= Velocity-alignment entropy production
{title2=$\frac{d}{dt}\int f\log f=d\iint K\rho\rho_\star\,dx\,dx_\star$}
For the <mean-field alignment equation> in $d$ position and $d$ <velocity> coordinates, put $\rho=\int f\,d^dv$. Its force has <velocity-space divergence> $-d\int K\rho_\star\,d^dx_\star$. Consequently <integration by parts> gives the nonnegative derivative in the title, assuming suitable positivity, decay and integrability. The fine-grained <Gibbs entropy> decreases because these dynamics contract <phase space>; the interaction is dissipative.
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