Self-similar inhomogeneous stationary states under constrained dynamics

TL;DR

This paper proposes a coarse-grained Markov process model based on contact counts to analyze the complex dynamics of inhomogeneous, constrained rigid body systems, demonstrated through a 1D array of curved squares undergoing biased diffusion.

Contribution

It introduces a novel Markov process framework based on contact states to simplify and analyze the dynamics of constrained rigid objects.

Findings

The contact-based Markov model effectively captures the system's dynamics.

The approach reduces high-dimensional configuration space to a single dimension.

The model accurately describes the biased diffusion of curved squares.

Abstract

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 render 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.