Boltzmann propagator, which is the correct description for ideal classical waves at finite temperature.

It is illuminating to compare the above framework directly with that for the overdamped Langevin evolution. Following the same, but somewhat simpler, procedure it can be shown [@Parisi] that the Fokker-Planck Hamiltonian takes the form $$\begin{aligned}
{\cal H}_{FP} = - {\delta \over \delta \phi} 
\left[ {\Omega  \over 2 \eta} {\delta \over \delta \phi}  - \nabla^2 \phi
+ {\delta V \over \delta \phi} \right],
\label{e20}\end{aligned}$$ leading to the eigenvalue functional equation $$\begin{aligned}
&&- \left\{ {\delta \over \delta \phi} 
\left[ {\Omega  \over 2 \eta} {\delta \over \delta \phi}  - \nabla^2 \phi + 
{\delta V \over \delta \phi} \right] + E_N \right\} P[\phi] = 0, \\
&& \eta \partial_t P[\phi] = - E_N P[\phi]. 
\label{e21}\end{aligned}$$ Again the canonical distribution is the eigenfunctional $P_0$ corresponding to a zero eigenvalue, i.e. $$\begin{aligned}
E_0=0; \qquad P_{eq}[\phi] = {\cal N}' e^{- \beta \int d^D x 
{1\over 2} (\nabla \phi)^2  + V(\phi)},
\label{e22}\end{aligned}$$ where as before the Einstein relation $\Omega= 2 \eta/\beta$ must hold.

Approach to equilibrium {#Sol}
=======================

A complete dynamical solution of the ensemble (under average over the noise) of fields obeying the Langevin equation (\[e4a\]-\[e4b\]) can be obtained in principle by 