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First question If we start with the Liouville equation, define entropy as $\mathcal{H}$, and take its time derivative, we obtain your last equation: $$ \frac{\mathrm{d} \mathcal{H}}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\int -\rho\log{\rho} \, \mathrm{d}\vec{p}\,\mathrm{d}\vec{q} = \int\mathrm{d}\vec{p}\,\mathrm{d}\vec{q} \, (-\log\rho - 1) \frac{\...


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