to obtain a reduced distribution in terms of $\phi$ alone. Performing the Gaussian integral over all $\pi$ we obtain $$\begin{aligned}
P_{eq}[\phi] = {\cal N}' \exp \left[ - \beta \int d^Dx ~S(\phi) \right],
\label{e19}\end{aligned}$$ which is the canonical Boltzmann distribution. $P_{eq}[\phi]$ could have been equally obtained from the static solution of the Fokker-Planck equation associated with the perhaps more usual first derivative Langevin equation (see below). In this respect we see that [*in equilibrium*]{} it makes no difference to perform the evolution, with or without the second time derivative in Eq. (\[e1\]). Away from equilibrium, or even if we simply wish to study the fluctuations around it, we should take the appropriate form of the evolution, choosing to keep or neglect the second time derivative according to the physical picture under consideration. The difference between both of these evolutions, is expressed in terms of the higher eigenvalues and eigenfunctionals associated with the different Hamiltonians, which I discuss below.

For a harmonic potential $V[\phi] = {m^2 \over 2} \phi^2$ we can also read the equilibrium thermal propagator directly from Eq. (\[e19\]). It is, in momentum space $$\begin{aligned}
\langle \phi_p \phi_{-p} \rangle = {T \over p^2 + m^2}.\end{aligned}$$ This is the canonical free 