Brownian motion fractional dynamics

After ruling out slow modulation as a possible approach to complexity, we are left with the search for a more satisfactory approach to complexity that accounts for the renewal BQD properties. Is it possible to propose a more exhaustive approach to complexity, which explains both non-Poisson statistics and renewal at the same time We attempt at realizing this ambitious task in Section XVII. In Section XVII.A we show that a non-Ohmic bath can regarded as a source of memory and cooperation. It can be used for a dynamic approach to Fractional Brownian Motion, which, is, however, a theory without critical events. In Section XVIII.B we show, however, that the recursion process is renewal and fits the requests emerging from the statistical analysis of real data afforded by the researchers in the BQD held. In Section XVII.C we explain why this model might afford an exhaustive approach to complexity. 

At very low shear rates (i.e., flow velocities), particles in a chemically stable suspension approximately follow the layers of constant velocities, as indicated in Fig. 2. But at higher shear rates hydro-dynamic forces drive particles out of layers of constant velocity. The competition between hydrodynamic forces that distort the microstructure of the suspension and drive particles together, and the Brownian motion and repulsive interparticle forces keeping particles apart, leads to a shear dependency of the viscosity of suspensions. These effects depend on the effective volume fraction of... 

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. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

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. From these analytic results we conclude that Xj is analogous to fractional Brownian motion. The analogy is complete if we set a = // 1/2 so that the... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

Equation (252) represents the generalization of the conventional result based on the Ornstein-Uhlenbeck (inertia-corrected Einstein) theory of the Brownian motion [87] to fractional dynamics. By way of illustration, we show in Fig. 20... 

Here y = x/r[ = =C, /2/IkT is chosen as the inertial effects parameter (y = /2/y is effectively the inverse square root of the parameter y used earlier in Section I). Noting e initial condition, Eq. (134), all the < (()) in Eq. (136) will vanish with the exception of n = 0. Furthermore, Eq. (136) is an example of hoyv, using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics. The normalized complex susceptibility /(m) = x ( ) z"( ) is given by linear response theory as... 

Equation (145) represents the generalization of the a.v.c.f. of the Ornstein Uhlenbeck [21] (inertia-corrected Einstein) theory of the Brownian motion to fractional dynamics. The long-time tail due to the asymptotic (t >> t) t -like dependence [72] of the ((j)(O)(j)(t))o is apparent, as is the stretched exponential behavior at short times (t t). Eor a > 1, ((j)(O)(t)(f))o exhibits oscillations (see Eig. 14) which is consistent with the large excess absorption occurring at high frequencies. 

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