Relaxation Transform for Iterations of the Sinusoidal Map and Its Physical Interpretation as a Memory-Based Process
Authors/Creators
Description
This research presents a novel mathematical framework for modeling relaxation phenomena in complex systems with memory. We develop a relaxation transformation that continuously interpolates discrete iterations of a sinusoidal mapping, providing a bridge between classical iterative dynamics and continuous relaxation processes.
The key innovation lies in interpreting the iteration parameter as a measure of accumulated system "experience" rather than physical time. By introducing a memory function that maps physical time to this experience parameter, the model can describe diverse non-exponential relaxation behaviors commonly observed in glassy materials, biological tissues, and geological media.
The approach offers a mathematically elegant alternative to traditional differential equation models, combining computational efficiency with clear physical interpretation. The framework shows particular promise for systems with hierarchical structure, aging phenomena, and multiple relaxation time scales.
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vozmishchev_relaxation_eng.pdf
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Dates
- Created
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2025-12-29/2026-01-14
References
- Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach // Physics Reports. 2000. Vol. 339, No. 1. P. 1-77.
- Debenedetti P.G., Stillinger F.H. Supercooled liquids and the glass transition // Nature. 2001. Vol. 410, No. 6825. P. 259-267.
- Götze W. Complex dynamics of glass-forming liquids: A mode-coupling theory. Ox- ford University Press, 2009.
- Zaslavsky G.M. Hamiltonian chaos and fractional dynamics. Oxford University Press, 2005.
- Podlubny I. Fractional differential equations. Academic Press, 1999.
- Ngai K.L. Relaxation and diffusion in complex systems. Springer, 2011.
- Kuczma M., Choczewski B., Ger R. Iterative functional equations. Cambridge Uni- versity Press, 1990.
- Volterra V. Theory of functionals and of integral and integro-differential equations. Dover Publications, 2005.
- Angell C.A. Formation of glasses from liquids and biopolymers // Science. 1995. Vol. 267, No. 5206. P. 1924-1935.
- Viterbi A.J. Principles of coherent communication. McGraw-Hill, 1966.
- Bouchaud J.P. Weak ergodicity breaking and aging in disordered systems // Journal de Physique I. 1992. Vol. 2, No. 9. P. 1705-1713.
- Hilfer R. (Ed.) Applications of fractional calculus in physics. World Scientific, 2000.
- Feigenbaum M.J. Quantitative universality for a class of nonlinear transformations // Journal of Statistical Physics. 1978. Vol. 19, No. 1. P. 25-52.
- Samorodnitsky K.A., Zaslavsky G.M. Nonlinear Physics: From Pendulum to Turbu- lence and Chaos. Moscow: Nauka, 2014. (In Russian)
- Schuster H.G., Just W. Deterministic chaos: an introduction. Wiley-VCH, 2005.