 Solution

ID: 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 equation

\partial_{t} ρ + \nabla \cdot (ρ u) = 0, ρ = \int f d^{3} v, ρ u = \int v f d^{3} v .

\int d^{3} v v_{i} \partial_{v_{j}} f_{2} = - δ_{ij} \int d^{3} v f_{2} .

Write

c = v - u

; then

\int c f d^{3} v = 0

, and the momentum flux splits into bulk transport and the pressure tensor:

\int v_{i} v_{j} f d^{3} v = ρ u_{i} u_{j} + P_{ij}, P_{ij} = \int (v_{i} - u_{i}) (v_{j} - u_{j}) f d^{3} v .

Therefore

\partial_{t} (ρ u_{i}) + \partial_{x_{j}} (ρ u_{i} u_{j} + P_{ij}) = - \int d^{3} x_{⋆} n_{2} (x, x_{⋆}, t) \partial_{x_{i}} ϕ (∣ x - x_{⋆} ∣),

where repeated Cartesian indices are summed and

n_{2} = \int f_{2} d^{3} v d^{3} v_{⋆}

. The negative sign is the attractive or repulsive force

- \nabla ϕ

, rather than an additional sign convention for the pressure tensor.

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