H-theorem ($H^{\prime }(t)=-\frac{1}{16\pi }\int (a-b)\log (a/b)\le 0$)

For the normalized isotropic Maxwell molecule collision operator, put $a=f^{\prime }{f}_{\ast }^{\prime }$ and $b=f{f}_{\ast }$. The integral is over position, both velocities and collision direction. The symmetrized weak identity yields the displayed derivative of the Boltzmann H functional. It is nonpositive because $(a-b)\log (a/b)\ge 0$ for positive $a,b$. Spatial transport contributes only a boundary flux, which must vanish for the total-space inequality.