Scaling Analysis and Systematic Extraction of Macroscopic Structures in Fluctuating Systems of Arbitrary Dimensions

TL;DR

This paper provides a comprehensive analytical framework for extracting macroscopic structures from complex fluctuating systems across arbitrary dimensions, enhancing understanding of their scaling behaviors and correlations.

Contribution

It introduces an exact analytical method for generalized detrended fluctuation analysis applicable to systems of any dimension, linking variance functions with correlation functions.

Findings

Derived explicit expressions for macroscopic structures.

Established exact relations between variance and correlation functions.

Applied the framework to interfacial superroughening phenomena.

Abstract

Many fluctuating systems consist of macroscopic structures in addition to noisy signals. Thus, for this class of fluctuating systems, the scaling behaviors are very complicated. Such phenomena are quite commonly observed in Nature, ranging from physics, chemistry, geophysics, even to molecular biology and physiology. In this paper, we take an extensive analytical study on the ``generalized detrended fluctuation analysis'' method. For continuous fluctuating systems in arbitrary dimensions, we not only derive the explicit and exact expression of macroscopic structures, but also obtain the exact relations between the detrended variance functions and the correlation function. Besides, we undertake a general scaling analysis, applicable for this class of fluctuating systems in any dimensions. Finally, as an application, we discuss some important examples in interfacial superroughening phenomena.