Boltzmann Equation - The Collision Term (Stosszahlansatz) and Molecular Chaos

The Collision Term (Stosszahlansatz) and Molecular Chaos

A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "Stosszahl Ansatz", and is also known as the "molecular chaos assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:

\left(\frac{\partial f}{\partial t} \right)_{\mathrm{coll}} = \iint gI(g, \Omega) \,d\Omega\,d^3\mathbf{p}_A.

where pA and pB are the momenta of any two particles (labeled as A and B for conveinience) before a collision, p′A and p′B are the momenta after the collision,

is the magnitude of the relative momenta (see relative velocity for more on this concept), and I(g, Ω) is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the solid angle dΩ, due to the collision.

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