A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes |
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Authors: | Daniel O’Malley John H Cushman |
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Institution: | (1) Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN 47907, USA;(2) Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA |
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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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Keywords: | |
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