Rigorous proof of a phase transition of parallelizability in a one-dimensional structure assembly

TL;DR

This paper rigorously proves a phase transition in the parallelizability of assembling one-dimensional chains, showing a non-analytic change in parallel efficiency that distinguishes between parallelizable and unparallelizable phases.

Contribution

It introduces a formal measure of parallel efficiency and demonstrates a phase transition in a one-dimensional assembly system with a rigorous proof.

Findings

Existence of a phase transition in parallelizability.

Parallel efficiency drops to zero at the transition point.

The transition is characterized by a non-analytic change in efficiency.

Abstract

In this paper, we prove the existence of a phase transition of parallelizability in the assembly of one-dimensional chains. By introducing the parallel efficiency that measures how efficiently the parallel assembly works, the parallelizable phase is defined by its positive value. The parallelizable/unparallelizable transition is then identified by the non-analytic change in the parallel efficiency from a positive value to zero. By evaluating the parallel efficiency on each side of the transition point, we show the existence of a phase transition in this system.