Here we are thinking of a catalyst surface like the ones that are present in the exhaust systems of many cars. The possible states in our model are 0 = vacant, 1 = carbon monoxide molecule, 2 = oxygen atom. The dynamics of the model, as formulated by Ziff, Gulari and Barshad (1986) are as follows:
(i) Carbon monoxide molecules land at vacant sites at rate p, a parameter which should be thought of as the fraction of carbon monoxide molecules in the atmosphere.
(ii) Diatomic oxygen molecules attempt to land at vacant sites at rate 1-p. When an attempt is made at x, a neighboring site y is selected. If y is also vacant, oxygen atoms attach at x and at y.
(iii) A carbon monoxide molecule adjacent to an oxygen atom reacts immediately producing a carbon dioxide modecule. In terms of the dynamics this means that each newly landed moecule must check its neighbors to see if a reaction can occur. If reactions with several neighbors are possible, a neighbor is picked at random from the set of possibilities.
This model has two trivial stationary distributions, "all sites 1" and "all sites 2", which represent poisoning of the catalyst surface so attention focusses on
Q. Is coexistence possible for p in an interval of values (p[1],p[2])?
Computer simulations of ZGB (1986) indicate that p[1] is about 0.389 while p[2] is about 0.525. At the first transition, the density of CO in equilibrium increases continuously from 0. However at the second, the density of CO jumps discontinuously from less than 0.25 to 1. Physicists call this a first order phase transition. Somewhat surprisingly theory predicts that at the critical value there is an equilibrium state with exponentially decaying correlations.
s3 Exercises. Because of rule (iii) one must start with all sites empty. Choose the "Full" initial condition and fill the grid with 0's.
1. When p = 0.55, 1's (i.e, CO) win. When p = 0.5, there is coexistence but the density of CO's is very small. In between there is the default simulation value of 0.53 which is larger than the critical value but has coexistence for a while in a "metastable" state. Physicists tell us this is common in the presence of a "first order" (i.e., discontinuous) phase transition.
2. When p = 0.33, 2's (i.e., O) win. It is interesting to set p = 0. O's of course are certain to win in this case but since they cannot fill single holes, the lattice only ends up being about 90% full.
Rigorous results. It is not hard to show that if p is close enough to 0, the system converges to all 2's, while if p > 2/3 then convergence to all 1's occurs. To explain the 2/3's note that one CO tries to land at rate p, while two O's attempt to land at rate 1-p.
Proving that coexistence occurs in the original ZGB model seems to be a difficult problem. Durrett and Swindle (1994) modified the dynamics to have a finite reaction rate r in (iii) for adjacent CO-O pairs and to add
(iv) stirring at rate nu for each pair of neighboring sites x and y we exchange the values at x and at y at rate nu.
To guess what would happen in the modified model they looked at the mean field ODE:
Conditions for an interior equilibrium involve solving a quadratic equation so formulas are a little messy but when one exists there are four equilbiria for the system: saddle points at (0,1) and (b,a) and locally attracting fixed points at (a,b) and (1,0). For example. when p = 1/4, r = 1, we have the following picture with a = 1/6 and b = 1/2:
For some parameter values the equilibrium at (a,b) will be stronger meaning that the travelling wave solution of the reaction diffusion equation
will lead to an advance of the (a,b) region. Durrett and Swindle (1994) showed
Theorem. If the (a,b) equilibrium is stronger, then when the stirring rate nu is large coexistence will occur. Conversely, if the (1,0) equilibrium is stronger, then when the stirring rate nu is large poisoning to all 1's will occur.
Other related work. Grannan and Swindle (1991) consider a version of the catalyst model in which diatomic oxygen was replaced by single oxygen atoms. In this case the one with the larger landing rate wins and coexistence is never possible. Bramson and Neuhauser (1997) replaced the diatomic oxygen molecule by a large square polymer and showed that under suitable conditions coexistence was possible.
Ziff, R.M., Gulari, E., and Barshad, Y. (1986) Kinetic phase transitions in an irreversible surface reaction model. Phys. Rev. Letters 56, 2553--2556
Grannan, E. and G. Swindle (1991). Rigorous results on mathematical models of catalyst surfaces. J. Stat. Phys. 61, 1085-1103
Durrett, R. and Swindle, G. (1994) Coexistence results for catalysts. Prob. Th. Rel. Fields, 98, 489-515
Bramson, M. and Neuhauser, C. (1997) Coexistence for a catalytic surface reaction model. Ann. Appl. Prob., to appear