Nonlinear Approximations to Critical and Relaxation Processes

Abstract

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.

Optimized Factor Approximants and Critical Index

“…with the critical index S = 2.08, rather close to 2, as we hoped. Application of the DLog Padé technique [17,18], gives a much larger result of 2.75.…”

“…with c = 1, κ = 5 2 . Iterative perturbation theory gives an expansion for permeability in powers of [2,6,17,60], to be truncated to make it applicable. For b = 0.5, the permeability can be expanded as follows, K( ) 1 − 3.14963 2 + 4.08109 4 , as → 0 , (45) and an inverse of permeability is given as follows, K( ) −1 1 + 3.14963 2 + 5.83908 4 , as → 0 .…”

“…One can think about the critical index as a control parameter needed to stabilize the divergent series, and find it from the optimization conditions, written as a minimal difference condition, or minimal derivative condition [1,14,15]. In such fashion, the so-called root approximants [16] were applied to the critical index calculations [6,17,18]. Root approximants are more suited technically to apply the minimal difference condition than minimal derivative condition.…”

“…Besides, the minimal derivative condition preserves uniqueness of the parameters definition. Optimization conditions could be applied to the exponential and super-exponential approximants as well [17].…”

Continued Roots, Power Transform and Critical Properties

“…Yukalov in [7,8], by Kadanoff and Houghton in [9], by Stevenson [10], by M. Suzuki [11,12] and H. Kleinert [13]. More information can be found in our recent papers [6,14,15]. Feynman's inspirational ideas were documented in [16].…”