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ID: past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2023/3/b/solution

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Past exam of the kinetic theory course of the University of Oxford / 2023 / 3 / b / Solution by

Codex  0 2026-10-25

Define the unperturbed orbital angular frequency vector by

Ω (J) = \frac{\partial H _{0}}{\partial J} .

It is the rate of advance of the angle variables along the unperturbed Hamiltonian flow; the action variables are constant. Expand the Collisionless Boltzmann equation to first order in the perturbations. Since

f_{0}

and

H_{0}

depend only on the action variables,

[f_{0}, H_{0}] = 0

, and

\partial_{t} δ f + [δ f, H_{0}] + [f_{0}, δ Φ] = 0, \partial_{t} δ f + Ω \cdot \partial_{θ} δ f - \partial_{J} f_{0} \cdot \partial_{θ} δ Φ = 0.

The omitted Poisson bracket

[δ f, δ Φ]

is quadratic in the perturbations. Use the Fourier series convention

δ f = \sum_{k \in Z^{3}} δ f_{k} (J, t) e^{i k \cdot θ}, δ Φ = \sum_{k \in Z^{3}} δ Φ_{k} (J, t) e^{i k \cdot θ} .

For each harmonic the linear ordinary differential equation is

\partial_{t} δ f_{k} + i k \cdot Ω δ f_{k} = i k \cdot \partial_{J} f_{0} δ Φ_{k} .

Multiplication by the integrating factor

e^{i k \cdot Ω t}

and use of

δ f_{k} (0) = 0

give the causal linear response in action-angle variables:

δ f_{k} (J, t) = i k \cdot \partial_{J} f_{0} \int_{0}^{t} δ Φ_{k} (J, t^{'}) e^{- i k \cdot Ω (t - t^{'})} d t^{'} .

The zero harmonic stays zero at this order. This formula retains the entire forcing history; replacing it immediately by a resonant Dirac delta function would discard the transient response.

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