Instability in a chemical network system as a model of reaction cascades

A chemical network system for use in the study of reaction cascades is described by the nonlinear rate equation $$\dot x = \varphi \left( x \right) \simeq - k_1 x - k_2 x^3 $$ , in which ϕ(x) is derived from the −∂G/∂x of Taylor's expansion of the order parameterx of the thermodynamic potential, Gibbs' functionG(T,P,x), at about the critical pointC(T C ,P C ) of the control variables (parameters)T andP. The behavior of the system around the stable pointx=0 is analyzed only with the sign ofk 1(T, P), becausek 2(T, P) is always positive. The system is not closed and is affected by physical and chemical changes in the neighbor systems. WhenT andP fluctuate ( $$\dot T \ne 0 and \dot P \ne 0$$ ) through the changes andk 1 passes zero, the system in the steady state $$\dot x = 0$$ becomes instable andk 1 jumps at the reaction threshold. Then, the products are formed at the number of moleculesx=(|k 1|/k 2)1/2. To describe such a transition,k 1 giving theG curvature atx=0 is represented phenomenologically by an approximate function, tanh(ΔG/RT), for a metastable state with a relatively smallk 1(>0). The reaction takes place in a cascade in accordance with a cubic state equation obtained from tanh(ΔG/RT), which describes a jump of the reaction energy ΔG at the reaction threshold.