uncommon questions - virial relationships for fields

Would there be a relationship for the average field gradient \(\langle (\nabla\phi)^2 \rangle\) and the average self-interaction \(\langle V(\phi)\rangle\) for a scalar field whose dynamics is described by the Lagrange-function \(\mathcal{L} = 1/2\partial_i\phi\partial^i\phi + V(\phi)\) with a power-law potential \(V(\phi)\propto \phi^n\)?