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I was looking at this paper which describes how the phase space volume changes with time in case a non-zero magnetic field and Berry curvature is present. On the first page, the authors state that the $\nabla_r \cdot \dot{r} + \nabla_k \cdot \dot{k}$ equals $-\frac{d}{dt} \ln(1 + e \vec{B}\cdot\vec{\Omega}/ \hbar) $ using the semiclassical equations of motion. I have tried to do the "straightforward but somewhat tedious" derivation but somehow run into an algebraic equation that I cannot manipulate any further. I would be really glad if someone can throw some light on the steps that lead to the derivation.

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