where Λ is the mean free path for phonon and ∂ ∂ ∂ ∂ T ϵ ϵ denotes the heat capacity. This equation is a result of combining the four previous equations with each other and knowing that ⟨ v x 2 ⟩ = 1 3 v 2 for cubic or isotropic systems and Λ Λ = v τ τ .