uncommon questions - virial gravitational fields

The Lagrange density of classical gravity, \(\mathcal{L} = 1/2 \partial_\mu\phi\partial^\mu\phi + \lambda\phi\) includes a term with the cosmological constant, which is effectively a homogeneous function of order one, implying again a virial relationship of the fields. What’s the relation between the the volume-averaged quadratic field gradient and the mean self-interaction? Could you think of a classical system where this is similar?