Kac model

This many-particle stochastic collision model rotates one uniformly selected velocity pair by a uniformly selected angle, conserving the sum of squared velocities. Its normalized angular averaging is the Kac collision operator, and its continuous-time evolution is the Kac master equation. In this version velocities range over ${\mathbb{R}}^N$ and initial energy is not restricted to one sphere; the evolution preserves the initial energy distribution.

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.

A permutation-symmetric sequence of probability laws is chaotic with one-particle law $f$ when, for every fixed $k$, its $k$-coordinate marginal distributions converge weakly to the product law $f^{\otimes k}$ as $N\to \infty$. This is an asymptotic independence condition, not a consequence of symmetry alone. A rigorous kinetic limit also requires propagation of this property under the dynamics and sufficient convergence to pass collision integrals.

The jump process with total collision rate $N$ and normalized pair-angle operator $Q$ evolves by ${\partial }_t{F}_N=N(Q-I)F_N$. It preserves mass and the total squared velocity. Its one-coordinate distribution satisfies the Kac marginal evolution equation, and its Gaussian relative entropy in Kac's model decreases at the rate given by Kac entropy production.

For a permutation-symmetric density in the Kac master equation, integrate over all but the first velocity. Collisions among integrated coordinates cancel. The $N-1$ pairs containing the first coordinate give equal terms, with coefficient $N(N-1)/(2\pi (\genfrac{}{}{0ex}{}{N}{2}))=1/\pi$. The resulting exact evolution depends on the two-coordinate marginal distribution; it is not a closed equation for the first marginal without additional asymptotic independence.

For $N\ge 2$, $U_{ij,\theta }F=F\circ R_{ij,\theta }$ uses the genuine two-coordinate rotation $(v_i,v_j)\mapsto (v_i\cos \theta +v_j\sin \theta ,-v_i\sin \theta +v_j\cos \theta )$. Average it uniformly over angles and pairs to obtain $Q$. Rotation invariance of Lebesgue measure makes $Q$ a bounded self-adjoint operator of norm one on $L^2({\mathbb{R}}^N)$. Each pair average is an orthogonal projection, but their full average is not in general a projection.

For the Kac collision operator, expanding the squared Hilbert space norm gives $⟨F,(I-Q)F⟩=(4\pi (\genfrac{}{}{0ex}{}{N}{2}){)}^{-1}{\sum }_{i<j}{\int }_0^{2\pi }{\int }_{{\mathbb{R}}^N}|F(R_{ij,\theta }v)-F(v){|}^{2}dvd\theta$. The nonnegative form measures failure of invariance under the pair rotations. It vanishes precisely on the radial kernel of the Kac collision operator.

Zero Dirichlet form of the Kac collision operator forces invariance under every pair rotation; strong continuity upgrades almost every angle to every angle. Coordinate-plane rotations generate the special orthogonal group, whose action is transitive on each sphere for $N\ge 2$. Averaging over normalized Haar measure therefore identifies the invariant $L^2$ functions with radial functions. Conversely radial functions are fixed by every pair rotation.