Boltzmann H functional ($H(f)=\int f\log fdxdv$)

The integral uses $0\log 0=0$ when it is defined. For the Boltzmann equation, the Boltzmann H theorem makes it nonincreasing under suitable regularity and integrability. This functional is distinct from the microstate-counting Boltzmann entropy; the physical kinetic entropy has the opposite sign.

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.

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.