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)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{00}(\tau) = \frac{1}{2\pi i} \oint_c \exp (-iz \tau) \psi^+_{00} (z)~ {\mathrm d}z }