Boltzmann entropy dissipation ($D(f)=\frac{1}{4}\int B(A-B_0)\log (A/B_0)dvd{v}_{\ast }dn$)

Put $A=f^{\prime }{f}_{\ast }^{\prime }$ and $B_0=f{f}_{\ast }$. The weak Boltzmann collision identity applied to $\log f$ gives $H^{\prime }(t)=-\int D(f)dx$ when transport boundary fluxes vanish. Monotonicity of the logarithm makes every integrand nonnegative. For positive sufficiently regular distributions and nondegenerate angular scattering, zero dissipation forces $\log f$ to be a collision invariant, hence $f$ to be a local Maxwellian. This equality statement alone does not establish convergence rates.