Slowing down of the relaxation of the fluctuations around equilibrium is investigated both by stochastic simulations and by analysis of master equation of reversible reaction networks consisting of reactions between a pair of resource and the corresponding high-energy product that works as a catalyst for another resource-product reaction. As the number of molecules $N$ is decreased, the relaxation time to equilibrium is prolonged due to the deficiency of catalysts, as demonstrated by the amplification compared to that by the continuum limit. This amplification ratio of the relaxation time is represented by a scaling function as $h=N\text{exp}\left(-\beta V\right)$, and it becomes prominent as $N$ becomes less than a critical value $h\sim 1$, where $\beta$ is the inverse temperature and $V$ is the energy required to the transformation from resources to the corresponding products.

• Received 1 March 2010

DOI:https://doi.org/10.1103/PhysRevE.81.051920