Mean-field alignment equation
= Mean-field alignment equation
{title2=$\partial_tf+\mathbf v\cdot\nabla_xf+\nabla_v\cdot(f\mathbf L[f])=0$}
The continuum <velocity alignment> equation with $\mathbf L[f](\mathbf x,\mathbf v)=\int K(\mathbf x,\mathbf x_\star)(\mathbf v_\star-\mathbf v)f_\star\,d^3x_\star\,d^3v_\star$. 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>.