  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.
