Abstract
“Odd” factor approximants of the special form suggested by Gluzman and Yukalov (J. Math. Chem. 2006, 39, 47) are amenable to optimization by power transformation and can be successfully applied to critical phenomena. The approach is based on the idea that the critical index by itself should be optimized through the parameters of power transform to be calculated from the minimal sensitivity (derivative) optimization condition. The critical index is a product of the algebraic self-similar renormalization which contributes to the expressions the set of control parameters typical to the algebraic self-similar renormalization, and of the power transform which corrects them even further. The parameter of power transformation is, in a nutshell, the multiplier connecting the critical exponent and the correction-to-scaling exponent. We mostly study the minimal model of critical phenomena based on expansions with only two coefficients and critical points. The optimization appears to bring quite accurate, uniquely defined results given by simple formulas. Many important cases of critical phenomena are covered by the simple formula. For the longer series, the optimization condition possesses multiple solutions, and additional constraints should be applied. In particular, we constrain the sought solution by requiring it to be the best in prediction of the coefficients not employed in its construction. In principle, the error/measure of such prediction can be optimized by itself, with respect to the parameter of power transform. Methods of calculation based on optimized power-transformed factors are applied and results presented for critical indices of several key models of conductivity and viscosity of random media, swelling of polymers, permeability in two-dimensional channels. Several quantum mechanical problems are discussed as well.
Highlights
Often, the limit-problems are characterized by power laws
We conclude that “odd” factor approximants of the special form represented by the Formula (11) are amenable to optimization by power transformation and can be successfully applied to the critical phenomena of various physical nature
The novelty of the current approach amounts to the idea that the critical index by itself should be optimized through the parameters of power transform calculated from the optimization condition
Summary
The limit-problems are characterized by power laws. Accurate analytical formulae for deterministic and random systems, such as composites, suspensions, and porous media, can be derived by employing approximants, when the low-concentration series are supplemented with information on the high-concentration regime near divergence points signifying the physical percolation effects [1,2]. Due to the self-similarity relation, one can look for the fixed point representing the sought sum in the analytical form, e.g., the power-transformed factor approximant can be considered as such a representation [3]. For such approximants, one can analytically express the critical index and amplitude, and optimize the critical index with respect to the parameters of power transform. Various self-similar techniques were applied to calculations with short series [5,6,18], but they all appear to be designed for providing lower and upper bounds for the index Such methods combine at least two separate methods and are applied through optimizing the expressions for critical amplitudes [2,5]. It appears to give more stable performance than all other methods based on finding the index as a control parameter from the critical amplitude optimization
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
