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Article

Nonlinear Approximations to Critical and Relaxation Processes

Materialica+ Research Group, Bathurst St. 3000, Apt. 606, Toronto, ON M6B 3B4, Canada
Axioms 2020, 9(4), 126; https://doi.org/10.3390/axioms9040126
Received: 5 September 2020 / Revised: 21 October 2020 / Accepted: 22 October 2020 / Published: 28 October 2020
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration. View Full-Text
Keywords: critical index; relaxation time; time-translation invariance breaking and restoration; market crash; COVID-19; Gompertz approximants critical index; relaxation time; time-translation invariance breaking and restoration; market crash; COVID-19; Gompertz approximants
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MDPI and ACS Style

Gluzman, S. Nonlinear Approximations to Critical and Relaxation Processes. Axioms 2020, 9, 126. https://doi.org/10.3390/axioms9040126

AMA Style

Gluzman S. Nonlinear Approximations to Critical and Relaxation Processes. Axioms. 2020; 9(4):126. https://doi.org/10.3390/axioms9040126

Chicago/Turabian Style

Gluzman, Simon. 2020. "Nonlinear Approximations to Critical and Relaxation Processes" Axioms 9, no. 4: 126. https://doi.org/10.3390/axioms9040126

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