Unified paradigm for interface dynamics

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 the 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.

Figure 1

Two ${256}^2$ simulation snapshots, with similar characteristic lengths, taken from ${1024}^2$ relativistic (left) and nonrelativistic (right) two-dimensional interface simulations of the ideal class of models with $N_{\varphi }=10$, with identical initial conditions. Different colors represent different minima.

Figure 2

The blue (light gray) stripe represents the interval $f_n\pm s_n$, where $f_n$ is the fraction of the number of domains with $n$ edges for either relativistic (upper panel) or nonrelativistic (lower panel) regimes and $s_n^2$ is the sample variance calculated using ten snapshots with $1024$ domains each, taken from ${1024}^2$ numerical simulations of the ideal class of models with $N_{\varphi }=10$. The red dots represent the value of $f_n$ in the nonrelativistic (upper panel) or relativistic cases (lower panel) for comparison.