Fibonacci family of dynamical universality classes

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 that exhibit some conservation law. Here we show that both classes are only part of an infinite discrete family of nonequilibrium universality classes. Remarkably, their dynamical exponents z_α are given by ratios of neighboring Fibonacci numbers, starting with either z₁ = 3/2 (if a KPZ mode exist) or z₁ = 2 (if a diffusive mode is present). If neither a diffusive nor a KPZ mode is present, all dynamical modes have the Golden Mean z=(1+5)/2 as dynamical exponent. The universal scaling functions of these Fibonacci modes are asymmetric Lévy distributions that are completely fixed by the macroscopic current density relation and compressibility matrix of the system and hence accessible to experimental measurement.The Golden Mean, φ=1/2+5/2≈1.61803..., also called Divine Proportion, has been an inspiring number for many centuries. It is widespread in nature; i.e., arrangements of petals of the flowers and seeds in the sunflower follow the golden rule (1). Being considered an ideal proportion, the Golden Mean appears in famous architectural ensembles such as the Parthenon in Greece, the Giza Great Pyramids in Egypt, or Notre Dame de Paris in France. Ideal proportions of the human body follow the Golden Rule.Mathematically, the beauty of the Golden Mean number is expressed in its continued fraction representation: All of the coefficients in the representation are equal to unity,φ=1+11+ 11+ 11+⋱.[1]Systematic truncation of the above continued fraction gives the so-called Kepler ratios, 1/1,2/1,3/2,5/3,8/5,..., which approximate the Golden Mean. Subsets of denominators (or numerators) of the Kepler ratios form the celebrated Fibonacci numbers, Fi=1,1,2,3,5,8,.., such that Kepler ratios are ratios of two neighboring Fibonacci numbers. As well as the Golden Mean, Fibonacci ratios and Fibonacci numbers are widespread in nature (1).The occurrence of the Golden Mean is not only interesting for aesthetic reasons but often indicates the existence of some fundamental underlying structure or symmetry. Here we demonstrate that the Divine Proportion as well as all of the truncations (Kepler ratios) of the continued fraction (Eq. 1) appear as universal numbers, namely, the dynamical exponents, in low-dimensional dynamical phenomena far from thermal equilibrium. The two well-known paradigmatic universality classes, Gaussian diffusion with dynamical exponent z = 2 (2, 3) and the Kardar−Parisi−Zhang (KPZ) universality class with z = 3/2 (4), enter the Kepler ratios hierarchy as the first two members of the family.The universal dynamical exponents in the present context characterize the self-similar space−time fluctuations of locally conserved quantities, characterizing, e.g., mass, momentum, or thermal transport in one-dimensional systems far from thermal equilibrium (5). The theory of nonlinear fluctuating hydrodynamics (NLFH) has recently emerged as a powerful and versatile tool to study space−time fluctuations, and specifically the dynamical structure function that describes the behavior of the slow relaxation modes, and from which the dynamical exponents can be extracted (6).The KPZ universality class has been shown to explain the dynamical exponent observed in interface growth processes as diverse as the propagation of flame fronts (7, 8), the growth of bacterial colonies (9), or the time evolution of droplet shapes such as coffee stains (10) where the Gaussian theory fails. For a nice introduction into the KPZ class and its relevance, we refer to ref. 11. Recent reviews (12, 13) provide a more detailed account of theoretical and experimental work on the KPZ class. The dynamical structure function originating from the one-dimensional KPZ equation has a nontrivial scaling function obtained exactly by Prähofer and Spohn from the totally asymmetric simple exclusion process (TASEP) and the polynuclear growth model (14, 15) and was beautifully observed in experiments on turbulent liquid crystals (16, 17). The theoretical treatment, both numerical and analytical, of generic model systems with Hamiltonian dynamics (18), anharmonic chains (19, 20), and lattice models for driven diffusive systems (21, 22) have demonstrated an extraordinary robust universality of fluctuations of the conserved slow modes in one-dimensional systems.Despite this apparent ubiquity, dynamical exponents different from z = 2 or z = 3/2 were observed frequently. Usually, it is not clear whether this corresponds to genuinely different dynamical critical behavior or is just a consequence of imperfections in the experimental setting. Moreover, recently, a new universality class with dynamical exponent z = 5/3 for the heat mode in Hamiltonian dynamics (18) was discovered, followed by the discovery of some more universality classes in anharmonic chains (19, 20) and lattice models for driven diffusive systems (21, 22). What is lacking, even in the conceptually simplest case of the effectively one-dimensional systems that we are considering, is the understanding of the plethora of dynamical nonequilibrium universality classes within a larger framework. Such a framework exists, e.g., for 2D critical phenomena in equilibrium systems where the spatial symmetry of conformal invariance together with internal symmetries give rise to discrete families of universality classes in which all critical exponents are simple rational numbers.It is the aim of this article to demonstrate that discrete families of universality classes with fractional critical exponents appear also far from thermal equilibrium. This turns out to be a hidden feature of the NLFH equations that we extract using mode coupling theory. It is remarkable that one finds dynamical exponents z_α, which are ratios of neighboring Fibonacci numbers {1,1,2,3,5,8,…} defined recursively as F_n = F_n?1 + F_n?2. The first two members of this family are diffusion (z = 2 = F₃/F₂) and KPZ (z = 3/2 = F₄/F₃). The corresponding universal scaling functions are computed and shown to be (in general asymmetric) z_α-stable Lévy distributions with parameters that can be computed from the macroscopic current density relation and compressibility matrix of the corresponding physical system and which thus can be obtained from experiments without detailed knowledge of the microscopic properties of the system. The theoretical predictions, obtained by mode coupling theory, are confirmed by Monte Carlo simulations of a three-lane asymmetric simple exclusion process, which is a model of driven diffusive transport of three conserved particle species.