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# On the Fibonacci universality classes in nonlinear fluctuating hydrodynamics

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### Abstract

We present a lattice gas model that without fine tuning of parameters is expected to exhibit the so far elusive modified Kardar-Parisi-Zhang (KPZ) universality class. To this end, we review briefly how non-linear fluctuating hydrodynamics in one dimension predicts that all dynamical universality classes in its range of applicability belong to an infinite discrete family which we call Fibonacci family since their dynamical exponents are the Kepler ratios $$z_i = F_{i+1}/F_{i}$$ of neighbouring Fibonacci numbers $$F_i$$, including diffusion ($$z_2=2$$), KPZ ($$z_3=3/2$$), and the limiting ratio which is the golden mean $$z_\infty=(1+\sqrt{5})/2$$. Then we revisit the case of two conservation laws to which the modified KPZ model belongs. We also derive criteria on the macroscopic currents to lead to other non-KPZ universality classes.

### Most cited references19

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(2014)
With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.
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### Dynamic relaxation of drifting polymers: A phenomenological approach

(1993)
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### Mode coupling and renormalization group results for the noisy Burgers equation

(1996)
We investigate the noisy Burgers equation (Kardar--Parisi--Zhang equation in 1+1 dimensions) using the dynamical renormalization group (to two--loop order) and mode--coupling techniques. The roughness and dynamic exponent are fixed by Galilean invariance and a fluctuation--dissipation theorem. The fact that there are no singular two--loop contributions to the two--point vertex functions supports the mode--coupling approach, which can be understood as a self--consistent one--loop theory where vertex corrections are neglected. Therefore, the numerical solution of the mode coupling equations yields very accurate results for the scaling functions. In addition, finite--size effects can be studied. Furthermore, the results from exact Ward identities, as well as from second--order perturbation theory permit the quantitative evaluation of the vertex corrections, and thus provide a quantitative test for the mode--coupling approach. It is found that the vertex corrections themselves are of the order one. Surprisingly, however, their effect on the correlation function is substantially smaller.
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### Author and article information

###### Journal
25 October 2017
###### Article
1710.09121