in Fig. 8 with its classical counterpart. When the temperature is high enough, QSCPT re- produces the classical behavior as shown in Fig. 4 and Fig. 5, where the mean free path and the specific heat are nearly constant and the temperature dependence of κ mainly follows that of the sound speed v$_{s}$ ∝ T 1 / $^{4}$. The classical and quantum results agree up to the Debye temperature. The deviation of the quantum result takes place around T$_{D}$ , which is followed by a peak at temperature T$_{max}$ ≈ 1, then the thermal conductivity drops to zero. T$_{max}$ here indicates the lower bound of the temperature range over which umklapp processes [17] yield the dominant contribution to thermal conductivity. The existence of this peak results from a competition between the increasing behavior of the mean free path and the strongly decreasing tendency of the specific heat. The latter property is a consequence of the partial thermalization of high frequency phonons, which is a pure quantum effect explaining the failure of classical approaches below the Debye temperature. At low temperatures, the mean free path and the sound speed are nearly constant in the quantum case. The difference for the temperature dependence of κ is mainly due to the difference of the specific heat. Our 12 FIG. 7: Specific heat as a function of temperature for (a) λ = 1; (b) λ = 0 . 1. The harmonic potential is given by V$_{0}$ = Kδx $^{2}$/ 2 + f$_{0}$x $^{2}$/ 2. K = 1 and f$_{0}$ = 10 − 7 for all cases. At low temperature regime, C ∝ T . GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s56> GLYPH<s49>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s55> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s49> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s56> GLYPH<s49>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s55> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s49> GLYPH<s40>GLYPH<s98> GLYPH<s41> GLYPH<s32> GLYPH<s67> GLYPH<s84> GLYPH<s32>GLYPH<s81>GLYPH<s83> GLYPH<s67>GLYPH<s80> GLYPH<s84> GLYPH<s32>GLYPH<s83> GLYPH<s67>GLYPH<s80> GLYPH<s84> GLYPH<s32>GLYPH<s81>GLYPH<s117>GLYPH<s97>GLYPH<s110>GLYPH<s116>GLYPH<s117>GLYPH<s109>GLYPH<s32>GLYPH<s104>GLYPH<s97>GLYPH<s114>GLYPH<s109>GLYPH<s111>GLYPH<s110>GLYPH<s105> GLYPH<s99> GLYPH<s32>GLYPH<s99> GLYPH<s104>GLYPH<s97>GLYPH<s105> GLYPH<s110> GLYPH<s40>GLYPH<s97> GLYPH<s41> GLYPH<s67> GLYPH<s32> GLYPH<s84> GLYPH<s32>GLYPH<s81>GLYPH<s83> GLYPH<s67>GLYPH<s80> GLYPH<s84> GLYPH<s32>GLYPH<s83> GLYPH<s67>GLYPH<s80> GLYPH<s84> GLYPH<s32>GLYPH<s81>GLYPH<s117>GLYPH<s97>GLYPH<s110>GLYPH<s116>GLYPH<s117>GLYPH<s109>GLYPH<s32>GLYPH<s104>GLYPH<s97>GLYPH<s114>GLYPH<s109>GLYPH<s111>GLYPH<s110>GLYPH<s105> GLYPH<s99> GLYPH<s32>GLYPH<s99> GLYPH<s104>GLYPH<s97>GLYPH<s105> GLYPH<s110>