= Solution
To obtain the <Boltzmann equation>, take a dilute <gas> with a short-range <pair potential>. Its interaction range is small compared with the <mean free path> and the spatial scale on which the <phase-space distribution function> changes. The <binary collision approximation> treats encounters as isolated <elastic collisions>; simultaneous encounters involving three or more particles can be neglected. The encounter duration is short compared with the <collision relaxation time> and macroscopic evolution time, and the external <force> changes negligibly during an encounter. This permits local, effectively instantaneous scattering at one position $\mathbf x$.
The essential statistical closure is <molecular chaos for the Boltzmann equation>: the distribution of \i[incoming] pairs factorizes as $ff_\star$. Outgoing pairs need not be independent. Follow each incoming pair through its two-body scattering, and use conservation of <momentum> and <kinetic energy>, together with time reversal and the invariant scattering measure, to express both gain and loss in the same variables. This gives
$$
C_B[f](\mathbf v)=\int d^3v_\star\int_{\mathrm{hemisphere}}d\Omega\,B(g,\theta)\left(f'f'_\star-ff_\star\right),\qquad g=|\mathbf v-\mathbf v_\star|.
$$
Here $d\Omega=\sin\theta\,d\theta\,d\varphi$; when the measure is written as $d\theta\,d\varphi$, its Jacobian is included in the <Boltzmann scattering kernel>. In a scattering-angle convention, that kernel contains relative <speed> times differential <scattering cross-section>. The hemisphere prevents double counting equivalent outcomes for indistinguishable particles; the precise factor is included in $B$. The primed <velocities> are connected to the unprimed ones by the reversible <elastic collision> map. This closure is appropriate for a dilute short-range <gas>, rather than a generic long-range interacting system with an important collective <mean-field theory> contribution.
The further replacement by the <BGK model> is a phenomenological relaxation closure. Replace the full <nonlinear Boltzmann collision operator> by relaxation towards a <local Maxwellian>, with a single, velocity-independent <collision relaxation time> $\tau>0$:
$$
C_{\mathrm{BGK}}[f]=-\frac{f-f^{(0)}}{\tau},\qquad f^{(0)}=\frac{\rho}{(2\pi\Theta)^{3/2}}\exp\left(-\frac{|\mathbf v-\mathbf u|^2}{2\Theta}\right).
$$
It models relaxation towards a <local Maxwellian> without resolving the detailed scattering. Treating $\tau$ as constant is an additional simplification; it is of the order of a mean time between effective collisions. Near a <local Maxwellian>, departures in the nonconserved <velocity moments> decay on this timescale.
Choose $f^{(0)}$ to match the <collision invariants> $1$, $\mathbf v$ and $|\mathbf v|^2$ of $f$. For unit particle <mass>, this means
$$
\boxed{\rho=\int f\,d^3v,\qquad \mathbf u=\rho^{-1}\int\mathbf v f\,d^3v,\qquad \Theta=\frac{1}{3\rho}\int|\mathbf v-\mathbf u|^2f\,d^3v.}
$$
Here $\rho$ is the <mass density>, $\mathbf u$ is the <mean velocity>, and $\Theta$ is the <temperature> in squared-<velocity> units, $\Theta=k_BT/m$ before setting $m=1$. In particular,
$$
\int (1,\mathbf v,|\mathbf v|^2)\,C_{\mathrm{BGK}}[f],d^3v=0.
$$
Thus relaxation preserves local <mass>, <momentum> and <kinetic energy> rather than relaxing them to externally prescribed values.
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