the real part of $E_n$ is positive. This follows from the requirement that $V(\phi)$ is bounded from below. For the $E_n$, solutions of the second time derivative eigenproblem there may be in addition an imaginary part. Thus the excited states decay away in time as an exponentially damped oscillator.

The eigenfunctional ${\cal P}_0$, characterizing the long time field probability is then interpreted as the equilibrium distribution. Taking $E_0=0$ in Eq. (\[e14\]) we find $$\begin{aligned}
{\cal P}_{\rm 0}[\pi,\phi]={\cal N} \exp  \left[ - \beta \int d^D x~\left( 
{\pi(x)^2 \over 2} + S(\phi) \right) \right] 
\label{e17}\end{aligned}$$ with $S(\phi)=\left( {1 \over 2} 
(\nabla \phi(x))^2 + V(\phi) \right)$. Here we took $\Omega=2 \eta/\beta$, which is the analog of Einstein’s relation for Brownian motion, and ensures the balance between fluctuations (sourced by the fields $\xi(x,t)$) and dissipation (through the $- \eta \pi(x,t)$ in Eq. (\[e4b\])) at long times.

It is interesting to note that it can be read directly from Eq. (\[e17\]) that the momentum equilibrium variance $$\begin{aligned}
\langle \pi(x)^2 \rangle = 1/\beta=T,
\label{e18}\end{aligned}$$ which expresses equipartition in a relativistic classical field theory. This quantity is also an excellent thermometer for the dynamical evolution.

Now, because the canonical distribution is Gaussian in $\pi$ we can proceed 