Fibonacci family of dynamical universality classesVladislav Popkova,b,1, Andreas Schadschneidera,2, Johannes Schmidta, and Gunter M. Schützc

aInstitut für Theoretische Physik, Universität zu Köln, 50937 Cologne, Germany; bCentro Interdipartimentale per lo Studio di Dinamiche Complesse,Università di Firenze, 50019 Sesto Fiorentino, Italy; and cTheoretical Soft Matter and Biophysics, Institute of Complex Systems II, Forschungszentrum Jülich,52425 Jülich, Germany

Edited by Giorgio Parisi, University of Rome, Rome, Italy, and approved August 31, 2015 (received for review June 23, 2015)

Universality is a well-established central concept of equilibriumphysics. However, in systems far away from equilibrium, a deeperunderstanding of its underlying principles is still lacking. Up tonow, a few classes have been identified. Besides the diffusiveuniversality class with dynamical exponent z= 2, another promi-nent example is the superdiffusive Kardar−Parisi−Zhang (KPZ)class with z= 3=2. It appears, e.g., in low-dimensional dynamicalphenomena far from thermal equilibrium that exhibit some con-servation law. Here we show that both classes are only part of aninfinite discrete family of nonequilibrium universality classes.Remarkably, their dynamical exponents zα are given by ratios ofneighboring Fibonacci numbers, starting with either z1 = 3=2 (if aKPZ mode exist) or z1 =2 (if a diffusive mode is present). If neithera diffusive nor a KPZ mode is present, all dynamical modes havethe Golden Mean z= (1+

ffiffiffi5

p)=2 as dynamical exponent. The uni-

versal scaling functions of these Fibonacci modes are asymmetricLévy distributions that are completely fixed by the macroscopiccurrent density relation and compressibility matrix of the systemand hence accessible to experimental measurement.

nonequilibrium physics | universality | dynamical exponent |driven diffusion | Golden Mean

The Golden Mean, φ= 1=2+ffiffiffi5

p=2≈ 1.61803..., also called Di-

vine Proportion, has been an inspiring number for many cen-turies. It is widespread in nature; i.e., arrangements of petals of theflowers and seeds in the sunflower follow the golden rule (1). Beingconsidered an ideal proportion, the Golden Mean appears in fa-mous architectural ensembles such as the Parthenon in Greece, theGiza 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 thecoefficients in the representation are equal to unity,

φ= 1+11 + 

11 + 

11 +

⋱

.[1]

Systematic truncation of the above continued fraction gives theso-called Kepler ratios, 1=1, 2=1, 3=2, 5=3, 8=5, ..., which approxi-mate the Golden Mean. Subsets of denominators (or numera-tors) of the Kepler ratios form the celebrated Fibonacci numbers,Fi = 1,1,2,3,5,8, .., such that Kepler ratios are ratios of two neigh-boring Fibonacci numbers. As well as the Golden Mean, Fibonacciratios 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 fun-damental underlying structure or symmetry. Here we demonstratethat the Divine Proportion as well as all of the truncations (Keplerratios) of the continued fraction (Eq. 1) appear as universalnumbers, namely, the dynamical exponents, in low-dimensionaldynamical phenomena far from thermal equilibrium. The two well-known paradigmatic universality classes, Gaussian diffusion withdynamical exponent z= 2 (2, 3) and the Kardar−Parisi−Zhang

(KPZ) universality class with z= 3=2 (4), enter the Kepler ratioshierarchy as the first two members of the family.The universal dynamical exponents in the present context char-

acterize the self-similar space−time fluctuations of locally con-served quantities, characterizing, e.g., mass, momentum, or thermaltransport 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 studyspace−time fluctuations, and specifically the dynamical structurefunction 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 dy-

namical exponent observed in interface growth processes as di-verse as the propagation of flame fronts (7, 8), the growth ofbacterial colonies (9), or the time evolution of droplet shapes suchas coffee stains (10) where the Gaussian theory fails. For a niceintroduction into the KPZ class and its relevance, we refer to ref.11. Recent reviews (12, 13) provide a more detailed account oftheoretical and experimental work on the KPZ class. The dy-namical structure function originating from the one-dimensionalKPZ equation has a nontrivial scaling function obtained exactly byPrähofer and Spohn from the totally asymmetric simple exclusionprocess (TASEP) and the polynuclear growth model (14, 15) andwas beautifully observed in experiments on turbulent liquid crys-tals (16, 17). The theoretical treatment, both numerical and ana-lytical, of generic model systems with Hamiltonian dynamics (18),anharmonic chains (19, 20), and lattice models for drivendiffusive systems (21, 22) have demonstrated an extraordinaryrobust universality of fluctuations of the conserved slow modes inone-dimensional systems.

Significance

Universality is a well-established central concept of equilibriumphysics. It asserts that, especially near phase transitions, theproperties of a physical system do not depend on its detailssuch as the precise form of interactions. Far from equilibrium,such universality has also been observed, but, in contrast toequilibrium, a deeper understanding of its underlying princi-ples is still lacking. We show that the two best-known exam-ples of nonequilibrium universality classes, the diffusive andKardar−Parisi−Zhang classes, are only part of an infinite discretefamily. The members of this family can be identified by theirdynamical exponent, which, surprisingly, can be expressed by aKepler ratio of Fibonacci numbers. This strongly indicates theexistence of a simpler underlying mechanism that determinesthe different classes.

Author contributions: A.S. and G.M.S. designed research; V.P., J.S., and G.M.S. performedresearch; V.P., A.S., J.S., and G.M.S. analyzed data; and V.P., A.S., and G.M.S. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1Present address: Helmholtz Institut für Strahlen- und Kernphysik, University of Bonn,53115 Bonn, Germany.

2To whom correspondence should be addressed. Email: as@thp.uni-koeln.de.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1512261112/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1512261112 PNAS | October 13, 2015 | vol. 112 | no. 41 | 12645–12650

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Despite this apparent ubiquity, dynamical exponents differentfrom z= 2 or z= 3=2 were observed frequently. Usually, it is notclear whether this corresponds to genuinely different dynamicalcritical behavior or is just a consequence of imperfections in theexperimental setting. Moreover, recently, a new universality classwith dynamical exponent z= 5=3 for the heat mode in Hamilto-nian dynamics (18) was discovered, followed by the discovery ofsome more universality classes in anharmonic chains (19, 20) andlattice models for driven diffusive systems (21, 22). What islacking, even in the conceptually simplest case of the effectivelyone-dimensional systems that we are considering, is the un-derstanding of the plethora of dynamical nonequilibrium uni-versality classes within a larger framework. Such a frameworkexists, e.g., for 2D critical phenomena in equilibrium systemswhere the spatial symmetry of conformal invariance togetherwith internal symmetries give rise to discrete families of uni-versality classes in which all critical exponents are simplerational numbers.It is the aim of this article to demonstrate that discrete families

of universality classes with fractional critical exponents appearalso far from thermal equilibrium. This turns out to be a hiddenfeature of the NLFH equations that we extract using modecoupling theory. It is remarkable that one finds dynamical ex-ponents zα, which are ratios of neighboring Fibonacci numbersf1,1,2,3,5,8,. . .g defined recursively as Fn =Fn−1 +Fn−2. The firsttwo members of this family are diffusion (z= 2=F3=F2) and KPZ(z= 3=2=F4=F3). The corresponding universal scaling functionsare computed and shown to be (in general asymmetric) zα -stableLévy distributions with parameters that can be computed fromthe macroscopic current density relation and compressibilitymatrix of the corresponding physical system and which thus canbe obtained from experiments without detailed knowledge of themicroscopic properties of the system. The theoretical predictions,obtained by mode coupling theory, are confirmed by Monte Carlosimulations of a three-lane asymmetric simple exclusion process,which is a model of driven diffusive transport of three conservedparticle species.

Nonlinear Fluctuating HydrodynamicsWe consider a rather general interacting nonequilibrium system oflength L described macroscopically by n conserved order param-eters ρλðx, tÞ with stationary values ρλ and associated macroscopicstationary currents jλðρ1, . . . , ρnÞ and compressibility matrix K withmatrix elements Kλμ = ð1=LÞhðNλ − ρλLÞðNμ − ρμLÞi where Nλ =R L0 dxρλðx, tÞ are the time-independent conserved quantities.The starting point for investigating density fluctuations

uλðx, tÞ := ρλðx, tÞ− ρλ in the nonequilibrium steady state are theNLFH equations (5)

∂t~u=−∂x�J~u+

12huj~Hjui−∂xD~u+B~ξ

�[2]

where J is the current Jacobian with matrix elements Jλμ = ∂jλ=∂ρμ,~H is a column vector whose entries ð~HÞλ =Hλ are the Hessianswith matrix elements Hλ

μν = ∂2jλ=ð∂ρμ∂ρνÞ, and the bra-ket no-tation represents the inner product in component spacehujð~HÞλjui=~uTHλ~u=

PμνuμuγH

λμν with huj=~uT and jui=~u. The

diffusion matrix D is a phenomenological quantity. The noiseterm B~ξ does not appear explicitly below, but plays an indirectrole in the mode coupling analysis. The product JK of the Jaco-bian with the compressibility matrix K is symmetric (23), whichguarantees a hyperbolic system of conservation laws (24). Weignore possible logarithmic corrections arising from cubic con-tributions (25).This system of coupled noisy Burgers equations is conveniently

treated in terms of normal modes~ϕ=R~u, where RJR−1 = diagðvαÞand the transformation matrix R is normalized such that RKRT = 1.

The eigenvalues vα of J are the characteristic velocities of the sys-tem. From Eq. 2, one thus arrives at ∂tϕα =−∂xðvαϕα + hϕjGαjϕi−∂xð~D~ϕÞα + ð~B~ξÞαÞ, with ~D=RDR−1, ~B=RB, and the mode couplingmatrices

Gα =12

Xβ

Rαβ

�R−1�THβR−1, [3]

whose matrix elements we denote by Gαβγ.

Computation of the Dynamical Structure FunctionThe dynamical structure function describes the stationary fluc-tuations of the conserved slow modes and is thus a key ingredientfor understanding the interplay of noise and nonlinearity andtheir role for transport far from equilibrium. We focus on thecase of strict hyperbolicity where all vα are pairwise different andstudy the large-scale behavior of the dynamical structure func-tion Sαβðx, tÞ= hϕαðx, tÞϕβð0,0Þi. Because all modes have differentvelocities, only the diagonal elements Sαðx, tÞ := Sααðx, tÞ arenonzero for large times. Mode coupling theory yields (5)

∂tSαðx, tÞ=�−vα∂x +Dα∂2x

�Sαðx, tÞ

+Z t

0

dsZR

dySαðx− y, t− sÞ∂2yMααðy, sÞ [4]

with the diagonal element Dα := ~Dαα of the phenomenologicaldiffusion matrix for the eigenmodes and the memory kernelMααðy, sÞ=

Pβ,γðGα

βγÞ2Sβðy, sÞSγðy, sÞ. The task therefore is to ex-tract for arbitrary n the large-time and large-distance behaviorfrom this nonlinear integro-differential equation.Remarkably, these equations can be solved exactly in the long-

wavelength limit and for t→∞ by Fourier and Laplace trans-formation (see Materials and Methods and SI Text). Using a suit-able scaling ansatz for the transformed structure function thenallows analysis of the small-p behavior from which the dynamicalexponents can be determined. We find that different conditionsarise depending on which diagonal elements of the mode couplingmatrices vanish.

Fibonacci Family of Dynamical Universality ClassesFibonacci Case. First, we consider the case where the self-couplingGα

αα is nonzero for one mode only, e.g., G111 ≠ 0. For all other

modes α > 1, we assume a single nonzero coupling to the previousmode, so Gα

α−1,α−1 ≠ 0, and Gαβ,β = 0 for β≠ α− 1. Then, as follows

from our analysis (see Materials and Methods), we find the fol-lowing recursion for the dynamical exponents:

zα = 1+1

zα−1[5]

with z1 = 3=2.The dynamical structure function in momentum space is pro-

portional to the zα -stable Lévy distribution with maximal asym-metry σα =±1; see ref. 26 and Eq. 13 below. The sign of theasymmetry depends on whether the mode ðα− 1Þ has bigger orsmaller velocity than the mode α, σα =−sgnðvα − vα−1Þ. The dy-namical exponents (Eq. 5) form a sequence of rational numbers,

zα =Fα+3

Fα+2, [6]

which are consecutive ratios of neighboring Fibonacci numbers Fα,defined by Fα =Fα−1 +Fα−2 with initial values F0 = 0, F1 = 1, whichconverge exponentially to the Golden Mean φ : = 1

2 ð1+ffiffiffi5

p Þ≈ 1.618,as first observed by Kepler in 1611 in a treatise on snowflakes. In a

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model with n conservation laws, one has the Fibonacci modes withdynamical exponents f3=2, 5=3, 8=5,  . . . ,   zng.Finally, we remark that if mode 1 is diffusive rather than KPZ,

then we find the same sequence (Eq. 6) of exponents, except thatit starts with z1 =F2 = 2.In Fig. 1, we show some representative examples of the scaling

functions that are quite different in shape. Furthermore, therelation between the exponents zα, determined by Eq. 11 inMaterials and Methods, and the mode coupling matrices Gα isillustrated for the case n= 2.

Golden Mean Case. As a second representative example, we con-sider the case where all self-coupling coefficients vanish, Gα

αα ≡ 0for all α, while each mode has at least one nonzero coupling toanother mode, Gα

ββ ≠ 0 for some β≠ α. Then, Eq. 5 reduces tozα = 1+ 1=zβ for all modes α,   β. The unique solution of thisequation is the Golden Mean zα =φ= ð1+ ffiffiffi

5p Þ=2 for all α. The

scaling functions (see SI Text) are proportional to φ-stableLévy distributions with parameters fixed by the collective ve-locities and the mode coupling coefficients. The asymmetry ofthe fastest right-moving (left-moving) mode is predicted to beβ=−1 (β= 1).

Simulation ResultsTo check the theoretical predictions for the two cases, we simulatemass transport with three conservation laws, i.e., three distinctspecies of particles. To maintain a far-from-equilibrium situation, adriving force is applied that leads to a constant drift superimposedon undirected diffusive motion. This is a natural setting for trans-port of charged particles in nanotubes (see Fig. 2 for an illustra-tion), where a direct measurement of the stationary particlecurrents is experimentally possible (27). However, due to the uni-versal applicability of NLFH, the actual details of the interaction ofthe particles with their environment and the driving field are ir-relevant for the theoretical description of the large-scale dynamics.Hence, for good statistics, we simulate a lattice model for trans-port that represents a minimal realization of the essential in-gredients, namely, a nonlinear current density relation for allthree conserved masses.Our model is the three-species version of the multilane

TASEP (28). Particles hop randomly in field direction on threelanes to their neighboring sites on a periodic lattice of 3×Lsites with rates that depend on the nearest-neighbor sites. Lanechanges are not allowed so that the total number of particles oneach lane is conserved. Due to excluded volume interaction,each lattice site can be occupied by at most one particle. Thus,the occupation numbers nðλÞk of site k on lane λ take only values0 or 1. The hopping rate rðλÞk from site k on lane λ to site k+ 1on the same lane is given by

rðλÞk = bλ +12

Xfμ:μ≠λg

γλμ

�nðμÞk + nðμÞk+1

�[7]

with a species-dependent drift parameter bλ and symmetric in-teraction constants γλμ = γμλ. Hopping attempts onto occupiedsites are rejected. The conserved quantities are the three total num-bers of particles Nλ on each lane with corresponding densitiesρλ =Nλ=L.The stationary distribution of our model factorizes (28) and

thus allows for the exact computation of the macroscopic cur-rent density relations jλðρ1, ρ2, ρ3Þ and the compressibility matrixKðρ1, ρ2, ρ3Þ. Furthermore, because there is no particle exchangebetween lanes, the compressibility matrix is diagonal, with ele-ments denoted by κλ. One has

Fig. 1. The scaling functions (Bottom) and dynamical exponents are relatedto the structure of the mode coupling matrices Gα (Top). The table shows thedynamical exponents zα in the case n= 2; see Eq. 11. The symbols * and ⋆denote nonzero elements. Red symbols correspond to self-coupling; blacksymbols correspond to couplings to other modes. Matrix elements not in-dicated can take any value. The colors in the table correspond to the colorsof the graphs of the scaling functions.

Fig. 2. Schematic drawing of three particle species drifting inside a nano-tube. Due to the interaction between the particles and with the walls, oneexpects a nonlinear current density relation.

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jλ = ρλð1− ρλÞ bλ +

Xfμ:μ≠λg

γλμρμ

![8]

κλ = ρλð1− ρλÞ. [9]

The diagonalization matrix R and the mode coupling matricesGα

are fully determined by these quantities.According to mode coupling theory, three different Fibonacci

modes with z1 = 3=2, z2 = 5=3, z3 = 8=5 occur, e.g., when G111 ≠ 0,

G211 ≠ 0, G3

22 ≠ 0, and G222 =G2

33 =G333 = 0. For our simulation, we

compute numerically densities, bare hopping rates, and interactionparameters to satisfy these properties as described inMaterials andMethods. For this choice, the velocities of the normal modes arev1 = 0.592315, v2 = 0.0281578, and v3 = 1.58226, which ensures agood spatial separation after quite small times. The propagation ofthe three normal modes (Fig. 3) with the predicted velocities isobserved with an error of less than 10−3. Moreover, the numericallyobtained dynamical structure function for mode 3 shows a startlingagreement with the theoretically predicted Lévy scaling functionwith z= 8=5 and maximal asymmetry (see Fig. 4). It takes longerfor the other two modes (KPZ mode and Lévy stable 5=3 mode) toreach their asymptotic form, which we argue is due to the muchsmaller respective couplings, ðG1

11=G322Þ2 � 1, ðG2

11=G322Þ2 � 1.

To observe three Golden Mean modes, it is sufficient to re-quire that each mode has zero self-coupling and at least onenonzero coupling to other modes. This can be achieved with theset of parameters given in Materials and Methods, which lead tothe velocities v1 = 1.83149, v2 = 0.762688, and v3 = 0.326778 of thenormal modes. The propagation of the three normal modes withthe predicted velocities is observed, approaching, for large times,a very small relative error of about 10−4. The structure functionfor the fastest mode 1 converges to its asymptotic shape fasterthan for the other modes, due to the large coupling coefficientG1

33. In Fig. 5, we show a scaling plot of the measured structurefunction for mode 1 with dynamical exponent z≡φ= ð1+ ffiffiffi

5p Þ=2

together with a fitted to a φ-stable Lévy function (Eq. 13) withmaximal asymmetry β=−1 as predicted by the theory. The datacollapse shows a striking agreement between the measured andtheoretical scaling function. Alternatively, the dynamical exponentzα can be derived from the maximum of the structure function,which scales as maxðS1ðx, tÞÞ= const · t−1=z. We obtain z≈ 1.63,which differs from the predicted value z≡φ= ð1+ ffiffiffi

5p Þ=2 by less

than 0.8%.

DiscussionOur work demonstrates that nonequilibrium phenomena aremuch richer than just the diffusive and KPZ universality suggest.We have established that in nonequilibrium phenomena gov-erned by NLFH with n conservation laws, mode coupling theorypredicts a family of dynamical universality classes with dynamicalexponents given by the sequence of consecutive Kepler ratios(Eq. 6) of Fibonacci numbers. With slightly modified initial con-ditions on G11

1 , this result is easily generalized for the case whenthe first mode α= 1 is diffusive. Then the sequence of dynamicalexponents becomes shifted by one unit with respect to Eq. 6. Onthe other hand, if all self-couplings vanish, but at least one otherdiagonal element Gα

ββ of the mode coupling matrix is non-zero,one has, as a unique solution for all modes α, the fixed point valuezα = z∞ =φ, which is the Golden Mean.For general mode coupling matrices, all critical exponents can

be computed (from Eq. 11 in Materials and Methods). The scalingfunctions of the nondiffusive and non-KPZ modes are asymmetricLévy distributions whose parameters are completely determinedby the macroscopic current density relation and compressibilitymatrix of the system.For 1+1 dimensional systems out of equilibrium, this is the first

time, to our knowledge, that an infinite family of discrete univer-sality classes is found. Recalling that 1 + 1 dimensional non-equilibrium systems with short-range interactions can be mappedonto 2D equilibrium systems (with the time evolution operatorplaying the role of the transfer matrix), one is reminded of thediscrete families of conformally invariant critical equilibrium sys-tems in two space dimensions (29, 30). We do not know whetherthere is any mathematical link, but the analogy is suggestive in sofar as conformal invariance is a local symmetry of spatially isotropicsystems with z= 1 (which happens to be the lowest-order Keplerratio) whereas z> 1 corresponds to strongly anisotropic systems forwhich local symmetry groups are also known to exist (31).Because an infinite number of lanes of coupled one-dimensional

systems correspond to a 2D system, it is intriguing to observe thatthe Golden Mean is close to the numerical value z= 1.612− 1.618of the dynamical exponent of the 2+ 1-dimensional KPZ equation

Fig. 3. Space−time propagation of three normal modes in the three-lanemodel. The modes (from left to right) are the Fibonacci mode with z= 8=5(mode 3), the KPZ mode with z= 3=2 (mode 1), and the Fibonacci mode withz= 5=3 (mode 2). The physical and simulation parameters are given in Ma-terials and Methods.

1200 1400 1600 18000

0.5

1

1.5

2

2.5

3

3.5

4x 10−3

k

S33 k

( t=

1000

)

Fig. 4. (Left) Vertical least squares fit of the numerically obtained dynam-ical structure function for the Fibonacci 8/5 mode (points), at time t =1,000with an 8/5-stable Lévy distribution, maximal asymmetry −1, and theoreticalcenter of mass (line), predicted by the mode coupling theory. The only fitparameter is the scale parameter of the Lévy stable distribution. The simu-lation results agree very well with the asymptotic theoretical result alreadyfor moderate times. (Right) Close-ups of the peak region and tail regions,according to a color code. Every tenth data point is plotted, to improve thevisibility of the data. The statistical error e99% with 99% confidence bound isfor every data point smaller than 1.6299 ·10−5.

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(32, 33). The scaling function of the 2+ 1-dimensional KPZ equa-tion, however, is not Lévy (34).To observe and distinguish between the different new classes,

highly precise experimental data will be required. For example,in the Fibonacci case, the dynamical exponents converge quicklyto the Golden Mean. A feature that might be easier to observeexperimentally is the scaling function itself, which, for higherFibonacci ratios 5=3, 8=5,  . . ., usually has a strong asymmetry(see Figs. 1, 4, and 5), whereas KPZ and Gauss scaling functionsare symmetric. Growth processes that can be mapped on ex-clusion processes with several conservation laws might be po-tentially suitable candidates for an experimental verification;see, e.g., refs. 16 and 17 for an example of a system with one con-servation law.

Materials and MethodsComputation of the Dynamical Structure Function. The mode coupling Eq. 4can be solved in the scaling limit by applying a Fourier transform (FT)fðxÞ→ ~fðpÞ and a Laplace transform (LT) fðtÞ→ ~fðωÞ. For more details, werefer to ref. 22, where the case n= 2 of two conservation laws has beentreated. After making the scaling ansatz

~Sαðp, ~ωαÞ=p−zαgαðζαÞ [10]

for the transformed dynamical structure function where Sαðp, 0Þ= 1=ffiffiffiffiffiffi2π

pand

ζα = ~ωαjpj−zα , we are in a position to analyze the small-p behavior. One has tosearch for dynamical exponents for which the limit p→0 is nontrivial, whichrequires a self-consistent treatment of all modes. We find that differentconditions arise depending on which diagonal elements of the mode cou-pling matrices vanish. To characterize the possible scenarios, we define theset Iα := fβ :Gα

ββ ≠ 0g of nonzero diagonal mode coupling coefficients. Throughpower counting one obtains

zα =

8>>>><>>>>:

2 if Iα = 0=

3=2 if α∈ Iα

minβ∈Iα

�1+

1zβ

�else

[11]

and the domain

1< zα ≤ 2 ∀α. [12]

for the possible dynamical exponents.In the Fibonacci case, the dynamical structure function of mode α in

momentum space has the scaling form

Sαðp, tÞ= 1ffiffiffiffiffiffi2π

p e−ivαpt−Eα jpjzα t

�1−iσαβp tan

�πzα2

��[13]

with inverse time scales Eα. The dynamical exponents then satisfy the re-cursion (Eq. 5). Up to the normalization 1=

ffiffiffiffiffiffi2π

pthe scaling form (Eq. 13) is an

α-stable Lévy distribution (26).

Simulation Algorithm. For the Monte Carlo simulation of the model, wechoose a large system size L≥ 5 ·105, which avoids finite-size effects. At timet = 0, Nλ particles are placed on each lane according to the desired initialstate. One Monte Carlo time unit consists of 3 · L · r* random sequentialupdate steps where r*=maxfrðλÞk g: In each update step, a bond ðkðλÞ, kðλÞ + 1Þis chosen randomly with uniform distribution. If nðλÞ

k ð1−nðλÞk+ 1Þ= 1, then the

particle at site k is moved to k+1 with probability rðλÞk =r * where r* is themaximal value that the rðλÞk can take among all possible particle configura-tions on the neighboring lanes. If nðλÞ

k ð1−nðλÞk+1Þ= 0, the particle configura-

tion remains unchanged.

Simulation of the Dynamical Structure Function. To determine the dynamicalstructure function, we initialize the system by placing Nλ particles uniformlyon each lane λ. This yields a random initial distribution drawn from thestationary distribution of the process. No relaxation is required.

Thenweuse translation invariance and compute the space and time average

σλμL,kðM, τ, tÞ= 1M

XMj=1

1L

XLl=1

nðλÞl+kðjτ+ tÞnðμÞ

l ðjτÞ− ρλρμ. [14]

To avoid noisy data of σλμL,k, we take in Eq. 14 the system size L and the timeaverage parameterM sufficiently large. To obtain Sλμk ðtÞ, we average over Pindependently generated and propagated initial configurations of σλμL,k.The error estimates for Sλμk ðtÞ are calculated from the P independent mea-surements. From Sλμk ðtÞ, we compute the structure function of the normalmodes by transformation with the diagonalizing matrix R determined by Eqs.8 and 9.

To obtain model parameters for three different Fibonacci modes withz1 = 3=2, z2 = 5=3, z3 = 8=5, we solve the equations given in the text after Eq. 9numerically with a C program performing direct minimization of the absolutevalues of the targeted G elements until the given tolerance value (10−6) isreached. The data shown here for the three-mode case have been obtainedfrom simulations with densities ρ1 = 0.2,   ρ2 = 0.25,   ρ1 = 0.3, bare hoppingrates b1 = 0.613185, b2 = 0.425714, b3 = 0.799831, and interaction parametersγ12 = 1.36145, γ23 = 3.69786, γ13 = 0.143082 for which the needed relationsare satisfied. This choice of parameters yields G1

11 = 0.322507, G211 =−0.15,

G322 =1.04547, while the absolute values of G2

22,  G233,  G

333 are smaller than

10−6. Besides these physical parameters, the simulation parameters for theFibonacci modes (Figs. 3 and 4) are L= 5 · 105, τ= 250, M= 1,400, P = 98.

For the Golden Mean case (Fig. 5), the set of parameters is ρ1 = 0.2,   ρ2 =ρ3 = 0.25, γ12 =0.0082334758646, γ23 = 1.68447706968, γ13 = 3.72140740146, andb1 = 0.905073261248, b2 = 0.86, and b3 =1.18875738638. This leads to G1

22 =0.405702, G1

33 = 0.929315, G211 =−0.104141, G2

33 =−0.208477, G311 =−0.182467,

G322 = 0.271246, while the absolute values of G1

11,G222,G

333 are smaller than

10−6. The simulation parameters for the Golden Mean case are L= 5 · 106

τ= 750, M= 30 and P = 303.

ACKNOWLEDGMENTS. We thank Herbert Spohn for helpful comments on apreliminary version of the manuscript. This work was supported by DeutscheForschungsgemeinschaft under Grant SCHA 636/8-1.

1. Livio M (2003) The Golden Ratio: The Story of PHI, the World’s Most Astonishing

Number (Broadway Books, New York).2. Landi GT, de Oliveira MJ (2014) Fourier’s law from a chain of coupled planar harmonic oscil-

lators under energy-conserving noise. Phys Rev E Stat Nonlin Soft Matter Phys 89(2):022105.3. Gendelman OV, Savin AV (2014) Normal heat conductivity in chains capable of dis-

sociation. EPL 106(3):34004.

4. Kardar M, Parisi G, Zhang Y-C (1986) Dynamic scaling of growing interfaces. Phys Rev

Lett 56(9):889–892.5. Spohn H (2014) Nonlinear fluctuating hydrodynamics for anharmonic chains. J Stat

Phys 154(5):1191–1227.6. Spohn H (2015) Fluctuating hydrodynamics approach to equilibrium time correlations

for anharmonic chains. arXiv:1505.05987.

−4 −3 −2 −1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

Sk11

( t )

* t1/

z ;

z =

( 1+5

1/2 )

/ 2

( k − v1 t ) * t−1/z

t=15000t=12750t=10500t=9000t=6750GM Lévy stable

Fig. 5. Scaling plot of the measured structure function of mode 1 withdynamical exponent z≡φ= ð1+ ffiffiffi

5p Þ=2 for the Golden Mean case, fitted to a

φ-stable Lévy distribution with maximal asymmetry −1 (see Eq. 13). The scaleparameter E1 for the Lévy-stable distribution and the center of mass velocityv1 are obtained by a vertical least square fit. Fitted parameters are v1,fit =1.83107±0.00009 and E1,fit = 1.1± 0.01. The fitted velocity v1,fit differs by0.02% from the theoretical velocity.

Popkov et al. PNAS | October 13, 2015 | vol. 112 | no. 41 | 12649

PHYS

ICS

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

1

7. Maunuksela J, et al. (1997) Kinetic roughening in slow combustion of paper. Phys RevLett 79(8):1515.

8. Miettinen L, Myllys M, Merikoski J, Timonen J (2005) Experimental determination ofKPZ height-fluctuation distributions. Eur Phys J B 46(1):55–60.

9. Wakita J, Itoh H, Matsuyama T, Matsushita M (1997) Self-affinity for the growinginterface of bacterial colonies. J Phys Soc Jpn 66(1):67–72.

10. Yunker PJ, et al. (2013) Effects of particle shape on growth dynamics at edges ofevaporating drops of colloidal suspensions. Phys Rev Lett 110(3):035501.

11. Buchanan M (2014) Equivalence principle. Nat Phys 10(8):543.12. Halpin-Healy T, Takeuchi KA (2015) A KPZ cocktail—Shaken, not stirred: Toasting 30

years of kinetically roughened surfaces. J Stat Phys 160(4):794–814.13. Quastel J, Spohn H (2015) The one-dimensional KPZ equation and its universality class.

J Stat Phys 160(4):965–984.14. Prähofer M, Spohn H (2004) Exact scaling functions for one-dimensional stationary

KPZ growth. J Stat Phys 115(1):255–279.15. Prähofer M, Spohn H (2002) Current fluctuations in the totally asymmetric simple

exclusion process. In and Out of Equilibrium, Progress in Probability, ed Sidoravicius V(Birkhauser, Boston), Vol 51, pp 185−204.

16. Takeuchi KA, Sano M (2010) Universal fluctuations of growing interfaces: evidence inturbulent liquid crystals. Phys Rev Lett 104(23):230601.

17. Takeuchi KA, Sano M, Sasamoto T, Spohn H (2011) Growing interfaces uncover uni-versal fluctuations behind scale invariance. Sci Rep 1:34.

18. van Beijeren H (2012) Exact results for anomalous transport in one-dimensionalhamiltonian systems. Phys Rev Lett 108(18):180601.

19. Mendl CB, Spohn H (2013) Dynamic correlators of Fermi-Pasta-Ulam chains andnonlinear fluctuating hydrodynamics. Phys Rev Lett 111(23):230601.

20. Spohn H, Stoltz G (2015) Nonlinear fluctuating hydrodynamics in one dimension: Thecase of two conserved fields. J Stat Phys 160(4):861–884.

21. Popkov V, Schmidt J, Schütz GM (2014) Superdiffusive modes in two-species drivendiffusive systems. Phys Rev Lett 112(20):200602.

22. Popkov V, Schmidt J, Schütz GM (2015) Universality classes in two-component drivendiffusive systems. J Stat Phys 160(4):835–860.

23. Grisi R, Schütz GM (2011) Current symmetries for particle systems with several con-servation laws. J Stat Phys 145(6):1499–1512.

24. Tóth B, Valkó B (2003) Onsager relations and Eulerian hydrodynamic limit for systemswith several conservation laws. J Stat Phys 112(3):497–521.

25. Devillard P, Spohn H (1992) Universality class of interface growth with reflectionsymmetry. J Stat Phys 66(3):1089–1099.

26. Durrett R (2010) Probability: Theory and Examples (Cambridge Univ Press, Cambridge,UK), 4th Ed.

27. Lee CY, Choi W, Han J-H, Strano MS (2010) Coherence resonance in a single-walledcarbon nanotube ion channel. Science 329(5997):1320–1324.

28. Popkov V, Salerno M (2004) Hydrodynamic limit of multichain driven diffusive mod-els. Phys Rev E Stat Nonlin Soft Matter Phys 69(4 Pt 2):046103.

29. Cardy J (1996) Scaling and Renormalization in Statistical Physics (Cambridge UnivPress, Cambridge, UK).

30. Henkel M (1999) Conformal Invariance and Critical Phenomena (Springer, Berlin).31. Henkel M (2002) Phenomenology of local scale invariance: From conformal invariance

to dynamical scaling. Nucl Phys B 641(3):405–486.32. Pagnani A, Parisi G (2015) Numerical estimate of the Kardar-Parisi-Zhang universality

class in (2+1) dimensions. Phys Rev E Stat Nonlin Soft Matter Phys 92(1-1):010101.33. Halpin-Healy T (2012) (2+1)-dimensional directed polymer in a random medium:

Scaling phenomena and universal distributions. Phys Rev Lett 109(17):170602.34. Kloss T, Canet L, Wschebor N (2012) Nonperturbative renormalization group for the

stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1,2+1, and 3+1 dimensions. Phys Rev E Stat Nonlin Soft Matter Phys 86(5 Pt 1):051124.

12650 | www.pnas.org/cgi/doi/10.1073/pnas.1512261112 Popkov et al.

Dow

nloa

ded

by g

uest

on

Dec

embe

r 31

, 202

1