Evolution in chemical systems

Example 12.6 Evolution in chemical systems Consider the following set of reactions  

At fixed concentrations of A and B, X is the only variable. At equilibrium, the detailed balance yields 

After expanding the cubic and quadratic binomial, we have the linearized equation in terms of perturbation x and the stationary solution Xs 

This equation has multiple solutions for some values of A and hence bounded values of B. So, nonequilibrium state can reveal the true properties that are disguised at equilibrium and near equilibrium nonlinearity combined with nonequilibrium constraints may allow the diversification of the behavior of a system. 

Macroscopic systems are composed of large numbers of interacting particles, and the state variables represent either averages of instantaneous states over a long time interval, or the most probable states. Most systems communicate with the environment by exchanging small quantities of matter, momentum, or energy, which are treated as experimental error and, confidence level. So, the instantaneous state of a system is not stationary state Xs but rather nearby state X related to Xs through the perturbation x(t) X(t) =Xs + x(t). 

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