Coexistence of ballistic and Fourier regimes in the β-FPUT lattice

Commonly, thermal transport properties of one-dimensional systems are found to be anomalous. Here, we perform a numerical and theoretical study of the $\beta$-FPUT chain, considered a prototypical model for one-dimensional anharmonic crystals, in contact with thermostats at different temperatures. We give evidence that, in steady state conditions, the {\it local} wave energy spectrum can be naturally split into modes that are essentially ballistic (non-interacting or scarcely interacting) and kinetic modes (interacting enough to relax to local thermodynamic equilibrium). We show numerically that the well-known divergence of the energy conductivity is related to how the transition region between these two sets of modes shifts in $k$-space with the system size $L$, due to properties of the collision integral of the system. Moreover, we show that the kinetic modes are responsible for a macroscopic behavior compatible with Fourier's law. Our work sheds light on the long-standing problem of the applicability of standard thermodynamics in one-dimensional nonlinear chains, testbed for understanding the thermal properties of nanotubes and nanowires.