Recall we defined the average height profile

.

If we approximated with the local equilibrium distribution

we showed that evolves approximately like

for and

and

Here, is the prefactor used in the hopping rate

where is the number of bonds the top atom at site has. If instead we wished to use the number of lateral neighbors, , then and the rates are given by

We make this point because in KMC simulations, we often define rates in terms of . For example, in our current simulation we set our rates such that , hence in our analysis we must fix . With , this means Note this is a large number.

Consider the average profile near equilibrium. Here the average profile does not change much in space () or time (). Then

Consider near :

Observe the are small even for (relatively) large difference in height profile. Then the are also small. Using the above graph as an example, we see that near equilibrium with high probability. In equilibrium all the $\mu_i$ are equal and so, by examining equations (1), (2), and (3) we see that in this case.

Now consider the system not near equilibrium. In particular, suppose we had the following height profile:

We wish to consider how our model predicts it will evolve with respect to the equations (1), (2), and (3) above. To that end, consider the plot of where the $\mu_i$ are calculated from this average profileAs we see, the values are somewhat close together — but not close enough! That is, consider , the unnormalized rates:

When we scale by we see that is very large! This could leads to some unstable behavior if we try to evolve using e.g Euler’s Method.

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