Past exam of the kinetic theory course of the University of Oxford / 2024 / 1 / b / Solution

Take the zeroth velocity moment of a. The right side is a velocity space divergence with vanishing boundary contribution, giving the continuity equationFor the first velocity moment, use integration by parts componentwise:Write $\mathbf{c}=\mathbf{v}-\mathbf{u}$; then $\int \mathbf{c}f{d}^{3}v=0$, and the momentum flux splits into bulk transport and the pressure tensor:Thereforewhere repeated Cartesian indices are summed and $n_2=\int f_2{d}^{3}v{d}^3{v}_{\star }$. The negative sign is the attractive or repulsive force $-\nabla \varphi$, rather than an additional sign convention for the pressure tensor.