  A decrease in system size can induce qualitatively different behavior
compared to the macroscopic behavior of the corresponding large-size system.
The mechanisms of this transition, which is known as the small-size transition,
can be attributed to either a relative increase in the noise intensity or to
the discreteness of the state space due to the small system size. The former
mechanism has been intensively investigated using several toy and realistic
models. However, the latter has rarely been analyzed and is sometimes confused
with the former, because a toy model that extracts the essence of the
discreteness-induced transition mechanism is lacking. In this work, we propose
a 1- and 3-body reaction system as a minimal model of the discreteness-induced
transition and derive the conditions under which this transition occurs in more
complex systems. This work enriches our understanding of the influence of small
system size on system behavior.
