= Solution
Use the correct Gaussian entropy decomposition
$$
H_N(F)=\int F\log F\,d\mathbf v+\frac N2\log(2\pi)
+\frac12\int|\mathbf v|^2F\,d\mathbf v.
$$
The last coefficient is $1/2$, as follows from $-\log\gamma_N=N\log(2\pi)/2+|\mathbf v|^2/2$; the printed hint omits it. Under the allowed differentiability and integrability assumptions, the supplied collision invariants conserve mass and energy. Differentiating therefore gives
$$
\frac d{dt}H_N(F)=N\int(Q-I)F\log F\,d\mathbf v,
$$
where the extra derivative term $\int\partial_tF$ vanishes by <mass conservation>. Thus the <Kac entropy production> is
$$
D_N(F)=\frac{N}{2\pi C_N}\sum_{i<j}\int_{\mathbb R^N}\int_0^{2\pi}
(F-F\circ R_{ij,\theta})\log F\,d\theta\,d\mathbf v.
$$
For one pair, call the double <integral> $A_{ij}$. The measure-preserving substitution $(\mathbf v,\theta)\mapsto(R_{ij,\theta}\mathbf v,-\theta)$ interchanges $F$ and $F\circ R_{ij,\theta}$, with angles taken modulo $2\pi$. Averaging the original and substituted expressions gives
$$
A_{ij}=\frac12\int_{\mathbb R^N}\int_0^{2\pi}
(F\circ R_{ij,\theta}-F)
(\log(F\circ R_{ij,\theta})-\log F)\,d\theta\,d\mathbf v.
$$
Since $N/(4\pi C_N)=1/[2\pi(N-1)]$, the desired <Kac entropy dissipation> formula is
$$
\boxed{D_N(F)=\frac1{2\pi(N-1)}\sum_{i<j}\int_{\mathbb R^N}\int_0^{2\pi}
(F(R_{ij,\theta}\mathbf v)-F(\mathbf v))
\log\!\frac{F(R_{ij,\theta}\mathbf v)}{F(\mathbf v)}\,d\theta\,d\mathbf v\geq0.}
$$
For positive values $a,b$, $(a-b)(\log a-\log b)\geq0$ because the logarithm is increasing. At two zeros use value zero; at one zero and one positive value use the nonnegative extended value $+\infty$. One may first use positive densities and then regularize by $(F+\varepsilon\gamma_N)/(1+\varepsilon)$; rotation invariance of $\gamma_N$ preserves the formula and permits the usual limit at zeros under the stated assumptions.
Thus \b[relative entropy is nonincreasing] along the evolution. When the dissipation is finite, zero dissipation means pairwise rotation invariance and hence radiality, by the preceding kernel argument. Radial normalized densities other than $\gamma_N$ can be stationary with positive relative entropy: vanishing dissipation is not a claim that the unique stationary density is Gaussian.
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