On generalized scaling laws with continuously varying exponents

TL;DR

This paper introduces a framework for understanding generalized scaling laws with continuously varying exponents, unifying power-law and logarithmic scaling through local scale invariance.

Contribution

It proposes a novel concept of local scaling invariance with variable exponents, deriving a set of differential equations that encompass both power-law and logarithmic scaling forms.

Findings

Logarithmic scaling emerges as a special case of the proposed framework.

Solutions include mixed scaling forms combining power-law and logarithmic behaviors.

The approach provides a unified explanation for different scaling phenomena in physical systems.

Abstract

Many physical systems share the property of scale invariance. Most of them show ordinary power-law scaling, where quantities can be expressed as a leading power law times a scaling function which depends on scaling-invariant ratios of the parameters. However, some systems do not obey power-law scaling, instead there is numerical evidence for a logarithmic scaling form, in which the scaling function depends on ratios of the logarithms of the parameters. Based on previous ideas by C. Tang we propose that this type of logarithmic scaling can be explained by a concept of local scaling invariance with continuously varying exponents. The functional dependence of the exponents is constrained by a homomorphism, which can be expressed as a set of partial differential equations. Solving these equations we obtain logarithmic scaling as a special case. The other solutions lead to scaling forms where logarithmic and power-law scaling are mixed.