Discreteness-Induced Slow Relaxation in Reversible Catalytic Reaction Networks

TL;DR

This paper investigates how decreasing molecule numbers in reversible catalytic reaction networks causes significant slowdowns in relaxation times, highlighting the impact of discreteness and resource scarcity on system dynamics.

Contribution

It introduces a scaling function for relaxation time amplification due to molecule number reduction, linking it to resource scarcity and energy gaps in catalytic networks.

Findings

Relaxation time increases as molecule number decreases.

Amplification ratio scales as h = N exp(-βV).

Critical behavior occurs when N < ~1/exp(-βV).

Abstract

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 resources and the corresponding products that work as catalysts. 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 $h = N \exp(-\beta V)$, 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 gap between a product and a resource.