Nonlinear Approximations to Critical and Relaxation Processes

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.

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Nonlinear Approximations to Critical and Relaxation Processes

10.20944/preprints202009.0141.v2 ◽

2020 ◽

Cited By ~ 1

Author(s):

Simon Gluzman

Keyword(s):

Stokes Flow ◽

Translation Invariance ◽

Relaxation Processes ◽

Type Solution ◽

Partial Restoration ◽

Nonlinear Approximations ◽

Time Translation ◽

Typical Time ◽

Flow Through ◽

2D And 3D

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.

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Nonlinear Approximations to Critical and Relaxation Processes

10.20944/preprints202009.0141.v1 ◽

2020 ◽

Author(s):

Simon Gluzman

Keyword(s):

Polynomial Regression ◽

Power Laws ◽

Translation Invariance ◽

Relaxation Processes ◽

Partial Restoration ◽

Nonlinear Approximations ◽

Time Translation ◽

Concrete Form ◽

Translation Symmetry ◽

2D And 3D

We discuss methods for calculation of critical indices and amplitudes from the perturbative expansions. They are demonstrated for the Stokes flow through 2D and 3D channels enclosed by two wavy walls. Efficient formulas for the permeability are derived in the form of series for small values of amplitude. Various power-laws are found in the regime of large amplitudes, based only on expansions at small amplitudes. Lubrication approximation is shown to break down, but accurate formulas for the effective permeability 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 time-translation 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 phenomenon, consisting of time-translation invariance breaking and restoration, is put forward. %Concrete form of symmetry breaking/restoration is suggested, using polynomial regression transformed into exponential and Gompertz approximants. COVID-19 related mini-crash in the time series for Shanghai Composite is discussed as an illustration.

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