Coarsening in one dimension: invariant and asymptotic states

TL;DR

This paper analyzes a one-dimensional coarsening process where cell boundary velocities depend on neighboring cell sizes, demonstrating invariant Poisson distributions and proposing a universality conjecture for the asymptotic behavior from various initial conditions.

Contribution

It establishes the invariance of the Poisson distribution under the coarsening dynamics and conjectures a universal convergence to this state from general initial conditions.

Findings

Poisson distribution is invariant under the process.

Average cell size grows exponentially at a prescribed rate.

Numerical evidence supports the universality conjecture.

Abstract

We study a coarsening process of one-dimensional cell complexes. We show that if cell boundaries move with velocities proportional to the difference in size of neighboring cells, then the average cell size grows at a prescribed exponential rate and the Poisson distribution is precisely invariant for the distribution of the whole process, rescaled in space by its average growth rate. We present numerical evidence toward the following universality conjecture: starting from any finite mean stationary renewal process, the system when rescaled by $e^{-2t}$ converges to a Poisson point process. For a limited case, this makes precise what has been observed previously in experiments and simulations, and lays the foundation for a theory of universal asymptotic states of dynamical cell complexes.