The equipartition theorem states that if $x_i$ is a canonical variable (either position or momentum), then
$$\left\langle x_i \frac{\partial \mathcal{H}}{\partial x_j}\right\rangle = \delta_{ij}\ k T.$$
Now, say one has a continuous medium (a field) $\phi(t,x)$ with ergodic properties.
What would the equipartition theorem be for this field?
My (naive?) guess:
since it is such that $x_i(t)\mapsto\phi(t,x)$, then
$$\left\langle\phi(x,t)\frac{\delta\mathcal H}{\delta \phi(y,t)}\right\rangle=\delta(y-x)kT,$$
$$\left\langle\dot\phi(x,t)\frac{\delta\mathcal H}{\delta \dot\phi(y,t)}\right\rangle=\delta(y-x)kT,$$
$$\left\langle\dot\phi(x,t)\frac{\delta\mathcal H}{\delta\phi(y,t)}\right\rangle=0=\left\langle\phi(x,t)\frac{\delta\mathcal H}{\delta \dot\phi(y,t)}\right\rangle,$$
where $\frac{\delta}{\delta \phi(y,t)}$ refers to functional derivative and $\delta(y-x)$ is the Dirac delta distribution.
The Hamiltonian in question is for Lorentz invariant 1+1 dim real scalar field with potential not differentable at minimum. Simplest case:
$$\mathcal H=\int dx\left[ \frac12\left(\frac{\partial\phi}{\partial t}\right)^2 + \frac12\left(\frac{\partial\phi}{\partial x}\right)^2 + |\phi| \right]$$
