Discreteness-induced transitions in multi-body reaction systems

TL;DR

This paper introduces a minimal reaction system model to analyze how small system sizes cause qualitative behavioral transitions due to state space discreteness, enhancing understanding of size-induced effects.

Contribution

The paper proposes a minimal 1- and 3-body reaction model to specifically investigate discreteness-induced transitions, filling a gap in existing analyses.

Findings

Derived conditions for discreteness-induced transitions in complex systems

Clarified the role of state space discreteness versus noise in small systems

Provided a minimal model to study size effects in reaction systems

Abstract

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.