Self-similar Inhomogeneous Stationary States under Constrained Dynamics

The dynamics of n rigid objects, each having d degrees of freedom, is played out in the configuration space of dimension nd. Being rigid, there are additional constraints at work that renders a portion of the configuration space inaccessible. In this paper, we make the assertion that treating the overall dynamics as a Markov process whose states are defined by the number of contacts made between the rigid objects provides an effective coarse-grained characterization of the otherwise complex phenomenon. This coarse graining reduces the dimensionality of the space from nd to one. We test this assertion for a one-dimensional array of curved squares each of which is undergoing a biased diffusion in its angular orientation.