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

Abstract

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.

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Acknowledgements

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

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Correspondence to Daniel O’Malley.

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O’Malley, D., Cushman, J.H. A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes. J Stat Phys 146, 989–1000 (2012). https://doi.org/10.1007/s10955-012-0448-3

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Keywords

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