A Simple Diffusion Model Showing Anomalous Scaling

G. Rowlands1 and J. C. Sprott2

1Department of Physics, University of Warwick, Coventry CV4 7AL, England
2Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA

Received 25 April 2008; accepted 21 July 2008; published online 13 August 2008

ABSTRACT

A number of iterated maps and one flow, which show chaotic behavior, have been studied numerically and their time evolution expressed in terms of higher-order moments Mm(t) All the cases show anomalous behavior with Mm(t) ~ tg(m) with g(m) <> am. A simple analytic treatment isgiven based on an effective diffusion that is dependent on both space and time. This leads to a form for g(m)/m = a − b/m, which is in good agreement with numerical results. This behavior is attributed to the presence of convective motion superimposed on the background diffusion, and hence this behavior is expected in a wide variety of maps and flows.

Ref: G. Rowlands and  J. C. Sprott, Physics of Plasmas 15, 082308-1 - 082308-7 (2008)

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Fig. 1. Intermittency in the Chirikov map for K = 2.2pi.
Figure 1


Fig. 2. Anomalous scaling in the Chirikov map as a function of K.
Figure 2


Fig. 3. Simple and anomalous scaling in the Chirikov map as a function of m.
Figure 3


Fig. 4. Normalized PDF for the Chirikov map after 1000 iterations.
Figure 4

Fig. 5. The Weiss map for k = 0.4.
Figure 5


Fig. 6. Scaling of moments in the Weiss map as a function of m.
Figure 6


Fig. 7. Scaling of moments in the labyrinth model as a function of m.
Figure 7


Fig. 8. Normalized PDFs for the Chirikov map at two different times.
Figure 8