FIG. 8: Temperature dependence of thermal conductivity for (a) λ = 1; (b) λ = 0 . 1. K = 1 and f$_{0}$ = 10 − 6 for both cases. result based on QSCPT, characterized by the existence of the umklapp peak, is qualitatively consistent with the experimental studies (see, e.g., [16, 18–20]). Finally, one should note that the scaling behavior (24) is not applicable for quantum case. In summary, SCPT and EPT are compared in the present study. The basic difference between SCPT and EPT lies in the different way to calculate the statistical average of physical quantities. Specifically, the canonical average is applied in EPT but Gaussian average is applied in SCPT. We show that both SCPT and EPT give the same behavior for the thermodynamic quantities of a classical system, which indicates their equivalence. It seems that gaussian average is preferable for analytical derivation, especially for models with an on-site potential, since the thermal averages should be computed in this case from numerical transfer matrix calculations. The additional gain is that SCPT can be extended to study the quantum system, as shown in section II. Note that EPT is based on the assumption 13 III. SUMMARY GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s50>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s52>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s54>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s56>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s49>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s53>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s49>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s52> GLYPH<s49>GLYPH<s46>GLYPH<s53>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s52> GLYPH<s50>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s52> GLYPH<s50>GLYPH<s46>GLYPH<s53>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s52> GLYPH<s51>GLYPH<s46>GLYPH<s48>GLYPH<s120>GLYPH<s49>GLYPH<s48> GLYPH<s52> 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<s40>GLYPH<s97> GLYPH<s41> GLYPH<s40>GLYPH<s98> GLYPH<s41> GLYPH<s84> GLYPH<s32>GLYPH<s81>GLYPH<s83>GLYPH<s67> GLYPH<s80>GLYPH<s84> GLYPH<s32>GLYPH<s83>GLYPH<s67> GLYPH<s80>GLYPH<s84>