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

${\mathcal {D}}_{0}\left(t,\rho _{\{k''\}}\left(\{{\mathbf {\upsilon } }\},0\right)\right)$

References

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