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Anomalous conduction and second sound in the Fermi-Pasta-Ulam-Tsingou chain: wave-turbulence approach

One-dimensional particle chains are fundamental models to explain anomalous thermal conduction in low-dimensional solids like nanotubes and nanowires. In these systems the thermal energy is carried by phonons, i.e. propagating lattice oscillations that interact via nonlinear resonance. The average energy transfer between the phonons is described by the wave kinetic equation (WKE), derived directly from the microscopic dynamics. Here, we use the spatially nonhomogeneous WKE of the prototypical $β-$ Fermi-Pasta-Ulam-Tsingou (FPUT) model, equipped with thermostats able to set different temperatures at the two ends. Our main findings are as follows: (i) The anomalous scaling of the conductivity with the system size, in close agreement with the known results from the microscopic dynamics, is due to a nontrivial interplay between high and low wavenumbers. (ii) The high-wavenumber phonons relax to local thermodynamic equilibrium transporting energy diffusively, {\it à la Fourier}. (iii) The low-wavenumber phonons are nearly noninteracting and transfer energy ballistically; this latter phenomenon is the analogous of the second sound emission, observed for example in superfluids.