Critical Indices as Limits of Control Functions

TL;DR

The paper introduces a novel self-similar approximation method using control functions to sum divergent series and relate critical indices to control function limits, demonstrating accurate results for complex problems.

Contribution

It presents a new approximation technique linking critical phenomena with control theory, enabling precise calculation of critical indices from limited series data.

Findings

Method accurately sums divergent series with few terms.

Critical indices are directly related to control function limits.

Results agree well with numerical data.

Abstract

A variant of self-similar approximation theory is suggested, permitting an easy and accurate summation of divergent series consisting of only a few terms. The method is based on a power-law algebraic transformation, whose powers play the role of control functions governing the fastest convergence of the renormalized series. A striking relation between the theory of critical phenomena and optimal control theory is discovered: The critical indices are found to be directly related to limits of control functions at critical points. The method is applied to calculating the critical indices for several difficult problems. The results are in very good agreement with accurate numerical data.