Saddle-Node Bifurcation to Jammed State for Quasi-One-Dimensional Counter Chemotactic Flow

The transition of a counter chemotactic particle flow from a free-flow state to a jammed state in a quasi-one-dimensional path is investigated. One of the characteristic features of such a flow is that the constituent particles spontaneously form a cluster that blocks the path, called a path-blocking cluster (PBC), and causes a jammed state when the particle density is greater than a threshold value. Near the threshold value, the PBC occasionally desolve itself to recover the free flow. In other words, the time evolution of the size of the PBC governs the flux of a counter chemotactic flow. In this paper, on the basis of numerical results of a stochastic cellular automata (SCA) model, we introduce a Langevin equation model for the size evolution of the PBC that reproduces the qualitative characteristics of the SCA model. The results suggest that the emergence of the jammed state in a quasi-one-dimensional counter flow is caused by a saddle-node bifurcation.Comment: 5pages, 8figure

"Glassy" Relaxation in Catalytic Reaction Networks

Relaxation dynamics in reversible catalytic reaction networks is studied, revealing two salient behaviors that are reminiscent of glassy behavior: slow relaxation with log(time) dependence of the correlation function, and emergence of a few plateaus in the relaxation. The former is explained by the eigenvalue distribution of a Jacobian matrix around the equilibrium state that follows the distribution of kinetic coefficients of reactions. The latter is associated with kinetic constraints, rather than metastable states, and is due to the deficiency of catalysts for chemicals in excess and negative correlation between the two chemical species. Examples are given, and generality is discussed

Discreteness-Induced Slow Relaxation in Reversible Catalytic Reaction Networks

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