content_image/372164.jpg content_image/372163.jpg content_image/372165.jpg content_image/372162.jpg content_image/372161.jpg content_image/372160.jpg content_image/372159.jpg content_image/372158.jpg content_image/372157.jpg content_image/372156.jpg
and 

The gaussian smearing (A.17) is a key characteristic of variational methods. 

The first order free energy per particle can be finally expressed as 

<img src='content_image/372164.jpg'>

<img src='content_image/372163.jpg'>

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 

<img src='content_image/372165.jpg'>

<img src='content_image/372162.jpg'>

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 

<img src='content_image/372161.jpg'>

<img src='content_image/372160.jpg'>

One can easily obtain the classical limit of (A.23) by letting ℏ → 0. The final result is 

<img src='content_image/372159.jpg'>

where ρ 2 and γ 2 in Eq. (A.23) are replaced by their classical counterpart, 

and 

For FPU- β model (4), 

<img src='content_image/372158.jpg'>

<img src='content_image/372157.jpg'>

<img src='content_image/372156.jpg'>

