of the equipartition theorem, whose validity in the quantum regime is broken. Thus QSCPT offers a simple way to explore for a given Hamiltonian model the low temperature physics related to the Bose-Einstein statistics. We studied the temperature dependence of the thermal conductivity of the FPU- β model for both the classical and quantum case. The temperature for the minimum of κ ( T ) was determined from the classical SCPT, which clearly shows the scaling behavior of thermal conductivity for this non-linear model. We also showed that non-equilibrium molecular dynamics simulations are in good agreement with the analytical results. At the low tem- perature regime, the semiclassical treatment of the Debye equation reproduced the umklapp peak, a well-known characteristic observed in experimental studies. This work was supported in part by grants from the Hong Kong Research Grants Council (RGC) and the Hong Kong Baptist University. In this appendix, we present a simple derivation of the first order self-consistent phonon theory for 1D Hamiltonian systems. The presented approach is similar to the derivation of the variational path integral method (see [21–23]). In the path integral representation of quantum statistical mechanics, the N -body partition function in the canonical ensemble can be expressed as a path integral over periodic trajectories, that is where is the Euclidean action and the first and second terms in the integral correspond to the total kinetic and potential energies. It is well known that an exact evaluation of the N -body path integral (A.1) for general anharmonic potentials is impossible. The idea of SCPT is basically approximating the original Euclidean action with a trial action that allows an exact evaluation of the trace (A.1). Since we deal in this work exclusively with 1D Hamiltonian 14 S = ∫ ℏ β 0 dτ ( m 2 ˙ x 2 + U [ x ] ) (A.2) Z = ∫ D x e − S [ x $^{]}$, (A.1) Appendix: Quantum Self-Consistent Phonon Theory Acknowledgments