Past exam of the kinetic theory course of the University of Oxford / 2025 / 3

For a regular integrable Hamiltonian system, action-angle variables offer two related simplifications.

These advantages assume regular or approximately integrable mean motion; globally defined action-angle variables need not exist in chaotic regions.

Use the Collisionless Boltzmann equation in action-angle variables,Define the orbital frequency vector byThe slow evolution of the angle average $f_0$ starts at second order in the perturbation. At first order, the Poisson bracket therefore givesFor the $\mathbf{k}$th term in the Fourier series, this readsThe lowercase $\delta {\varphi }_{\mathbf{k}}$ here denotes the Fourier coefficient of the uppercase $\delta \Phi$ in the Hamiltonian; these are the same perturbing potential. Multiply by the integrating factor $e^{i\mathbf{k}\cdot \mathbf{\Omega }t}$ and set the initial homogeneous response to zero. One obtainsRetaining the slow time dependence of $f_0$ inside this integral is compatible with the subsequent second-order calculation; changes in $\mathbf{\Omega }$ caused by that slow evolution affect only higher orders in this approximation.

Let $⟨\cdot ⟩$ denote $(2\pi {)}^{-2}\int d^2\theta$. Average the Collisionless Boltzmann equation to second order. All first-order angle derivatives average to zero, leavingAn angle integration by parts shows thatThus the angle-averaged galactic distribution function obeys an action-space continuity equation,This sign also follows from Hamilton's equations, ${\dot{J}}_i=-{\partial }_{{\theta }_i}\delta \Phi$: the flux is $⟨\delta f{\dot{J}}_i⟩$.

Reality gives $\delta {\varphi }_{-\mathbf{k}}=\delta {\varphi }_{\mathbf{k}}^{\ast }$ and $\delta f_{-\mathbf{k}}=\delta f_{\mathbf{k}}^{\ast }$. Orthogonality of the Fourier series impliesInsert b to obtainAlthough individual terms are complex, pairing $\mathbf{k}$ with $-\mathbf{k}$ makes the flux real.

Now take the marginalization specified by $F_0=2\pi {\int }_0^{\infty }{f}_{0}d{J}_R$. Assume vanishing radial-action flux at $J_R=0$ and $J_R\to \infty$: the first boundary does not permit negative radial action, and the second has no escaping flux in this calculation. ThenwithIf radial-action boundary flux were present, its explicit boundary term would have to be retained.

The restriction $k_{\phi }=\pm m$ describes an $m$-fold rotating perturbation of a galactic disk, for example a transient spiral density wave or, for $m=2$, a galactic bar. Its pattern speed is ${\Omega }_{\mathrm{p}}$, its amplitude peaks at $t_{\mathrm{peak}}$, and its Gaussian function envelope has duration of order $\tau$. The coefficients $u_{\mathbf{k}}(\mathbf{J})$ describe its strength along different stellar orbits. A real perturbing potential requires $u_{-\mathbf{k}}=u_{\mathbf{k}}^{\ast }$.

Write $g(t)=\exp [-(t-t_{\mathrm{peak}}{)}^2/(2{\tau }^2)]$. The product entering c isThe phase therefore measures detuning from the rotating pattern:Substitution givesThe strongest long-lived response occurs near an orbital resonance, ${\omega }_{\mathbf{k}}=0$, where successive force contributions do not rapidly cancel in phase.

The required time separation is $\tau \ll t_{\sec }$, where $t_{\sec }$ is the evolution timescale of $f_0$, together with $t_{\mathrm{peak}}\gg \tau$ and an observation time well after the pulse. The first condition permits freezing $f_0(\mathbf{J},t^{\prime })$ at its pre-pulse value; its neglected change would enter beyond the retained second order. The second makes the original initial time lie in a negligible tail of the Gaussian function, permitting replacement of the lower time limits by $-\infty$. Short duration alone would not justify that replacement for a pulse centered at the initial time.

Integrate the continuity equation in c over the whole pulse. For fixed $\mathbf{J}$, the time factor is the ordered Gaussian response integralA common translation by $t_{\mathrm{peak}}$ removes the peak time. Pair the positive and negative azimuthal harmonics: $(-m,-k_R)$ is paired with $(m,k_R)$. Both $k_{\phi }$ and $\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0$ change sign, so their product does not; ${\omega }_{-\mathbf{k}}=-{\omega }_{\mathbf{k}}$ makes the two time factors complex conjugates. Hence the pair contributes $2\mathrm{Re}\mathcal{I}$.

For completeness, the real part of the integrand is symmetric under $t\leftrightarrow t^{\prime }$. The triangular time domain therefore gives half of the full plane, and the Fourier transform of a Gaussian function givesThis derives the supplied identity rather than requiring a separate time-ordering calculation.

DefineUsing the paired harmonics and $\Delta F_0=-{\partial }_{J_{\phi }}\int Q_{0}dt$ now yieldsBy the stated assumption, ${\omega }_{m{k}_R}$ is independent of $J_R$, so its exponential can be taken outside the radial integral. Put $\Gamma =(\sqrt{2}\tau {)}^{-1}$ and useThereforeThe outer derivative acts on the Gaussian resonance broadening factor as well as on the radial integral. The result describes quasilinear diffusion in action space weighted by the squared pulse amplitude, with a detuning width $\Gamma$. A shorter pulse has broader frequency selectivity, while a longer pulse selects narrower orbital resonances. If the action-space boundary flux vanishes, integrating $\Delta F_0$ over $J_{\phi }$ gives zero, as required by conservation of mass.

For $k_R=0$, the detuning is ${\omega }_{m0}=m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})$. The Gaussian resonance broadening factor selects the corotation resonance, where the star's azimuthal frequency matches the pattern speed. The derivative driving the flux is just $m\partial f_0/\partial J_{\phi }$. In fact this contribution alone has the formA single Fourier series harmonic changes the action variables in the direction $(m,k_R)$, because $\dot{\mathbf{J}}=-i\mathbf{k}\delta {\varphi }_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{\theta }}$ plus its conjugate. For this harmonic, $\Delta J_R=0$: stars change angular momentum and shift their epicyclic guiding center without a leading-order increase of random radial motion. This is stellar radial migration driven by the corotation resonance. The resonance condition explains the absence of leading heating: $\Delta E\simeq {\Omega }_{\phi }\Delta J_{\phi }={\Omega }_{\mathrm{p}}\Delta J_{\phi }$ at the corotation resonance.

For $k_R=\pm 1$, the selected orbital resonances satisfyFor near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. Thus $k_R=-1$ is the inner Lindblad resonance, $m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})={\Omega }_R$, and $k_R=+1$ is the outer Lindblad resonance, $m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})=-{\Omega }_R$.

The flux is driven by $m{\partial }_{J_{\phi }}{f}_0\pm {\partial }_{J_R}{f}_0$, and each resonant harmonic exchanges both angular momentum and radial action in the ratioThese terms can therefore produce galactic disk heating as well as angular momentum transport. In the usual absorbing response, stars losing angular momentum at the inner Lindblad resonance gain radial action, and stars gaining angular momentum at the outer Lindblad resonance also gain radial action. The actual direction of flux follows from the displayed derivatives of $f_0$; it is not fixed by the resonance condition alone. A finite pulse replaces exact Lindblad resonance selection by a band of detunings of width of order $1/\tau$.