Polymer Dynamics Through Group Invariance of SL(2R) - Type in a Fractal Paradigm

Constantin Placinta,Tudor Cristian Petrescu,Maricel Agop,Viorel-Puiu Paun,Decebal Vasincu,Vlad Ghizdovat,Stefan Andrei Irimiciuc

doi:10.37358/mp.20.1.5323

Abstract

We analyze polymer dynamics in a fractal paradigm. Then, it is shown that polymer dynamics in the form of Schrödinger - type regimes imply synchronization processes of the polymers� structural units, through joint invariant function of two simultaneous isomorphic groups of SL(2R) - type, as solutions of Stoka equations. In this context, period doubling, damped oscillations, self - modulation and chaotic regimes emerge as natural behaviors in the polymer dynamics. The present model can also be applied to a large class of materials, such as biomaterials, biocomposites and other advanced materials.

Highlights

To describe polymer dynamics in the fractal paradigm, but remaining faithful to the differentiable mathematical procedures, it is necessary to explicitly introduce scale resolutions, both in the expression of the variables and in the fundamental equations which govern polymer dynamics. This means that, instead of “working” with a single variable described by a strict non-differentiable function, it is possible to “work” only with approximations of these mathematical functions obtained by averaging them on different scale resolutions
Any variable purposed to describe polymer dynamics will perform as the limit of a family of mathematical functions, this being non – differentiable for null scale resolutions and differentiable otherwise [1, 2]
Period doubling, damping oscillations, self – modulation and chaotic regimes emerge as natural behaviors in the polymer dynamics. (Figures 1 a – l for α = ωt, tanh φ = 0.1 and Real [(z − u)/ v] ≡Amplitude at various scale resolutions, given by means of the maximum value of ω)

Summary

Introduction

To describe polymer dynamics in the fractal paradigm, but remaining faithful to the differentiable mathematical procedures, it is necessary to explicitly introduce scale resolutions, both in the expression of the variables and in the fundamental equations which govern polymer dynamics. In the present paper, considering the fractal paradigm as being functional [1-4], a non – differentiable model describing various polymer dynamics is proposed. 1.Mathematical Model The polymer is a set of entities (or structural units) that, through their interactions, relationships or dependencies form a unified whole.

Results

Conclusion