Anomalous diffusion field model

In a Hamiltonian system having mean field interaction, referred to as the HMF model, we have investigated two features that must reflect self-similar hierarchy of phase space power-type distribution and anomalous diffusion. They have been reported in the same model for one type of initial condition, and we used a different type of initial condition to check generality. 

Besides the above simplified models, more interesting is the understanding of the anomalous diffusion in incompressible velocity fields or deterministic maps. In this direction, Avellaneda, Majda, and Vergassola [22, 23] obtained a very important and general result about the character of the asymptotic diffusion in an incompressible velocity field u(x). If the molecular diffusivity D is nonzero and the infrared contribution to the velocity field are weak enough, namely,... 

Anomalous diffusion of a continuous concentration field can be modelled in terms of fractional differential equations. To see how they arise we can write Eq. (2.9) for normal diffusion in terms of the spatial Fourier transform of the concentration field C(k, t). This can be easily done under periodic boundary conditions or in unbounded space as... 

In polymers, it is always observed that a packet of carriers spreads faster with time than predicted by Eq. (30). Thus, the spatial variance of the packet yields an apparent diffusivily that exceeds the zero-field diffusivity predicted by the Einstein relationship. Further, the pholocurrent transients frequently do not show a region in which the photocurrent is independent of time. As a result, inflection points, indicative of the arrival of the carrier packet at an electrode, can only be observed by plotting the time variance of the photocurrent in double logarithmic representation. The explanation of this behavior, as originally proposed by Scher and Lax (1972, 1973) and Scher and Montroll (1975), is that the carrier mean velocity decreases continuously and the packet spreads anomalously with time, if the time required to establish dynamic equilibrium exceeds the average transit time. Under these conditions, the transport is described as dispersive. There have been many models proposed to describe dispersive transport. Of these, the formalism of Scher and Montroll has been the most widely used. 

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7], This diffusion equation allows one to include explicitly in Frohlich s model as generalized to fractional dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion (0rotation about fixed axis in a potential Vo(< >)- We suppose that a uniform field Fi (having been applied to the assembly of dipoles at a time t = oo so that equilibrium conditions prevail by the time t = 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., pFj linear response condition). 

So far we have discussed random walks with a finite mean waiting time and a finite variance of the jump length. These models lead to the classical parabolic scaling X x/e, t t/e. The governing macroscopic equation for the density p becomes the standard diffusion equation. Let us now consider two cases for which the scaling is anomalous and the mean-field equations for p are fractional diffusion equations. 

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