  A biological system consists of a variety of polymers that are synthesized
from monomers, by catalysis that exists only for some long polymers. It is
important to elucidate the emergence and sustenance of such autocatalytic
polymerization. We analyze here the stochastic polymerization reaction
dynamics, to investigate the transition time from a state with almost no
catalysts to a state with sufficient catalysts. We found an optimal volume that
minimizes this transition time, which agrees with the inverse of the catalyst
concentration at the unstable fixed point that separates the two states, as is
theoretically explained. Relevance to the origin of life is also discussed.
