Dirichlet form of the Kac collision operator
= Dirichlet form of the Kac collision operator
{c}
{title2=$\langle F,(I-Q)F\rangle\geq0$}
For the <Kac collision operator>, expanding the squared <Hilbert space> <norm> gives $\langle F,(I-Q)F\rangle=(4\pi\binom N2)^{-1}\sum_{i<j}\int_0^{2\pi}\int_{\mathbb R^N}|F(R_{ij,\theta}v)-F(v)|^2dv\,d\theta$. The nonnegative form measures failure of invariance under the pair rotations. It vanishes precisely on the <radial kernel of the Kac collision operator>.