Nakajima Zwanzig Equation d t χ = L χ χ = P χ + Q χ d t ( P Q ) χ = ( P Q ) L ( P Q ) χ + ( P Q ) L ( Q P ) χ ⇒ Q χ = e Q L t Q χ 0 + ∫ ′ e Q L t Q L P χ ( t − t ′ ) ⇒ d t P χ = P L P χ + P e Q L t Q χ 0 ⏟ = 0 + P L ∫ ′ e Q L t Q L P χ ( t − t ′ ) {\displaystyle {\begin{aligned}&{{d}_{t}}\chi =L\chi \\&\chi ={\mathcal {P}}\chi +{\mathcal {Q}}\chi \\&{{d}_{t}}\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)\chi =\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)L\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)\chi +\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)L\left({\begin{matrix}{\mathcal {Q}}\\{\mathcal {P}}\\\end{matrix}}\right)\chi \\&\Rightarrow {\mathcal {Q}}\chi ={{e}^{{\mathcal {Q}}Lt}}Q{{\chi }_{0}}+\int '{{e}^{{\mathcal {Q}}Lt}}{\mathcal {Q}}L{\mathcal {P}}\chi (t-{t}')\\&\Rightarrow {{\text{d}}_{t}}{\mathcal {P}}\chi ={\mathcal {P}}L{\mathcal {P}}\chi +\underbrace {{\mathcal {P}}{{e}^{{\mathcal {Q}}Lt}}Q{{\chi }_{0}}} _{=0}+{\mathcal {P}}L\int '{{e}^{{\mathcal {Q}}Lt}}{\mathcal {Q}}L{\mathcal {P}}\chi (t-{t}')\\\end{aligned}}}
