The fractal geometry of growth: fluctuation-dissipation theorem and hidden symmetry

TL;DR

This paper explores the fractal geometry of crystal growth interfaces, linking fluctuations, responses, and hidden symmetries to the universality of growth exponents in KPZ-type models.

Contribution

It introduces a novel connection between fractal interface structures, fluctuation-response relations, and hidden symmetries in growth dynamics.

Findings

Fractal dimension relates to KPZ growth exponents.

Identification of a hidden symmetry in growth universality.

Fluctuation-response relations are tied to fractal geometry.

Abstract

Growth in crystals can be { usually } described by field equations such as the Kardar-Parisi-Zhang (KPZ) equation. While the crystalline structure can be characterized by Euclidean geometry with its peculiar symmetries, the growth dynamics creates a fractal structure at the interface of a crystal and its growth medium, which in turn determines the growth. Recent work (The KPZ exponents for the 2+ 1 dimensions, MS Gomes-Filho, ALA Penna, FA Oliveira; \textit{Results in Physics}, 104435 (2021)) associated the fractal dimension of the interface with the growth exponents for KPZ, and provides explicit values for them. In this work we discuss how the fluctuations and the responses to it are associated with this fractal geometry and the new hidden symmetry associated with the universality of the exponents.