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

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.