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On the Fibonacci Universality Classes in Nonlinear Fluctuating Hydrodynamics

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From Particle Systems to Partial Differential Equations (PSPDE 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 258))

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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.

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Notes

  1. 1.

    If some collective velocities are equal then the crucial assumption of spatial separation of the normal modes at large times breaks down. The predictions of NLFH for this case have not been studied yet in great detail.

  2. 2.

    When the time t is irrelevant we drop the dependence on t.

  3. 3.

    It seems to have gone unnoticed that, quite remarkably, this symmetry relates a purely static property of the invariant measure (the covariances \(K_{\alpha \beta }\)) with the microscopic dynamics which give rise to the currents \(\bar{j}^\alpha \). This restricts severely the possible microscopic dynamics for which a given measure can be invariant.

  4. 4.

    We stress that for more than one conservation law this expectation is mathematically very difficult to prove. Not only is on macroscopic level existence and uniqueness of global solutions in time a major open problem in pde theory, but also the derivation of the hydrodynamic limit after the occurrence shocks, which is a generic property of hyperbolic systems of conservation laws, is largely an open problem, with some results only for the Leroux system [9].

  5. 5.

    The product measure (23) remains invariant also for different interaction strength \(\alpha _1\ne \alpha _2\) which leaves the currents unchanged. However, equal interaction asymmetry is required.

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Authors and Affiliations

  1. Institute of Complex Systems II, Forschungszentrum Jülich, 52425, Jülich, Germany

    G. M. Schütz

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  1. G. M. Schütz
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  1. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

    Patrícia Gonçalves

  2. Center of Mathematics, University of Minho, Braga, Portugal

    Ana Jacinta Soares

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Schütz, G.M. (2018). On the Fibonacci Universality Classes in Nonlinear Fluctuating Hydrodynamics. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations . PSPDE 2016. Springer Proceedings in Mathematics & Statistics, vol 258. Springer, Cham. https://doi.org/10.1007/978-3-319-99689-9_2

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