By William T Coffey; Yuri P Kalmykov; John Wiley & Sons.; Wiley InterScience (Online service)
Fractals, Diffusion and leisure in Disordered advanced platforms 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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Extra info for Fractals, Diffusion and Relaxation in Disordered Complex Systems, Part B
So let us consider the case where such moments do not exist and therefore are of no help in determining the scaling exponents. B. Dichotomous Fluctuations with Memory The time interval between steps were assumed to be a constant finite value in the simple random walk model. If, however, we explicitly take the limit where the time interval vanishes, then the discrete walk is replaced with a continuous rate. We begin our discussion of the scaling of statistical processes by considering one of the simplest stochastic rate equations and follow the development of Allegrini et al.
3. Late Time Behavior Now let us now consider the time asymptotic behavior of the exact solution. In the late time domain t ! 1, we have s ! 0, so examining the behavior of fractal physiology, complexity, and the fractional calculus Eq. (67) in this domain we obtain ~ x ðsÞ % Àð1 À bÞTW È ðsTÞ1Àb 2 " ðsTÞ1Àb 1À Àð1 À bÞ 41 # ð72Þ Note that as s ! 0 the leading term in this expansion diverges for b < 1, corresponding to the fact that there is no correlation time for this process. Inserting this expression for the Laplace transform of the correlation function into Eq.
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 !