By writing the density matrix in path integral notation and making use of Feynman–Vernon influence functional, all the bath coordinates in the interaction terms can be grouped into this influence functional which in some specific cases can be calculated in closed form. Assuming a high temperature heat bath with the Drude spectral distribution J ( ω ω ) = ℏ ℏ η η γ γ 2 ω ω / π π ( γ γ 2 + ω ω 2 ) and taking the time derivative of the path integral form density matrix the equation and writing it in hierarchal form yields