10 Symmetrized Weak Form
Integration by parts applied to \(\int Q(f,f) \log f\), rewriting the collision operator in divergence form and integrating by parts in \(v\):
\[ D(f) = -\int \int \langle \nabla _v \log f, A(v-w)[f(w)\nabla f(v) - f(v)\nabla f(w)] \rangle \, dw\, dv. \]
The double integral from Landau IBP can be symmetrized by swapping \(v \leftrightarrow w\) and using \(A(v-w) = A(w-v)\), yielding the PSD integrand form with \((\nabla \log f(v) - \nabla \log f(w))\).