Nakajima-Zwanzig-Gleichung

Nakajima Zwanzig Equation $\begin{array}{rl}& d_t\chi =L\chi \\ & \chi =\mathcal{𝒫}\chi +\mathcal{𝒬}\chi \end{array}$$\begin{array}{rl}& d_t\left(\begin{array}{c}\mathcal{𝒫}\\ \mathcal{𝒬}\\ \end{array}\right)\chi =\left(\begin{array}{c}\mathcal{𝒫}\\ \mathcal{𝒬}\\ \end{array}\right)L\left(\begin{array}{c}\mathcal{𝒫}\\ \mathcal{𝒬}\\ \end{array}\right)\chi +\left(\begin{array}{c}\mathcal{𝒫}\\ \mathcal{𝒬}\\ \end{array}\right)L\left(\begin{array}{c}\mathcal{𝒬}\\ \mathcal{𝒫}\\ \end{array}\right)\chi \\ & \Rightarrow \mathcal{𝒬}\chi =e^{\mathcal{𝒬}Lt}Q{\chi }_0+{\int }^{\prime }{e}^{\mathcal{𝒬}Lt}\mathcal{𝒬}L\mathcal{𝒫}\chi (t-{t^{\prime }}^{\prime })\\ & \Rightarrow {\text{d}}_t\mathcal{𝒫}\chi =\mathcal{𝒫}L\mathcal{𝒫}\chi +\underset{=0}{\underbrace{\mathcal{𝒫}{e}^{\mathcal{𝒬}Lt}Q{\chi }_0}}+\mathcal{𝒫}L{\int }^{\prime }{e}^{\mathcal{𝒬}Lt}\mathcal{𝒬}L\mathcal{𝒫}\chi (t-{t^{\prime }}^{\prime })\\ \end{array}$