from the existence of nonlinearity in the system. The dispersion relation of effective phonons corresponding to the effective Hamiltonian (5) can be written in the form It is then straightforward to calculate the dimensionless nonlinearity ϵ and the specific heat C from gaussian averages of SCPT, which yields The derivation of the effective phonon spectrum is based on the generalized equipartition theorem, where the bracket 〈·〉 c stands for a thermal average in the canonical ensemble and q$_{k}$ denotes the Fourier transform of the coordinate x$_{i}$ . The next step consists of an approximative transformation of the right hand side of Eq. (9) into a more compact form (see [3, 12] for details), Specifically, the effective phonon spectrum ˜ ω$_{k}$ for the FPU- β model reads as Here α is given by where Y$_{1}$$_{/}$$_{4}$ ( x ) and Y$_{3}$$_{/}$$_{4}$ ( x ) are the modified Bessel functions of the second kind and x ≡ K $^{2}$/ (8 λT ). The parameters for the nonlinearity ˜ ϵ and the specific heat ˜ C that follow from EPT can be expressed in the form 4 ˜ C = 3 4 + K 2 16 λT Y$_{3}$$_{/}$$_{4}$ ( x ) Y$_{1}$$_{/}$$_{4}$ ( x ) + K 4 64 λ $^{2}$T $_{2}$(1 − Y$_{3}$$_{/}$$_{4}$ ( x ) 2 Y$_{1}$$_{/}$$_{4}$ ( x ) $_{2}$) . (14) ˜ ϵ = K $^{2}$[ Y$_{1}$$_{/}$$_{4}$ ( x ) − Y$_{3}$$_{/}$$_{4}$ ( x )] + 2 λT Y$_{1}$$_{/}$$_{4}$ ( x ) K $^{2}$[ Y$_{3}$$_{/}$$_{4}$ ( x ) Y$_{1}$$_{/}$$_{4}$ ( x )] + 2 λT Y$_{1}$$_{/}$$_{4}$ ( x ) , (13) α = K + λ 〈 ∑ $_{i}$( x$_{i}$$_{+1}$ − x$_{i}$ ) $^{4}$〉 c $_{i}$( x$_{i}$$_{+1}$ x$_{i}$ ) $^{2}$〉 c = 2 λT Y$_{1}$$_{/}$$_{4}$ ( x ) K [ Y$_{3}$$_{/}$$_{4}$ ( x ) − Y$_{1}$$_{/}$$_{4}$ ( x )] , (12) ˜ ω$_{k}$ $_{= 2}$√$_{α}$$_{sin}$ k 2 . (11) k$_{B}$T = ˜ ω 2 k 〈 q 2 k 〉 $_{c}$. (10) k$_{B}$T = 〈 q$_{k}$ ∂H ∂q$_{k}$ 〉 c , (9) ϵ = f − K f + K , (7) C = 3 4 + K $_{4}$√$_{K}$ 2 + 12 λT . (8) ω$_{k}$ = 2 √ f sin k 2 . (6)