= Boltzmann entropy dissipation
{c}
{title2=$D(f)=\frac14\int B(A-B_0)\log(A/B_0)\,dv\,dv_*\,dn$}
= Boltzmann entropy production
{c}
{synonym}
Put $A=f'f_*'$ and $B_0=ff_*$. The <weak Boltzmann collision identity> applied to $\log f$ gives $H'(t)=-\int D(f)\,dx$ 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.
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