Boltzmann Collision Integral

The Boltzmann collision integral, also called the Boltzmann collision operator, gives the change caused by binary elastic collisions in a nonnegative function that describes particle density over velocity space. For particles of equal mass, one common form is

Here lies on the unit sphere, and the pairs of velocities and are related by and . The nonnegative function weights collisions according to their relative velocity and scattering direction. The first product in the integrand accounts for particles entering velocity , while the second product accounts for particles leaving it.