Qualitative change in dynamical behaviors due to finite system size attracts much attention in various interdisciplinary fields such as systems biology (stochastic chemical reactions), ecology (population dynamics), game theory, and social networks (agent dynamics). Actually, increase in multiplicative noise intensity due to smaller system size can induce qualitative transition of stationary distribution, which has been known as noise-induced transition since 1970s [1]. Although finite system size not only increases noise intensity but also emphasizes state space discreteness, the role of such discreteness of state space has been rarely studied [2]. Thus, we identify such discreteness-induced transition in a simple 1-and 3 body reaction system[3]. This model enables us to derive a sufficient condition under which the discreteness-induced transition arises. Furthermore, we find that the critical size of discreteness-induced transition can become quadratically large, if multi-body reactions are allowed.

References:
[1] W. Horsthemke and R. Lefever, Noise-Induced Transitions in Physics, Chemistry, and Biology (Springer, 2006); J. Ohkubo, N. Shnerb, and D. A. Kessler, J. Phys. Soc. Jpn. 77 (2008); T. J. Kobayashi, Phys. Rev. Lett. 106, 228101 (2011).
[2] Y. Togashi and K. Kaneko, Phys. Rev. Lett. 86, 2459 (2001).
[3] Y. Saito, Y. Sughiyama, K. Kaneko and T. J. Kobayashi, arXiv:1605.06273.