Velocity alignment (${\dot{\mathbf{v}}}_i={\sum }_j{K}_{ij}({\mathbf{v}}_j-{\mathbf{v}}_i)$)

An interaction making particle velocities approach those of their neighbors. Symmetric nonnegative weights $K_{ij}$ conserve total momentum for equal particle masses and dissipate relative kinetic energy. Connectivity and quantitative lower bounds on the interaction matter for eventual consensus; nonnegativity alone permits noninteracting groups.

Neglect of pair correlations closes the first marginal equation of the dissipative Liouville equation. For unit-normalized marginal distributions, the acceleration is $(N-1)\mathbf{L}[p_1]$. With number-normalized $f=N{p}_1$, it is $(1-1/N)\mathbf{L}[f]$. The usual mean-field alignment equation neglects this relative $1/N$ correction. Independence at one time does not imply that interactions preserve independence.

The continuum velocity alignment equation with $\mathbf{L}[f](\mathbf{x},\mathbf{v})=\int K(\mathbf{x},{\mathbf{x}}_{\star })({\mathbf{v}}_{\star }-\mathbf{v})f_{\star }{d}^3{x}_{\star }{d}^3{v}_{\star }$. For symmetric nonnegative $K$, it conserves total mass and momentum while dissipating kinetic energy. Its force is affine in velocity, with a nonzero velocity-space divergence.

For the mean-field alignment equation in $d$ position and $d$ velocity coordinates, put $\rho =\int f{d}^{d}v$. Its force has velocity-space divergence $-d\int K{\rho }_{\star }{d}^d{x}_{\star }$. 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.

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.