Kinetic theory ($f(t,x,v)$)

Describe many-particle systems through distributions on phase space. Transport moves particles along trajectories, while collision or diffusion terms modify their velocity distributions. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation are central models.

For a dilute phase-mixed population, an identical-species collision rate counts unordered pairs: ${\Gamma }_{ii}=K_{ii}{N}_i(N_i-1)/2$. Different species have rate ${\Gamma }_{ij}=K_{ij}{N}_i{N}_j$. Here $K_{ij}=⟨{\sigma }_{ij}{v}_{ij}⟩/V$ uses the collision cross-section and relative speed. Destroying both identical particles gives ${\dot{N}}_i=-2{\Gamma }_{ii}$, not $-{\Gamma }_{ii}$. This distinction prevents inconsistent factors when using one kinetic kernel for several particle sizes.

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.

On the unit-period spatial circle, free streaming has solution $f_t(x,v)=f_0(x-tv,v)$. With Fourier phase $e^{-2\pi i(kx+\xi v)}$, its mixed Fourier transform is ${\hat{f}}_t(k,\xi )={\hat{f}}_0(k,\xi +kt)$. A nonzero spatial mode of the velocity-integrated density therefore samples increasingly large velocity frequencies. The conserved spatial zero mode remains, while extra velocity regularity supplies quantitative decay of the other modes.

For $m\ge 2$ integrable weak velocity derivatives on a unit-period spatial circle, the Fourier transform of a derivative gives $|{\hat{f}}_0(k,kt)|\le \Vert {\partial }_v^m{f}_0{\Vert }_1/(2\pi |kt|{)}^m$ for $k\ne 0$. Summing these modes uses ${\sum }_{k\ne 0}|k{|}^{-m}=2\zeta (m)$. The resulting absolute Fourier summability gives the displayed uniform bound for the continuous representative of the density. For $m=1$ the same summation diverges, so this argument needs additional input.

The mass-normalized first velocity moment of a distribution on phase space. For a velocity-only distribution, omit the position integral. Nonzero finite mass and a finite first moment are required. For a probability distribution this is its expected value; its unnormalized counterpart is the momentum.

Transport a phase-space density along Hamilton's equations. The Hamiltonian vector field is $b=({\nabla }_{v}H,-{\nabla }_{x}H)$ and the equation is ${\partial }_{t}f+b\cdot {\nabla }_{x,v}f=h$. Its zero divergence gives volume-preserving transport. For $h=0$, the density is constant along characteristics; a source is integrated by the Duhamel formula for Hamiltonian transport. This is distinct from the elliptic Liouville equation.

The Hamiltonian $(v^2-x^2)/2$ gives $\dot{x}=v$, $\dot{v}=x$. Its characteristic flow map is multiplication by $A_t$, with $A_t{A}_s=A_{t+s}$ and determinant one. The backward map is $A_{-t}$. This expanding-contracting linear flow preserves phase-space Lebesgue measure and every finite-$p$ integral of a transported density, despite stretching its level sets.

If $f_0=0$ and a nonzero time-independent source is invariant along the Hamiltonian flow, then the Duhamel formula for Hamiltonian transport gives $f_t=th$. For the nonzero-frequency oscillator, $h=e^{-H_{\omega }}$ is smooth and belongs to all finite Lp spaces, yet its resulting solution has infinite space-time norm on $(0,\infty )$.

The Duhamel formula for Hamiltonian transport, phase space preservation and Minkowski integral inequality give the displayed bound. A time-independent source in an Lp space yields finite space-time norm on every bounded interval; it need not yield integrability on infinite time.

Smooth forcing need not be integrable in phase space. For the hyperbolic characteristic flow for an inverted oscillator, the source $x^2+v^2$ accumulates as $q_t=\frac{1}{2}\sinh (2t)(x^2+v^2)-(\cosh (2t)-1)xv$. Its smallest quadratic-form eigenvalue is $(1-e^{-2t})/2>0$ for $t>0$. Starting with a nonnegative integrable Gaussian function, the solution is therefore unbounded and outside every finite Lp space at positive times. Locally time-integrable $L^p$ forcing prevents this failure by the Minkowski integral inequality.

For a complete invertible Hamiltonian flow $\Phi (t,s)$, integrate the source along the backward characteristic ending at $z$ at time $t$. The displayed formula solves the Hamiltonian Liouville equation. Volume preservation makes each composition with $\Phi (s,t)$ an isometry of Lp spaces.