The gaussian smearing (A.17) is a key characteristic of variational methods. The first order free energy per particle can be finally expressed as and By minimizing this expression with respect to ω$_{p}$ , one obtains the optimal pseudo-phonon frequency On the other hand, the variation of Eq. (A.20) with respects to η yields Note that η = 0 for pair on site potentials ( V ( x ) = V ( − x )). Using (A.17), (A.18) and (A.21) in Eq. (A.20), we can further simplify the first order free energy in the form The specific heat per mode can be then defined by One can easily obtain the classical limit of (A.23) by letting ℏ → 0. The final result is where ρ 2 and γ 2 in Eq. (A.23) are replaced by their classical counterpart, and For FPU- β model (4), 17 W ( δx$_{i}$ ) = K 2 δx 2 i + λ 4 δx 4 $_{i}$, V = 0 . (A.28) γ 2 c ≡ 〈 ( x$_{k}$ − x$_{k}$$_{−}$$_{1}$ ) $^{2}$〉 ℏ → 0 = 1 Nmβ ∑ p 4 sin 2 ( pπ N ) ω 2 p . (A.27) ρ 2 c ≡ 〈 x 2 k 〉 ℏ → 0 = 1 Nmβ ∑ p ω − 2 p (A.26) F$_{1}$ = 1 Nβ ∑ p ln ( βω$_{p}$ ) + V$_{ρ}$$_{c}$ − ρ 2 c ∂V$_{ρ}$$_{c}$ ∂ρ 2 c + W$_{γ}$$_{c}$ − γ 2 c ∂W$_{γ}$$_{c}$ ∂γ 2 c , (A.25) C$_{p}$ = − T ∂ $^{2}$F$_{1}$ ∂T 2 . (A.24) F$_{1}$ = F$_{0}$ + V$_{ρ}$ − ρ $_{2}$∂V$_{ρ}$ ∂ρ 2 + W$_{γ}$ − γ $_{2}$∂W$_{γ}$ ∂γ 2 . (A.23) ∂V$_{ρ}$ ∂η = 0 . (A.22) F$_{1}$ = F$_{0}$ − ∑ p { β ℏ ω$_{p}$ 4 coth ( β ℏ ω$_{p}$ 2 )} + V$_{ρ}$ ( η ) + W$_{γ}$. (A.20) γ 2 ≡ 〈 ( x$_{k}$ − x$_{k}$$_{−}$$_{1}$ ) $^{2}$〉 = ℏ 2 Nm ∑ p 4 sin 2 ( pπ N ) ω$_{p}$ coth ( β ℏ ω$_{p}$ 2 ) . (A.19) ω 2 p = 2 m { ∂V$_{ρ}$ ∂ρ 2 + 4 sin 2 ( pπ N ) ∂W$_{γ}$ ∂γ 2 } . (A.21)