BGK model ($C[f]=-(f-\mathcal{M}[f])/\tau$)

Replace the nonlinear Boltzmann collision operator by relaxation to the local Maxwellian $\mathcal{M}[f]$ with the same mass density, mean velocity and random kinetic energy as $f$. Matching those velocity moments ensures vanishing collision contributions to mass, momentum and kinetic energy balance. The collision relaxation time $\tau$ is often treated as a constant, although microscopic collision rates generally depend on the state and particle velocity.

The timescale on which a kinetic relaxation model damps departures from a matching local Maxwellian. In the spatially homogeneous BGK model with constant $\tau$, conserved velocity moments fix $\mathcal{M}$, and $f(t)=\mathcal{M}+[f(0)-\mathcal{M}]e^{-t/\tau }$. The parameter is of the order of an effective mean time between collisions rather than an exact universal microscopic collision time.