Reaction-Diffusion Fronts on Fractals

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, 

Second, the PDF P(r, t) must agree with the scaling form (6.4). Third, the equation has to recover the form of the standard diffusion equation for = 2, df = I, and r/min = 1. We have shown that only the CMF equation (6.16) meets all three requirements. As expected, this result remains true for the description of front propagation on fractals. 

Following the argument by Bunde and Drager [62], front propagation is well defined in the chemical distance space. The front has to propagate with constant velocity, i.e., the front position in the chemical distance space behaves like / t. Euclidean and chemical distances are related by (6.1). The front position in real 

Since d 1, the front is always decelerated. Clearly, the models that do not explicitly consider d cannot fulfill the scaling law (6.18) for the front velocity on a fractal only the CMF equation can satisfy (6.18). To obtain the CMF equation with reaction, we combine the constitutive equation for the flux (6.15) with the conservation equation for the particle density p  

The reaction-CMF equation predicts a front velocity with the appropriate scaling (6.18). 

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