It should be emphasized that both EPT and SCPT cannot give the microscopic definition of the relaxation time τ$_{k}$ for phonons of mode k . A traditional perturbative way consists in studying the single mode relaxation time based on three- and four- phonon processes [7]. Great efforts have been devoted to obtain the relaxation time of the heat current correla- tions [8, 9], which in general might be related to the relaxation time of phonons. However, the relation was so far unclear. For an anharmonic chain, a simple but physically appealing assumption has been proposed in Ref. [4], where ω$_{k}$ is the phono frequency of mode k . The dimensionless nonlinearity ϵ is defined as the ratio between the average of the anharmonic potential energy and the total potential energy: One can see that, when ϵ vanishes, τ approaches to infinity, leading to the expected diver- gence of the thermal conductivity. In the following, we will apply the assumption of Eq. (2) to Eq. (1) in order to study the thermal conduction of an anharmonic chain, which gives surprising agreement with non-equilibrium simulations. In this section we will compare the SCPT (see, e.g., [10, 11]) and EPT [12, 13]. To make this concrete, we mainly focus on the temperature dependence of the physical quan- tities of the Fermi-Pasta-Ulam β (FPU- β ) model, which is a classic example to study heat conduction. The Hamiltonian of the FPU- β model is given by Within the SCP approximation, the Hamiltonian (4) is approximated by a trial Hamiltonian where the effective harmonic coupling constant f is given by Eq. (A.29) in the appendix with the Boltzmann constant k$_{B}$ = 1. Note that f is temperature dependent, which stems 3 H$_{0}$ = ∑ p 2 i 2 + f 2 ( x$_{i}$$_{+1}$ − x$_{i}$ ) $^{2}$, (5) H = ∑ p 2 i 2 + K 2 ( x$_{i}$$_{+1}$ − x$_{i}$ ) 2 + λ 4 ( x$_{i}$$_{+1}$ − x$_{i}$ ) $^{4}$. (4) ϵ = |〈 E$_{n}$ 〉| 〈 E$_{l}$ + E$_{n}$ 〉 . (3) τ − 1 k ∝ ω$_{k}$ϵ (2) I. EFFECTIVE PHONON APPROACHES experimental studies.