Boltzmann collision term irreversibility

[lalbatros]

Hello,

Here is the Boltzmann collision term as it is expressed on http://en.wikipedia.org/wiki/Boltzmann_equation" :

Deriving the H theorem from this equation is the way to usually prove of the irreversibility of this term.
Of course, this collision term should not be symmetric by time reversal.
I guess it is easy to prove that without recourse to the H theorem.
How could I do (see) that?

Thanks,

Michel

 

[GeorgeRaetz]

First find an equilibruim distribution f_eq such that the collision term is zero. Next show that the sign of the collision term is the opposite of any deviation of f from f_eq so that f restores back to f_eq.
To start off, show that the collision term is zero for f_eq~exp[-(p-p₀)²]. Where p₀ is a constant average momentum of the gas (in dimensionless units scaled by sqrt(2 m E_avg) ). p₀could be set to zero for simplicity); m is mass of the particles.
Keep in mind that the vector q must be constrained so that only the angle of the relative velocity changes during a collision.