By Yuri P. Kalmykov, William T. Coffey, Stuart A. Rice
Fractals, Diffusion and leisure in Disordered advanced structures is a unique guest-edited, two-part quantity of Advances in Chemical Physics that keeps to record contemporary advances with major, up to date chapters by means of the world over well-known researchers.
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Additional resources for Advances in Chemical Physics, Vol.133, Part B. Fractals, Diffusion, and Relaxation (Wiley 2006)
22). However, whereas the spectrum of ﬂuctuations is ﬂat, since it is white noise, the spectrum of the system response is inverse power law. From these analytic results we conclude that Xj is analogous to fractional Brownian motion. The analogy is complete if we set a ¼ H À 1=2 so that the spectrum in Eq. 0 ð33Þ Taking the inverse Fourier transform of the exact expression in Eq. (31) yields the autocorrelation coefﬁcient  rk ¼ hXj Xjþk i Àð1 À aÞ 2aÀ1 k % ÀðaÞ hXj2 i Àð1:5 À HÞ 2HÀ2 k ÀðH À 0:5Þ ð34Þ as the lag time increases without limit k !
Simple Random Walks and Scaling We deﬁne the variable of interest as Xj, where j ¼ 0,1,2, . . indexes the time step, and in the simplest model a step is taken in each increment of time, which for convenience we set to one. The operator B lowers the index by one unit such that BXj ¼ XjÀ1 so that a simple random walk can be written ð1 À BÞXj ¼ xj ð11Þ where xj is þ1 or À1 and is selected according to some random process characterized by the probability density pðxÞ. The solution to this discrete equation is given by the position of the walker after N steps, the sum over the sequence of steps XðNÞ ¼ N X j¼1 xj ð12Þ fractal physiology, complexity, and the fractional calculus 29 and the total number of steps N can be interpreted as the total time t over which the walk unfolds, since we have set the time increment to one.
30) and substituting that expression into Eq. (29), we obtain 1 SðoÞ / ð31Þ ð2sino=2Þ2a fractal physiology, complexity, and the fractional calculus 33 for the spectrum of the fractional-differenced white noise process. In the lowfrequency limit we therefore obtain the inverse power-law spectrum SðoÞ / 1 o2a ð32Þ Thus, since the fractional-difference dynamics are linear, the system response is Gaussian, the same as the statistics for the white noise process on the right-hand side of Eq. (22). However, whereas the spectrum of ﬂuctuations is ﬂat, since it is white noise, the spectrum of the system response is inverse power law.