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Front propagation in reactive systems with anomalous diffusion. (English) Zbl 1058.80004
Summary: We study front propagation of reactive fields in systems whose diffusive behavior is anomalous (both superdiffusive and subdiffusive). The features of the front propagation depend, not only on the scaling exponent $\nu \left(〈x{\left(t\right)}^{2}〉\sim {t}^{2\nu }\right)$, but also on the detailed shape of the probability distribution of the diffusive process. From the analysis of different systems we have three possible behavior of front propagation: the usual (Fisher-Kolmogorov like) scenario, i.e., the field has a spatial exponential tail moving with constant speed, ${v}_{f}$, and thickness, $\lambda$; the field has a spatial exponential tail but ${v}_{f}$ and $\lambda$ change in time (as a power law); and finally the field has a spatial power law tail and ${v}_{f}$ increases exponentially in time. A linear analysis of the front tail is in quantitative agreement with the numerical simulations. It is remarkable the fact that anomalous diffusion is neither necessary nor sufficient condition for the linear front propagation. Moreover, if the probability distribution of the transport process follows the scaling relation given by the Flory argument, the front propagation is standard (Fisher-Kolmogorov like) even in presence of super (or sub) diffusion.
##### MSC:
 80A30 Chemical kinetics (thermodynamic aspects) 82C70 Transport processes (time-dependent statistical mechanics)
##### Keywords:
scaling exponent; probability distribution shape