Symmetry in Self-Similarity in Space and Time—Short Time Transients and Power-Law Spatial Asymptotes

Ken Sekimoto,Takahiko Fujita

doi:10.3390/sym11121489

Abstract

The self-similarity in space and time (hereafter self-similarity), either deterministic or statistical, is characterized by similarity exponents and a function of scaled variable, called the scaling function. In the present paper, we address mainly the self-similarity in the limit of early stage, as opposed to the latter one, and also consider the scaling functions that decay or grow algebraically, as opposed to the rapidly decaying functions such as Gaussian or error function. In particular, in the case of simple diffusion, our symmetry analysis shows a mathematical mechanism by which the rapidly decaying scaling functions are generated by other polynomial scaling functions. While the former is adapted to the self-similarity in the late-stage processes, the latter is adapted to the early stages. This paper sheds some light on the internal structure of the family of self-similarities generated by a simple diffusion equation. Then, we present an example of self-similarity for the late stage whose scaling function has power-law tail, and also several cases of self-similarity for the early stages. These examples show the utility of self-similarity to a wider range of phenomena other than the late stage behaviors with rapidly decaying scaling functions.

Highlights

When the evolution of state shows some scaling behavior involving both space and time without fixed characteristic length or time, it is said to have self-similarity in space and time [1]
Self-similarity is a symmetry of the time-dependent fields which remain invariant under certain scale transformations of space and time
The basic reason for this is the fact that the group velocity in the parabolic or hyperbolic evolution equation is vanishing or finite at large length scale. This implies that the self-similarity with algebraic asymptotes of the scaling function is reduced to static algebraic tails in the far field

Summary

Introduction

When the evolution of state shows some scaling behavior involving both space and time without fixed characteristic length or time, it is said to have self-similarity in space and time Most studies on the self-similarity in physics have focused on the late-stage evolution, i.e., well after the initial transient, and at the same time focused on the case where the scaling function. The basic reason for this is the fact that the group velocity in the parabolic or hyperbolic evolution equation is vanishing or finite at large length scale This implies that the self-similarity with algebraic asymptotes of the scaling function is reduced to static (with steady flux) algebraic tails in the far field. The other examples are the short-time self-similarity of 1D diffusion from parabolic or cusp-like initial field configuration and the short-time self-similarity of 1D linearized capillary-driven thin-film equation Through these studies, we shed light on the family of self-similarity behaviors and the relationship among them. Symmetry Properties in the Family of Scaling Functions for the 1D Diffusion Process

Scaling Functions

Symmetry in the Family of Scaling Functions

Self-Similarity in Space and Time with Scaling Function Having Algebraic Tail

Late-Stage Self-Similarity in Space and Time of Gel Network

Self-Similarity in Space and Time in the Short-Time Limit of 1D Diffusion

Conclusions