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

Coexistence of ballistic and Fourier regimes in the $\beta$-FPUT lattice