Master equation

The master equation describes the exact behavior of the velocity distribution for any time (Ref. 1 Eq. 3-11)

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 \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)

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 }

Equations of evolution

The equations of evolution for the distribution function 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 \rho} for the diagonal fragments(Ref. 1 Eq. 3-1)

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 \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)

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 \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)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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 }

References

1. I. Prigogine and P. Résibois "On the kinetics of the approach to equilibrium", Physica 27 pp. 629-646 (1961)