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 Incoming links: Phase-space divergence

Past exam of the kinetic theory course of the University of Oxford / 2023 / 1 / b / Solution 2026-10-25

Let

A_{i} = \sum_{j \neq = i} K_{ij} (v_{j} - v_{i})

. Conservation of probability in the full phase space gives the dissipative Liouville equation

\partial_{t} ϱ + i = 1 \sum N \nabla_{x_{i}} \cdot (v_{i} ϱ) + i = 1 \sum N \nabla_{v_{i}} \cdot (A_{i} ϱ) = 0.

Normalize

\int ϱ \prod_{i} d^{3} x_{i} d^{3} v_{i} = 1

. Decay at infinity makes all boundary fluxes vanish, so the equation preserves this normalization of a probability distribution.

Unlike the Hamiltonian Liouville equation, this transport has nonzero phase-space divergence. In three dimensions,

\nabla_{v_{i}} \cdot A_{i} = - 3 \sum_{j \neq = i} K_{ij}, \partial_{t} ϱ + \sum_{i} (v_{i} \cdot \nabla_{x_{i}} ϱ + A_{i} \cdot \nabla_{v_{i}} ϱ) = 3 ϱ \sum_{i} \sum_{j \neq = i} K_{ij} .

The probability density increases along a contracting characteristic flow map; setting its material derivative to zero would lose conservation of probability. The

j = i

force is identically zero, including its derivative, which is why the last sums exclude it.

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