3/4-Fractional Superdiffusion in a System of Harmonic Oscillators Perturbed by a Conservative Noise

Abstract

We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4.

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

“…For quadratic V (r), i.e harmonic chains, there are a macroscopic number of conserved quantities and transport becomes ballistic. In this case a number of studies have considered augmenting the Hamiltonian dynamics with a stochastic component such that the system again has only three conserved quantities [9,[29][30][31]. In this case one again recovers the typical features of anomalous transport and several exact results are possible.…”

“…The evolution of the density fields corresponding to these conserved quantities at the macroscopic length and time scales was studied in [62] using NFH, where it has been shown that this model has two normal modes -one diffusive sound mode and a 3 2 -asymmetric Lévy heat mode. Subsequently, it was rigorously shown that the local energy density e(x, t) satisfies a (3/4)-skew-fractional equation [31] ∂ t e(x, t)…”

Fluctuating Hydrodynamics Approach to Equilibrium Time Correlations for Anharmonic Chains

“…For them the couplings can be more easily adjusted than for anharmonic chains, which offers the possibility to test the dynamical phase diagram. Also anharmonic chains with a stochastic collision mechanism, respecting the conservation laws, have been studied in considerable detail [67,68,42]. …”

Anomalous diffusion phenomena: A kinetic approach

“…See in particular Basile-Bernardin-Olla [3,4], Basile-Olla-Spohn [5], Jara-Komorowski-Olla [18,19], Olla [30], Bernardin-Gonçalves-Jara [8].…”

“…There is no reason to believe that the two limits should commute, and it is thus not clear that the scaling of this Fourier law is consistent with the scaling of the microscopic Hamiltonian dynamics. In particular other approaches, that do not rely on the kinetic description, lead to different powers (see Spohn [35], Bernardin-Gonçalves-Jara [8], Jara-Komorowski-Olla [18]). We also point out that in higher dimensions numerical simulations as well as theoretical arguments point to anomalous behavior in dimension 2 and normal diffusion in dimension 3.…”