Master equation
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The master equation describes the exact behavior of the velocity distribution for any time (Ref. 1 Eq. 3-11)
\[\partial_{t \rho_0} \left( \{ {\mathbf \upsilon} \},t \right) = {\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right) + \int_0^t G_{00}(t-t') \rho_0 \left( \{ {\upsilon} \},t' \right) {\mathrm d}t'\]
where the time dependent functional of the initial conditions is given by (Ref. 1 Eq. 3-9)
\[{\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right) = \frac{-1}{2\pi} \oint_c \exp (-izt) \sum_{ \{k'' \} \neq 0} {\mathcal D}^+_{0 \{k'' \}} (z) \rho_{\{k'' \}} \left( \{ {\mathbf \upsilon} \},0 \right) \]
and the diagonal fragment is given by (Ref. 1 Eq. 3-10)
\[G_{00}(\tau) = \frac{1}{2\pi i} \oint_c \exp (-iz \tau) \psi^+_{00} (z)~ {\mathrm d}z \]
[edit] Equations of evolution
The equations of evolution for the distribution function \(\rho\) for the diagonal fragments(Ref. 1 Eq. 3-1)
\[\psi_{ \{k\}\{k\}}(z) = \sum_{n=2}^\infty (-\lambda)^n \langle \{k\} \vert \delta L \left[ \frac{1}{L_0-z} \delta L \right]^n \vert \{k\} \rangle \]
for the creation fragments (Ref. 1 Eq. 3-2)
\[\tilde{C}_{ \{k\}\{k'\}}(z) = \sum_{n=1}^\infty (-\lambda)^n \langle \{k\} \vert \left[ \frac{1}{L_0-z} \delta L \right]^n \vert \{k'\} \rangle \]
and for the destruction regions (Ref. 1 Eq. 3-3)
\[\mathcal{D}_{ \{k'\}\{k''\}}(z) = \sum_{n=1}^\infty (-\lambda)^n \langle \{k'\} \vert \left[ \delta L \frac{1}{L_0-z} \right]^n \vert \{k''\} \rangle \]
