Master equation

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)

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

References

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