= Velocity-alignment energy dissipation
{title2=$\dot{\mathcal E}=-\tfrac12\iint K|\mathbf v_\star-\mathbf v|^2ff_\star\,dz\,dz_\star$}
For the <mean-field alignment equation>, multiply by $|\mathbf v|^2/2$ and use <integration by parts>. Exchanging $z$ with $z_\star$ and averaging the two results gives the displayed nonpositive <kinetic energy> derivative. The factor $1/2$ accounts for counting ordered pairs twice. Total <momentum> is conserved, so the dissipated part is motion relative to the global <mean velocity>.
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