  In this paper we develop a common theoretical framework for the dynamics of
thin featureless interfaces. We explicitly demonstrate that the same phase
field and velocity dependent one-scale models characterizing the dynamics of
relativistic domain walls, in a cosmological context, can also successfully
describe, in a friction dominated regime, the dynamics of nonrelativistic
interfaces in a wide variety of material systems. We further show that a
statistical version of von Neumann's law applies in the case of scaling
relativistic interface networks, implying that, although relativistic and
nonrelativistic interfaces have very different dynamics, a single simulation
snapshot is not able to clearly distinguish the two regimes. We highlight that
crucial information is contained in the probability distribution function for
the number of edges of domains bounded by the interface network and explain why
laboratory tests with nonrelativistic interfaces can be used to rule out
cosmological domain walls as a significant dark energy source.
