Fractals Diffusion And Relaxation In Disordered Complex by Yuri P. Kalmykov, William T. Coffey, Stuart A. Rice

By Yuri P. Kalmykov, William T. Coffey, Stuart A. Rice

Fractals, Diffusion, and leisure in Disordered complicated structures is a unique guest-edited, two-part quantity of Advances in Chemical Physics that keeps to document fresh advances with major, up to date chapters by means of across the world well-known researchers.

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The best physical model is the simplest one that can ‘‘explain’’ all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion.

By solving Eq. (52) and placing the solution into Eq. (51), we obtain ð 1 X qs0 ðx; tÞ q q2 s0 ðx; tÞ ¼ W2 am jam j2 dt0 exp½ÀÃm ðt À t0 ފ qt qx qx2 m6¼0 t ð53Þ 0 From now on we shall focus on the reduced density matrix s0 ðx; tÞ and for the sake of simplicity we omit the subscript zero. Using Eq. (48), we can rewrite Eq. (53) in the form ðt qsðx; tÞ q2 sðx; t0 Þ ¼ dt0 hxðtÞxðt0 Þi qt qx2 ð54Þ 0 where again the brackets denote an average over an ensemble of realizations of the statistical fluctuations.

35] and involved observations of healthy individuals, as well as of subjects having certain diseases that affect gait and also the elderly. Additional experiments and analyses that both verified and extended the earlier results were subsequently done by West and Griffin [11,36]. Human gait is a complex process, since the locomotor system synthesizes inputs from the motor cortex, the basal ganglia, and the cerebellum, as well as fractal physiology, complexity, and the fractional calculus 19 feedback from vestibular, visual, and proprioceptive sources.

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