Is statistical equilibrium a necessary condition for the unitarity property of the transition probability in transport theory, which states that:

$$ \int w(\Gamma_1^\prime,\Gamma_2^\prime; \Gamma_1, \Gamma_2) d\Gamma_1^\prime d\Gamma_2^\prime = \int w(\Gamma_1, \Gamma_2; \Gamma_1^\prime,\Gamma_2^\prime) d\Gamma_1^\prime d\Gamma_2^\prime $$

I am asking this because according to my knowledge you use this to rewrite the collision integral in the Boltzmann equation:

$$ I[f]= \int (w^\prime f_1^\prime f_2^\prime - w f_1 f_2)\ d\Gamma_2 d\Gamma_1^{\prime} d\Gamma_2^\prime = \int w^\prime (f_1^\prime f_2^\prime - f_1 f_2)\ d\Gamma_2 d\Gamma_1^{\prime} d\Gamma_2^\prime $$ $$ w^\prime \equiv w(\Gamma_1, \Gamma_2; \Gamma_1^\prime,\Gamma_2^\prime),\ w \equiv w(\Gamma_1^\prime,\Gamma_2^\prime; \Gamma_1, \Gamma_2) $$

The Boltzmann equations applies especially to systems not being in statistical equilibrium. Accordingly statistical equilibrium should not be a necessary condition, right? There is a very unclear passage in one of my scripts that seems to imply that statistical equilibrium is a necessary condition and leads to my confusion.

Thanks in advance!


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