Fibonacci family of dynamical universality classes

TL;DR

This paper introduces an infinite family of non-equilibrium universality classes with dynamical exponents related to Fibonacci numbers, expanding understanding beyond known diffusive and KPZ classes.

Contribution

It reveals a new hierarchy of universality classes characterized by Fibonacci ratios of dynamical exponents in non-equilibrium systems.

Findings

Dynamical exponents are ratios of Fibonacci numbers.

Universal scaling functions are asymmetric Lévy distributions.

All modes have exponents linked to the Golden Mean when no specific modes are present.

Abstract

Universality is a well-established central concept of equilibrium physics. However, in systems far away from equilibrium a deeper understanding of its underlying principles is still lacking. Up to now, a few classes have been identified. Besides the diffusive universality class with dynamical exponent $z=2$ another prominent example is the superdiffusive Kardar-Parisi-Zhang (KPZ) class with $z=3/2$. It appears e.g. in low-dimensional dynamical phenomena far from thermal equilibrium which exhibit some conservation law. Here we show that both classes are only part of an infinite discrete family of non-equilibrium universality classes. Remarkably their dynamical exponents $z_\alpha$ are given by ratios of neighbouring Fibonacci numbers, starting with either $z_1=3/2$ (if a KPZ mode exist) or $z_1=2$ (if a diffusive mode is present). If neither a diffusive nor a KPZ mode are present, all dynamical modes have the Golden Mean $z=(1+\sqrt{5})/2$ as dynamical exponent. The universal scaling functions of these Fibonacci modes are asymmetric L\'evy distributions which are completely fixed by the macroscopic current-density relation and compressibility matrix of the system and hence accessible to experimental measurement.