Model Schlogl

M. Vellela, H. Qian, Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system the Schlogl model revisited. J. R Soc. Interface. (doi 10.1098/rsif.2008.0476) (2008)... 

The first Schlogl model [384] is a generalization of the branching-coalescence scheme and is defined by the mechanism... 

The original version of the first Schlogl model contains a fourth step, the back reaction C —> B -I- U. We assume that the product C is immediately removed from the system. We consider two different ways of nondimensionalizing this rate equation, (i) Set t = k2 PnP ) t and p = 2/°/( iPa k- P, ). This is acceptable for all situations where k p > k p. Then we obtain the following nondimensionalized version of (1.66) ... 

As the previous models, the second Schlogl model relies on the pool chemical assumption to account for the inflow of reactants. If we assume instead that this reaction scheme occurs in a CSTR, we obtain the Gray-Scott model [170, 171] ... 

Solve the kinetic equation (1.69) for the first Schlogl model. Confirm the results of the linear stability analysis for /r > 0 and /u. < 0. Determine the stability of... 

The Schlogl model of second-order phase transition... 

Fig. 5.7 Second-order transition in the deterministic (Schlogl) model.

The Schlogl model of the first-order phase transition is given by the reaction... 

We take the probability distribution to obey the master equation which has been used extensively. For the cubic Schlogl model ((2.7) with r = 3, s = 1) the master equation is [1,5]... 

In regions of multistability the stationary probability distribution is bi-modal and is shown in Fig. 2.1 for the cubic Schlogl model. 

Let us take a one-variable system, such as the Schlogl model... 

This procedure is easy for a one-variable system because we know the solution of the stationary master equation to this approximation. For example, for the one-variable Schlogl model we have the elementary reaction steps... 

Figure 3 Steady states of the Schlogl model. The steady-state concentration is plotted...

Figure 4 Formation of domains of the two stable states in the Schlogl model color coded in gray shades.

Thus, we find the macroscopic chemical rate law for the Schlogl model. Mesoscopic simulations of the Schlogl model have been carried out using a Markov chain model.Figure 3 shows the results for the steady-state concentrations derived from such a mesoscopic simulation along with the deterministic steady-state concentrations discussed earlier. The stochastic model yields results that are close to those of the mass action rate law. However, in the vicinities of points where the deterministic stable and unstable fixed points meet, so that one of the stable states loses its stability, fluctuations play an important role. 

We consider behaviour in the Schlogl model [2] as a function of flow rate (inverse residence time). This result is in better agreement with experimental findings [3] than that presented in our previous paper [1]. We conclude that the interaction between reactant streams and finite stirring is the dominant mechanism underlying the phenomena observed in experiment [3]. 

The Schlogl Model [2] is a generic model for bistability in a chemical reaction described by the evolution of one intermediate species. The reaction scheme... 

Schlogl Model Stationary state mean and variance vs. flow rate (k = 1,... 

The Schlogl model [2] yields results that are closer to experimental findings [3] when the homogeneity breaking mechanism is reactant flows than when it is fluctuations. We conclude that the observed phenomena are essentially the effects of reactant streams and finite stirring rates in the CSTR. 

Bistable chemical reactions are the object of increasing interest from the experimental and theoretical points of view. The simplest abstract example is the Schlogl model well known experimental cases of bistability are the chlorite-iodide reaction or the iodate oxydation of the arsenous acid. 

Deterministic description of a closed system Let us take as an example the Schlogl model 1 ]... 

AN ELEMENTARY BISTABLE REACTION THE SCHLOGL MODEL 2.1 Schlogl reaction... 

The numerical investigation of this equation shows that in the bistable regime the minimum in the diffusional dependence of the transition time between stationary states also occurs Cas in the case of the multivariate master equation ). Fig. 2 shows the results for the Schlogl model Cthe system size is 100 and 200D. 

Figure 4 Kinetic potential U(x) versus concentration x for thermal explosion model (a), and for Schlogl model (b).

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