The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Applic

Parameter Dependence of Ballistic Velocity in Deterministic Diffusion in the Form of Devil’s Staircase

Syuji Miyazaki, Masaomi Yoshida and Hirokazu Fujisaka

Graduate School of Informatics, Kyoto University, Kyoto 606-8501, JAPAN,

E-mail: syuji@acs.i.kyoto-u.ac.jp 1 Introduction

The diffusion coefficient defined via conventional mean square displacements is not sufficient for describing non-Gaussian statistics of velocity caused by ballistic motion in diffusive processes. In order to remedy this deficiency, we introduce large-deviation statistics also known as thermo-dynamical formalism. 2 Large-deviation statistics of velocity in diffusive dynamics

Let us briefly describe large-deviation statistics following the series of studies by Fujisaka and Inoue [1]. Consider a stationary time series of velocity . The average over time interval is given by this formula, which distributes when

is finite. When is much larger than the correlation time of , the distribution of coarse-grained is assumed to be an exponential form . Here we can as When is introduce the fluctuation spectrum

comparable to the correlation time, correlation cannot be ignored, so non-exponential or non-extensive statistics will be a problem, but here we do not discuss this point further.

be a real parameter. We introduce the generating function of by this Let

definition. We can here also assume the

as exponential distribution and introduce a characteristic function

The Legendre transform holds between fluctuation spectrum

and characteristic function , which is obtained from saddle-point calculations:

In this transform a derivative

appears, and it is a weighted average of , so we find that is a kind of weight index. We can also introduce susceptibility as a weighted variance. These statistical structure functions

constitute the framework of statistical thermodynamics of temporal

fluctuation, which characterize static properties of chaotic dynamics.

© 2005, S. Miyazaki

Diffusion Fundamentals 2 (2005) 31.1 - 31.2

1

3 Conclusion

We deal with extracting the non-Gaussian characteristics of the phenomenon of diffusion. Then, we refer to the mapping system in which Klages and Dorfman discovered complex dependence of diffusion coefficients on the parameter [2]. A graph plotting the weighted

of diffusive particles against parameter is numerically obtained. mean velocity

corresponding to the velocity of the ballistic This demonstrates that the graph

trajectory has a structure resembling the devil’s staircase. It is found that the unstable periodic orbit corresponding to the ballistic motion with the largest velocity in this system changes in a complex manner depending on the value of parameter . This seems to be

have complex structures. one of the factors explaining why the graphs of

Fig.1: Parameter dependence of ballistic velocity

References

[1] H. Fujisaka and M. Inoue, Prog. Theor. Phys. 77, 1334 (1987);

Phys. Rev. A 39, 1376 (1989).

[2] R. Klages and J. R. Dorfman, Phys. Rev. Lett. 74, 387 (1995).

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