The Fractal Geometry of Growth: Fluctuation–Dissipation Theorem and Hidden Symmetry

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 by Gomes-Filho et al. (Results in Physics, 104,435 (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.

Restoring the Fluctuation–Dissipation Theorem in Kardar–Parisi–Zhang Universality Class through a New Emergent Fractal Dimension

“…The attempt to obtain exact height fluctuations for the stationary KPZ equations, as well as for most of KPZ growth physics in dimensions, is still in its beginning. These theoretical methods will benefit from the fixed points obtained by precise KPZ exponents, and from the idea of a fractal geometry that must be associated with them [ 31 ]. We also expect that new methods would confirm our results.…”

“…Our starting point is to try to understand the fluctuation–dissipation theorem (FDT) in KPZ growth systems. Since there is a long history of violation of the FDT in some complex systems such as structural glasses [ 18 , 19 , 20 , 21 ], proteins [ 22 ], mesoscopic radioactive heat transfer [ 23 ], and ballistic diffusion [ 24 , 25 , 26 , 27 , 28 ], it has been suggested that, for KPZ, the FDT should always fail at dimension [ 6 , 8 , 29 , 30 , 31 ] (for a review, see [ 32 ]).…”

“…More recently, we demonstrated the existence of an FDT for KPZ growth in 1 + 1 dimensions [ 30 ], leading us to find the corresponding KPZ exponents for dimensions analytically [ 31 ]. We explored the idea that the fractal dimension of the surface, denoted by , is connected to the KPZ exponents at the saturation of the growth process.…”

Editorial: The Fluctuation-Dissipation Theorem Today

“…This observation became an impediment to determining the KPZ exponents. Recently Anjos et al [28] proposed that the growth dynamics builds up an interface with a fractal dimension d f , which filters the original fluctuations given origin to new fluctuations which, by it turns, yields a new FDT in the fractal space. This allows a possible solution for the KPZ exponents [29].…”

The Multiscale Principle in Nature (Principium luxuriæ): Linking Multiscale Thermodynamics to Living and Non-Living Complex Systems

“…The multiscale properties of a system refer to the existence of multiple scales of organization within the system [1][2][3][4], usually during phase transition, where each scale could be characterized by different mathematical physics [5][6][7], biochemical [8,9], or biological [10][11][12][13][14][15] properties. A wide range of varied systems, such as cosmology [16], zoology [17], networks [18], cities [19], ecology [20], computational imaging [21], interface physics [22,23], geophysics [24], emergent processes [25], genetic [26], or complex systems [27] often exhibit hierarchical organization [28][29][30], where smaller-scale components or subsystems interact to create emergent behavior at larger scales. These emergent behaviors are typically described as a type of dynamic that cannot be described as the sum of its parts [31][32][33].…”