Subdiffusion equation

This model for subdiffusion in the external force field F(x) — — ( ) provides a basis for fractional evolution equations, starting from Langevin dynamics that is combined with long-tailed trapping events possessing a... 

In the special case where the site energies are random fluctuations, this is the Anderson model [20,21]. It is well known that Anderson used this model to prove that randomness makes a crystal become an insulating material. Anderson localization is subtly related to subdiffusion, and consequently this important phenomenon can be interpreted as a form of anomalous diffusion, in conflict with the Markov master equation that is frequently adopted as the generator of ordinary diffusion. It is therefore surprising that this is essentially the same Hamiltonian as that adopted by Zwanzig for his celebrated derivation of the van Hove and, hence, of the Pauli master equation. 

Let us consider now the case p < 2. In this case the third generalized diffusion equation, derived from a CTRW, is well established [43] as a paradigmatic case of subdiffusion. The correlation function of the... 

Let s consider the physical basis of the change of diffusion regime from slow (subdiffusive) up to rapid (superdiffusive). Anomalous (strange) transport processes are described by the general equation [8] ... 

Here V(<(), t) = pF(t) cos 4> is the potential arising from an external applied electric field F(f). Here, just as with the translational diffusion equation treated in Ref. 7, we consider subdiffusion, 0 < ct < 1 phenomena only. Here, the internal field effects are ignored, which means that the effects of long-range torques due to the interaction between the average moments and the Maxwell fields are not taken into account. Such effects may be discounted for dilute systems in first approximation. Thus, the results obtained here are relevant to situations where dipole-dipole interactions have been eliminated by extrapolation of data to infinite dilution. 

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). 

As shown above, the standard diffusion equation (2.1) has a fractional diffusion equation (2.59) as its analog in the subdiffusive case. As in the case of reaction-transport equation with inertia, see Sect. 2.2, the question arises how to combine reactions and subdiffusion in the activation-controlled regime. (For a discussion of the subdiffusion-limited case, which is outside the scope of this monograph as mentioned on page 34, see for example [491-493, 369, 391, 392, 389, 409, 410, 390, 411, 203, 187].) In some schemes, [188, 189, 186, 187], reactions terms are simply added to the fractional diffusion equation, in a manner similar to the ad hoc HRDEs (2.16), assuming at the outset that the effects of subdiffusion and reactions are separable as in the standard reaction-diffusion (2.11). However, it is easy to... 

The memory kernel in (2.59), recall that v] represents a nonlocal-in-time integral operator, is a clear indication that subdiffusive transport is non-Markovian. Incorporating kinetic terms into a non-Markovian transport equation requires great care and is best carried out at the mesoscopic level. We show in Sect. 3.4 how to proceed directly at the level of the mesoscopic balance equations for non-Markovian CTRWs. Here we pursue a different approach. As stated above, if all processes are Markovian, then contributions from different processes are indeed separable and simply additive. As is well known, processes often become Markovian if a sufficiently large and appropriate state space is chosen. For the case of reactions and subdiffusion, the goal of a Markovian description can be achieved by taking the age structure of the system explicitly into account as done by Vlad and Ross [460,461]. This approach is equivalent to Model B, see Sect. 3.4. 

Remark 2.4 In the derivation of the generalized reaction-diffusion equation (2.82) we do not explicitly refer to the particular form of the waiting time PDF. Equation (2.82) is valid for arbitrary waiting time PDFs < (t) and has much wider applicability than subdiffusive transport. 

The generalized reaction-diffusion equation (2.82) can be written in a form using fractional derivatives for subdiffusive transport, where the waiting PDF of species i is given in Laplace space by (2.52), (i) 1 —. In that case... 

In this section we show how to obtain subdiffusive transport by using the idea of inverse subordination [278, 371]. Assume that the density 0 obeys the Kolmogorov-Feller equation... 

In the previous section we presented some of the equations proposed in the literature for describing diffusion on fractal structures. These equations must meet three requirements to be considered valid. First, the MSD must display subdiffusion,... 

In this section, we use Model B, see Sect. 3.4.2, to explore the effects of subdiffusion on the Turing instability. We consider the two-variable generalized reaction-diffusion equation [484],... 

Equation (10.167) is the general condition for the occurrence of a Turing instability in a two-component system when one of the components subdiffuses. By choosing y = 1 in (10.167), we obtain the classical Turing condition, (10.30), for the case when both entities undergo normal diffusion ... 

We are interested in the effect of subdiffusive motion of the activator on Turing instabilities. We focus on this mechanism exclusively and consider the LE model in its original form without a substrate, i.e., a = 1 in (1.152) or (1.160). Equation (10.180) results in the following expression ... 

These equations describe subdiffusion-limited chemical reactions. As stated in Chap. 2, such reactions are outside the scope of this monograph. For results on fronts and Turing instabilities in systems with subdiffusion-controlled chemical reactions, see for example [490,240, 147, 148, 235, 318,191]. 

At 0 < p < 1 it is said about subdiffusive transport processes, at 1 < < 2 - about superdiffusive ones and p = 1 corresponds to classical (Gaussian) diffusion. In its turn, the exponent p is coimected with Hurst ejqtonent H by the equation [42] ... 

Equation 49 is subdiffusive with y = 1/2 in Eq. 1. This is due to the fact that the trajectory of a bead in a chain is spatially confined by its neighbours. In the limit t tr the chain diffuses as one entity and the bead s trajectory follows mainly the motion of the chain s center of mass. The chain s center of mass obeys... 

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