FIG. 4: Temperature dependence of thermal conductivity. Here K = 1, λ = 1, and system size N = 1024 for simulation. Analytical results (solid line and dot line) are rescaled (divided by a constant), which is consistent with simulation result in whole temperature range. The simulation result shows the temperature for the minimum of κ at T$_{0}$ ≈ 5, in agreement with Eq. (19). Eq. (19) can be understood by scaling the Hamiltonian (4) as done in Ref. [14]. Let where the dimensionless Hamiltonian ˇ H reads as one obtains Eq. (21) leads to p$_{i}$ = K √$_{λ}$ ˇ p$_{i}$, (21a) x$_{i}$ = √ K λ ˇ x$_{i}$, (21b) H = K 2 λ ˇ H, (22) ˇ H = ∑ ˇ p 2 i 2 + 1 2 (ˇ x$_{i}$$_{+1}$ − ˇ x$_{i}$ ) 2 + 1 4 (ˇ x$_{i}$$_{+1}$ − ˇ x$_{i}$ ) $^{4}$. (23) T = 〈 p 2 i 〉 = K 2 λ 〈 ˇ p 2 i 〉 = K 2 λ ˇ T. (24) 8 GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s50> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s50> GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s49>GLYPH<s48> GLYPH<s50> GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s84> GLYPH<s32>GLYPH<s32>GLYPH<s83>GLYPH<s105> GLYPH<s109> GLYPH<s117> GLYPH<s108> GLYPH<s97> GLYPH<s116>GLYPH<s105> GLYPH<s111> GLYPH<s110> GLYPH<s32>GLYPH<s83>GLYPH<s67> GLYPH<s80>GLYPH<s84> GLYPH<s32>GLYPH<s69>GLYPH<s80>GLYPH<s84>