Velocity-alignment energy dissipation ($\dot{\mathcal{E}}=-\frac{1}{2}\iint K|{\mathbf{v}}_{\star }-\mathbf{v}{|}^{2}f{f}_{\star }dzd{z}_{\star }$)

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.