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)

${\mathcal {D}}_{0}\left(t,\rho _{\{k''\}}\left(\{{\mathbf {\upsilon } }\},0\right)\right)={\frac {-1}{2\pi }}\oint _{c}\exp(-izt)\sum _{\{k''\}\neq 0}{\mathcal {D}}_{0\{k''\}}^{+}(z)\rho _{\{k''\}}\left(\{{\mathbf {\upsilon } }\},0\right)$

and the diagonal fragment is given by (Ref. 1 Eq. 3-10)

$G_{00}(\tau )={\frac {1}{2\pi i}}\oint _{c}\exp(-iz\tau )\psi _{00}^{+}(z)~{\mathrm {d} }z$

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)

${\mathcal {D}}_{\{k'\}\{k''\}}(z)=\sum _{n=1}^{\infty }(-\lambda )^{n}\langle \{k'\}\vert \left[\delta L{\frac {1}{L_{0}-z}}\right]^{n}\vert \{k''\}\rangle$

References

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