trivially that this basis is complete.

The non-zero Eigenvalues $E_N$ set the time scales for the approach to thermal equilibrium, which we will discuss in more detail below.

The second order Fokker-Planck equation {#s2FP}
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For the second time-derivative harmonic evolution the Langevin equation becomes $$\begin{aligned}
&& \partial \phi_k = \pi_k \\
&& \partial \pi_k = - \eta \pi_k - (k^2 +m^2) \phi_k  + \xi_k,\end{aligned}$$ which as in subsection \[sOFP\] can be written in terms of real and imaginary components. The Fokker-Planck Hamiltonian for the latter is $$\begin{aligned}
{\cal H}_k =&& -{\Omega \over 4} {\delta^2 \over \delta {\pi_k^{R,I}}^2}
+ \pi_k^{R,I} { \delta \over \delta \phi_k^{R,I}} \\
&& - {\delta \over \delta \pi_k^{R,I}} \left[ \eta  \pi_k^{R,I} 
+ (k^2 +m^2)  \phi_k^{R,I} \right]. \nonumber\end{aligned}$$

To find the excited states we proceed as before to factor out the canonical distribution $P= F_N P_0$ to get $$\begin{aligned}
&& \left\{ -{\Omega \over 4} {\delta^2 \over \delta {\pi_k^{R,I}}^2}
+ \left[ \eta \pi_k^{R,I} -  (k^2 +m^2)\phi_k^{R,I} \right]  
{ \delta \over \delta \phi_k^{R,I}} \right. \nonumber \\
&& \left. \qquad \qquad \qquad 
+ {\delta \over \delta \phi_k^{R,I}}  - E_N \right\} F_N=0.
\label{eigen2}\end{aligned}$$ We can bring Eq. (\[eigen2\]) to the form of Eq. (\[Hermite\]) by making the change of variables $X_\pm= 