Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed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

A peer-reviewed article of this Preprint also exists.

Gluzman, S. Nonlinear Approximations to Critical and Relaxation Processes. Axioms 2020, 9, 126. Gluzman, S. Nonlinear Approximations to Critical and Relaxation Processes. Axioms 2020, 9, 126.


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.


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


Physical Sciences, Mathematical Physics

Comments (1)

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

We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.

Leave a public comment
Send a private comment to the author(s)
* All users must log in before leaving a comment
Views 0
Downloads 0
Comments 1
Metrics 0

Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.