4 H-Theorem and Maxwellian Characterization
If \(\Psi \ge 0\) and \(f {\gt} 0\) is smooth, then the entropy dissipation is non-positive: \(D(f) \le 0\).
If \(\Psi {\gt} 0\), \(f {\gt} 0\), \(f\) smooth and integrable, and \(D(f) = 0\), then \(f\) is Maxwellian. This is the core analytic step: \(D = 0\) forces the PSD integrand to vanish, which by the nullspace characterization of \(A(z)\) forces \(\nabla \log f\) to be affine, hence \(f = \exp (\text{quadratic})\).
If \(\log f(v) = a_0 + b \cdot v + c_0 |v|^2\) with \(c_0 {\lt} 0\), then \(Q(f,f)(v) = 0\) for all \(v\).