 Solution

ID: past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2024/3/e/solution

Represent the perturber by the periodic Dirac delta function at

x_{p} + v_{p} t

. With spatial Fourier series coefficient normalized by

L^{- 3}

ρ_{p, k} (t) = \frac{m _{p}}{L ^{3}} e^{- i k \cdot x_{p}} e^{- i k \cdot v_{p} t}, t \geq 0.

δ Φ_{k}^{ext} (t) = C_{k} e^{- ia t}, C_{k} = - \frac{4 π G m _{p}}{L ^{3} k ^{2}} e^{- i k \cdot x_{p}}, a = k \cdot v_{p} .

The perturbation is zero before switch-on. For

Im ω > 0

\int_{0}^{\infty} e^{i (ω - a) t} d t = - \frac{1}{i ( ω - a )} = \frac{1}{i ( a - ω )} .

Hence

δ Φ_{k}^{ext} (ω) = - \frac{4 π G m _{p}}{L ^{3} k ^{2}} e^{- i k \cdot x_{p}} \frac{1}{i ( k \cdot v _{p} - ω )} .

The switch-on creates a forcing pole at the real frequency

a

, approached causally from the original upper inversion contour.

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