and scaling in high frequency foreign exchange markets. Int. Rev. Financ. Anal. 20, 121–126 (2011) CrossRefGoogle Scholar
  13. 13.
    Montroll, E.W., Weiss, G.H.: Random walks on lattices. II. J. Math. Phys. 6, 167 (1965) MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Scher, H., Montroll, E.W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455 (1975) ADSCrossRefGoogle Scholar
  15. 15.
    Klafter, J., Blumen, A., Shlesinger, M.F.: Stochastic pathway to anomalous diffusion. Phys. Rev. A 35, 3081 (1987) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Meerschaert, M.M., Scheffler, H.P.: Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41, 623–638 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P.: Limit theorems for coupled continuous time random walks. Ann. Probab. 32, 730 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P.: Limit theorems for continuous-time random walks with two scales. J. Appl. Probab. 41, 455 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York (1994) zbMATHGoogle Scholar
  20. 20.
    Kolmogorov, A.N.: Wienersche spiralen und einige andere interessante kurven im hilbertschen raum. C. R. (Dokl.) Acad. Sci. URSS 26, 115 (1940) Google Scholar
  21. 21.
    Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968) MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    He, Y., Burov, S., Metzler, R., Barkai, E.: Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. 101, 058101 (2008) ADSCrossRefGoogle Scholar
  23. 23.
    Lubelski, A., Sokolov, I.M., Klafter, J.: Nonergodicity mimics inhomogeneity in single particle tracking. Phys. Rev. Lett. 100, 250602 (2008) ADSCrossRefGoogle Scholar
  24. 24.
    Deng, W., Barkai, E.: Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E 79, 011112 (2009) MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Cushman, J.H., Park, M., O’Malley, D.: A stochastic model for anomalous diffusion in confined nano-films near a strain-induced critical point. Adv. Water Resour. (2011). doi: 10.1016/j.advwatres.2011.01.005 Google Scholar
  26. 26.
    Burov, S., Jeon, J.-H., Metzler, R., Barkai, E.: Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking. Phys. Chem. Chem. Phys. 13, 1800–1812 (2011) CrossRefGoogle Scholar
  27. 27.
    Kong, X.P., Cohen, E.G.D.: Anomalous diffusion in a lattice-gas wind-tree model. Phys. Rev. B 40, 4838 (1989) ADSCrossRefGoogle Scholar
  28. 28.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (2003) zbMATHCrossRefGoogle Scholar
  29. 29.
    Park, M., Cushman, J.H.: The complexity of Brownian processes run with non-linear clocks. Mod. Phys. Lett. B 25, 1–10 (2011) MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Taqqu, M.S.: Fractional Brownian motion and long-range dependence. In: Doukhan, P., Oppenheim, G., Taqqu, M.S. (eds.) Theory and Applications of Long Range Dependence. Birkhauser, Cambridge (2003) Google Scholar
  31. 31.
    Jeon, J., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sorenson, K., Oddershede, L., Metzler, R.: In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011) ADSCrossRefGoogle Scholar
  32. 32.
    Condamin, S., Tejedor, V., Voituriez, R., Bénichou, O., Klafter, J.: Probing microscopic origins of confined subdiffusion by first-passage observables. Proc. Natl. Acad. Sci. USA 105, 5675 (2008) ADSCrossRefGoogle Scholar
  33. 33.
    Magdziarz, M., Weron, A., Burnecki, K., Klafter, J.: Fractional Brownian motion versus the continuous-time random walk: A simple test for subdiffusive dynamics. Phys. Rev. Lett. 103, 180602 (2009) ADSCrossRefGoogle Scholar
  34. 34.
    Tejedor, V., Bénichou, O., Voituriez, R., Jungmann, R., Simmel, F., Selhuber-Unkel, C., Oddershede, L.B., Metzler, R.: Quantitative analysis of single particle trajectories: Mean maximal excursion method. Biophys. J. 98, 1364 (2010) ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Earth and Atmospheric SciencesPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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