Relative entropy in Kac's model ($H_N(F)=\int F\log (F/{\gamma }_N)$)

Use ${\gamma }_N=(2\pi {)}^{-N/2}{e}^{-|v{|}^2/2}$ as the reference probability density. The relative entropy is nonnegative by writing $F={\gamma }_{N}u$ and integrating $u\log u-u+1$. Its expansion is $\int F\log F+(N/2)\log (2\pi )+(1/2)\int |v{|}^{2}F$. Equality holds only at $F={\gamma }_N$, whereas other radial densities may still be stationary for the Kac master equation.

For the Kac master equation and finite differentiable entropy, mass and energy conservation remove the non-logarithmic entropy terms. Symmetrizing each pair rotation then gives $D_N(F)=[2\pi (N-1){]}^{-1}{\sum }_{i<j}\int \int (F\circ R_{ij,\theta }-F)\log ((F\circ R_{ij,\theta })/F)dvd\theta$. The integrand is nonnegative by monotonicity of the logarithm, with the usual extended convention at zeros. Zero dissipation corresponds to radiality; it need not imply zero relative entropy.