Quest for the golden ratio universality class

Using mode coupling theory the conditions for all allowed dynamical universality classes for the conserved modes in one-dimensional driven systems are presented in closed form as a function of the stationary currents and their derivatives. With a view on the search for the golden ratio universality class the existence of some families of microscopic models is ruled out a priori by using an Onsager-type macroscopic current symmetry. At equal mean densities of the conserved quantities the golden modes can only appear if the currents are antisymmetric under interchange of the conserved densities and if these densities are correlated, but not in the symmetric case where at equal densities one mode is always diffusive and the second may be either Kardar-Parisi-Zhang (KPZ), modified KPZ, 3/2-L\'evy, or also diffusive. We also show that the predictions of mode coupling theory for a noisy chain of harmonic oscillators are exact.