I am currently rereading J. Krug’s paper “Atom Mobility for the Solid-on-Solid Model.” I think I understand more of it than last time, so I will now attempt to discuss the continuum limit for the evolution of a height profile. (more…)

## November 15, 2010

## October 12, 2010

### The H-Theorem for a Time-Reversible Markov Chain.

For a countable set of states and an evolving distribution on , we may define the **entropy** at time by

The **H-Theorem** asserts that (with some assumptions) ; the entropy is always non-decreasing. Calculating this quantity, we see

Since we see we have

We assume describes a **Continuous Time Markov Chain**. That is, we assume there exists transition rates for distinct such that evolves according to:

If we write

we may write the above as

If we consider the as an infinite dimensional matrix and similarly consider as an infinite dimensional vector, we may write the above compactly in matrix notation:

.

We also assume the chain is **time reversible.** That is, there exists a distribution on such that the **detailed balance property** holds:

.** **

For example,suppose was finite and the rates were symmetric. Then the corresponding chain is time-reversible and the requisite is the uniform distribution on . It is easy to see the H-theorem holds in this context, since we haveso that

Note that in this sum, both the terms

and

occur. Grouping them together, and summing over , we get:

Each term in the sum is non-negative, so that .

We wish to consider the general time reversible case. Instead of being symmetric, we have

Then

so that

and as before, we observe the terms

and

occur in the above summation. Therefore

## October 8, 2010

### Marginal Distributions – Part 6

### Marginal Distributions – Part 5

Now I compare marginal distributions for a system with initial profile

That is, like in the previous post, but with an even more compressed sine function. The motivation behind this is to provide a buffer between the boundary and the mass in the middle, which we hope will lead to a continuous profile at the boundary. (more…)

### Marginal Distributions – Part 4

## October 6, 2010

### Comparing Marginal Distributions – Part 3

This is a follow up of this post.

Recall we are evolving a height profile that is initially sinusoidal for and we have fixed for all . We simulate the evolution with atom hopping rates

where is the number of lateral neighbors of the top atom at site (always counting half bonds with the wall for ) and . We also set the rates and to zero. We noted two things in the previous post. First, we had observed a discontinuity between and for small . Second, comparing the expected and empirical marginal distributions and

we observed that the two distributions did not agree for small .

We had discussed that this was probably due to the discontinuity between and . To avoid this, we decided to use an initial height profile that was very smooth at , e.g. . So we repeat the previous experiment with initial profile

Here is a movie of the simulated evolution of with rates specified above and total simulated time of seconds. The predicted average height profile was solved using Euler method with a time step of seconds.

As we can see, we still get a discontinuity between and . Here are the predicted vs. observed height profiles along with marginals at several times in the interval . (more…)

## October 4, 2010

### Comparing Marginal Height Profiles – Part 2

This is a follow up to this post. Recall, in our KMC simulations, we had originally assigned atom hopping rates

where , and is the number of lateral bonds of the top atom at site (and we count the atom-wall bond at site as half a bond). Here and . The different cases here in arise due to how the simulations sample a hop event — the code first samples a site according to rate , then hops the atom to the left or right with equal probability (hence the ) if and directly to the right or left if or , respectively (and hence no factor). This is incongruous to our analysis of the average height profile and its evolution.

We can rectify this in two ways. First, we can change the simulation so that and . Second, we can change the evolution of by replacing expressions with and similarly with . We opt to make the first change. Then we have the usual expressions for the evolution of :

where .

This allows us to approximate the evolution of the average height profile. In this run, we have 4000 KMC simulations of an initially sinusoidal height profile with period 64 and amplitude 64. We measure the height profile every 0.01 second. This allows to calculate the empirical average height profile and the empirical distributions for the height differences. We compare this to the expected height profile , which we compute using the evolution equation above and initial conditions . We also compare empirical vs. predicted marginal distributions.

We plot the empirical data along with the predicted data below. Each row corresponds to . The first plot in a row is the plot of and . Subsequent plots in a row are expected and observed plots of the marginal distribution for for . In these graphs, blue plots are data observed from the simulations while red plots are the expected average height profile and marginal distributions. We note two things. First, the average height profile predicts the simulation data very well. Second, except for the case where , we get a good match in the marginal distributions as well. When or , the marginal distribution for is not good. Note these correspond to the case when is much larger than . (more…)

## September 29, 2010

### How the average height profile evolves over time.

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.

## September 24, 2010

### Comparing Marginal Distributions

Recall: We wish to compare the distributions and , where

and

for energy functions and partition functions . Because of the form of and , we may instead (and perhaps more naturally) write these probabilities in terms of the height differences latex . We do so below and write for this probability, where . For example

One way to visualize this is to consider the marginals

and similarly for . The marginal distribution may be simplified as

where and is a function of the average height profile at time . We compare this distribution with the empirical estimation for , which we obtained by KMC simulation.