Anomalous diffusion with memory in criticality theory

The application of the hereditarian anomalous diffusion in the theory of critical phenomena is considered. The process modes are investigated depending on the fractional parameters of the derivatives of the initial diffusion equation. The critical indices determining the changes of the process modes are found from the conditions of circulation to infinity of the statistical moments of the power-law space-time distribution of the diffusion process. The changes of process modes depending on the critical indices can be considered as a sequence of phase transitions. The relationship of fractional derivatives and critical indices of the process with its fractal dimension is shown, which determines the evolution of moments and the associated classification of types of hereditarian anomalous diffusion. It is concluded that the features of anomalous phenomena are due to spatiotemporal dispersion and resonant effects determined by the properties of power-law spatiotemporal distributions of the diffusion process. This is connected with the structural restructuring of the process and the renormalization of its sources. The changes in the modes of the diffusion process, in which fractional diffusion turns into advection or wave process, are discussed. A generalization of the hereditarian anomalous diffusion is proposed for the case of power-law nonstationarity and spatial heterogeneity of the process. The presented fractional diffusion model can be used to describe the modes of activation and fading of deformation processes accompanied by the generation of acoustic and electromagnetic emissions.