Skip to main content
SpringerLink
Log in

Random Renormalization Group Operators Applied to Stochastic Dynamics

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Let X(t) be a fixed point the renormalization group operator (RGO), R p,r X(t)=X(rt)/r p. Scaling laws for the probability density, mean first passage times, finite-size Lyapunov exponents of such fixed points are reviewed in anticipation of more general results. A generalized RGO, \(\mathcal{R}_{P,n}\) where P is a random variable, is introduced. Scaling laws associated with these random RGOs (RRGOs) are demonstrated numerically and applied to subdiffusion in bacterial cytoplasm and a process modeling the transition from subdiffusion to classical diffusion. The scaling laws for the RRGO are not simple power laws, but are a weighted average of power laws. The weighting used in the scaling laws can be determined adaptively via Bayes’ theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. O’Malley, D., Cushman, J.H., Johnson, G.: Scaling laws for fractional Brownian motion with a power-law clock. J. Stat. Mech. Theory Exp. 2011, L01001 (2011)

    Article  Google Scholar 

  2. Gitterman, M.: Mean first passage time for anomalous diffusion. Phys. Rev. E 62, 6065–6070 (2000)

    Article  ADS  Google Scholar 

  3. Parashar, R., Cushman, J.H.: Finite-size Lyapunov exponent for Lévy processes. Phys. Rev. E 76, 017201 (2007)

    Article  ADS  Google Scholar 

  4. Kleinfelter, N., Moroni, M., Cushman, J.H.: Application of the finite-size Lyapunov exponent to particle tracking velocimetry in fluid mechanics experiments. Phys. Rev. E 72, 056306 (2005)

    Article  ADS  Google Scholar 

  5. 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. Birkhäuser, Cambridge (2003)

    Google Scholar 

  6. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York (1994)

    MATH  Google Scholar 

  7. Golding, I., Cox, E.C.: Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96(9), 098102 (2006)

    Article  ADS  Google Scholar 

  8. Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  9. Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Shraiman, B.I.: Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A 33, 1141–1151 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Stanley, H.E., Meakin, P.: Multifractal phenomena in physics and chemistry. Nature 335, 405–409 (1988)

    Article  ADS  Google Scholar 

  11. Paladin, G., Vulpiani, A.: Anomalous scaling laws in multifractal objects. Phys. Rep. 156, 147–225 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  12. Hoel, P.G., Port, S.C., Stone, C.J.: Introduction to Probability Theory. Houghton Mifflin, Boston (1971)

    MATH  Google Scholar 

  13. O’Malley, D., Cushman, J.H.: A renormalization group classification of nonstationary and/or infinite second moment diffusive processes. Journal of Statistical Physics 146(5), 989–1000 (2012). doi:10.1007/s10955-012-0448-3

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. 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)

    Article  ADS  Google Scholar 

  15. Debabrata, P.: Probabilistic phase space trajectory description for anomalous polymer dynamics. J. Phys. Condens. Matter 23(10), 105103 (2011)

    Article  Google Scholar 

  16. Jeon, J.H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sorensen, K., Oddershede, L., Metzler, R.: In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011)

    Article  ADS  Google Scholar 

  17. Schoen, M., Cushman, J.H., Diestler, D.J.: Anomalous diffusion in confined monolayer films. Mol. Phys. 81(2), 475–490 (1994)

    Article  ADS  Google Scholar 

  18. 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. 34(4), 490–494 (2011)

    Article  ADS  Google Scholar 

  19. Kantor, Y., Kardar, M.: Anomalous diffusion with absorbing boundary. Phys. Rev. E 76, 061121 (2007)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors wish to thank Ido Golding and Edward C. Cox for the mRNA data, and the NSF for supporting this work under contracts CMG-0934806 and EAR-0838224.

Author information

Authors and Affiliations

  1. Department of Earth, Atmospheric, and Planetary Sciences, Purdue University, West Lafayette, IN, 47907, USA

    Daniel O’Malley & John H. Cushman

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    John H. Cushman

Authors
  1. Daniel O’Malley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John H. Cushman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel O’Malley.

Rights and permissions

Reprints and permissions

About this article

Cite this article

O’Malley, D., Cushman, J.H. Random Renormalization Group Operators Applied to Stochastic Dynamics. J Stat Phys 149, 943–950 (2012). https://doi.org/10.1007/s10955-012-0630-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0630-7

Keywords

Search

Navigation