Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Nonlinear Approximations to Critical and Relaxation Processes

Version 1 : Received: 5 September 2020 / Approved: 6 September 2020 / Online: 6 September 2020 (15:18:36 CEST)
Version 2 : Received: 10 September 2020 / Approved: 11 September 2020 / Online: 11 September 2020 (08:44:02 CEST)

How to cite: Gluzman, S. Nonlinear Approximations to Critical and Relaxation Processes. Preprints 2020, 2020090141 (doi: 10.20944/preprints202009.0141.v2). Gluzman, S. Nonlinear Approximations to Critical and Relaxation Processes. Preprints 2020, 2020090141 (doi: 10.20944/preprints202009.0141.v2).

Abstract

We discuss methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D and 3D channels are brought up. Power series for the permeability derived for small values of amplitude are employed to 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. The technique developed for critical phenomena is applied then for relaxation phenomena. The concept of time-translation invariance is discussed, its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the timetranslation symmetry complete (or partial) restoration. We estimate typical time extent, amplitude and direction for such restorative process. The new technique is based on explicit introduction of origin in time. After some transformations we come to the exponential and generalized, exponential-type solution with explicit finite time scale, which was only implicit in initial parametrization with polynomial approximation. The concept of crash as a relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is put forward. COVID-19 related mini-crash in the time series for Shanghai Composite is discussed as an illustration.

Subject Areas

nonlinearity; approximants; critical phenomena; permeability; relaxation phenomena; time series; crashes

Comments (1)

Comment 1
Received: 11 September 2020
Commenter: Simon Gluzman
Commenter's Conflict of Interests: Author
Comment: Improvements to the text are made.
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