Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 49. no. 4. P. 220-230. ISSN 2079-6641

PHYSICS
https://doi.org/10.26117/2079-6641-2024-49-4-220-230
Research Article
Full text in Russian
MSC 60K50

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Anomalous Diffusion with Memory in Criticality Theory

B. M. Shevtsov^{\ast}, O. V. Sheremetyeva

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS 684034, Kamchatka Krai, Paratunka village, Mirnaya str., 7, Russia

Abstract. 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.

Key words: hereditarian anomalous diffusion, criticality theory, random processes, fractional derivatives, critical indices, resonances, charge renormalization, phase transitions, anomalous phenomena.

Received: 09.11.2024; Revised: 18.11.2024; Accepted: 26.11.2024; First online: 28.11.2024

For citation. Shevtsov B. M., Sheremetyeva O. V. Anomalous diffusion with memory in criticality theory. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 49: 4, 220-230. EDN: PVYICK. https://doi.org/10.26117/2079-6641-2024-49-4-220-230.

Funding. The work was supported by IKIR FEB RAS State Task (subject registration No. 124012300245-2)

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: bshev@ikir.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Shevtsov B. M., Sheremetyeva O. V., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)

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Information about the authors

Shevtsov Boris Mikhailovich – D. Sci. (Phys. & Math.), Professor, Major Researcher, Electromagnetic Radiation Laboratory, IKIR FEB RAS, Paratunka, Kamchatka, Russia, ORCID 0000-0003-0625-0361.

Sheremetyeva Olga Vladimirovna – Ph. D. (Tech.), Research Scientist, Laboratory of Physical Process Modeling, IKIR FEB RAS, Paratunka, Kamchatka, Russia, ORCID 0000-0001-9417-9731.

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