Classical harmonic oscillator Green's functions

A recent paper (Modeling heat transport in crystals and glasses
from a unified lattice-dynamical approach) derived expressions for thermal conductivity in a system of harmonic oscillators that decay with individual time constants determined by their anharmonic interactions.

In this paper, they wrote the Hamiltonian of a classical harmonic operator in complex amplitude coordinates:

$$ H=sum_nomega_n|alpha_n|^2,$$

where $omega_n$ is the frequency of oscillator $n$ and

$$ alpha_n =sqrt{frac{omega_n}{2}}xi_n + frac{i}{sqrt{2omega_n}}pi_n$$

is a "complex amplitude," where $xi_n$ and $pi_n$ are the mode amplitudes and momenta, respectively.

Now, the authors take an ensemble average, $langlealpha^*_n(t)alpha_m(0)rangle$, and claim that

$$ langlealpha^*_n(t)alpha_m(0)rangle=delta_{nm}frac{k_BT}{omega_n}e^{iomega_nt},$$

where the term on the right-hand side is a "single-mode Green’s function".

I’m confused about this ensemble average average, why it equals the right-hand side, and why it’s called a "single-mode Green’s function". Has anyone seen anything like this before?

• Why does $ langlealpha^*_n(t)alpha_m(0)rangle=delta_{nm}frac{k_BT}{omega_n}e^{iomega_nt}$?
• Why is $frac{k_BT}{omega_n}e^{iomega_nt}$ called a Green’s function?
• Where should I read to learn more about this stuff?