The <Fluctuation-dissipation theorem> in this convention reads
$$
S(\omega,k)=\frac{\varrho(\omega,k)}{1-e^{-\omega/T}}.
$$
In the <classical limit> $|\omega|\ll T$, this becomes
$$
S(\omega,k)\simeq\frac{T}{\omega}\varrho(\omega,k)
=\frac{\pi T}{s\omega_k}
\left[\delta(\omega-\omega_k)+\delta(\omega+\omega_k)\right].
$$
The inverse temporal <Fourier transform> is therefore
$$
C(t,k)=\int\frac{d\omega}{2\pi}e^{-i\omega t}S(\omega,k)
=\frac{T}{s\omega_k}\cos(\omega_kt)
=\frac{T}{\rho_sk^2}\cos(\omega_kt).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 2 / f / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-337/2/f/solution