Diffusion and Front Propagation

To characterize diffusion on these structures, we calculate the MSD using the CTRW formalism with the waiting time PDFs (6.29), (6.31), (6.34), (6.36), and the dispersal PDF (6.27). In Laplace-Fourier space, the MSD reads 

Introducing (6.34) into (6.37), we find anomalous diffusion in the limit f - oo for case (c)  

We plot in Fig. 6.5 the dimensionless front velocity vt/u vs the reaction rate r on a log-log scale. The front velocity increases with r. For the cases / = a and I = 2a, the slope is very similar, but for / = oo it is steeper. In all cases the front velocity increases as a power law of r, straight line in a log-log plot, for small and moderate values of r and saturates to 1 for larger values, the slope in the log-log plot tends to 0. This behavior is due to the fact that an increase of the reaction rate r leads to an increase of the front velocity. However, the front cannot travel faster than the jump velocity of the particles if all of them jump in the backbone direction, i.e., V ajx. For I = a and / = 2a the transport is diffusive, and the diffusion coefficient is properly defined. If this transport is combined with a KPP reaction, a Fisher velocity is expected, i.e., in both cases v fr. Computing numerically the slope from a linear fit in Fig. 6.5 we obtain and for / = a and I = 2a, respectively. The case / oo is quite different, because the transport is anomalous. Equation (5.36) with y = 1/2 yields v while the linear fit of the numerical results yields Numerical and analytical results are in good agreement. 

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