# Kris's Research Notes

## October 4, 2010

### Comparing Marginal Height Profiles – Part 2

Filed under: Stat. Mech for SOS Model — Kris Reyes @ 3:58 pm

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

$r_{i, i\pm1} =\begin{cases} \frac{1}{2}R_i & i = 2, \hdots , N-1 \\ R_i & i = 1, N. \end{cases}$

where $R_i = \Omega^\prime e^{-\beta \gamma n_i}$, and $n_i$ is the number of lateral bonds of the top atom at site $i$ (and we count the atom-wall bond at site $N$ as half a bond). Here $\Omega^\prime = 5\times 10^7$ and $\gamma = 0.25 eV$. The different cases here in $r_{i, i\pm1}$ arise due to how the simulations sample a hop event — the code first samples a site according to rate $R_i$, then hops the atom to the left or right with equal probability (hence the $1/2$) if $i \neq 1, N$ and directly to the right or left if $i = 1$ or $N$, respectively (and hence no $1/2$ 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 $r_{N, N-1} = \frac{1}{2} R_N$ and $r_{1, 2} = \frac{1}{2} R_1$.  Second, we can change the evolution of $\bar h(t)$ by replacing expressions $\frac{1}{2} R_1$ with $R_1$ and similarly $\frac{1}{2} R_N$ with $R_N$. We opt to make the first change. Then we have the usual expressions for the evolution of $\bar h_k$:

$\displaystyle \frac{d \bar h_k}{dt} = \begin{cases}\frac{\Omega}{2}\left(-e^{\beta\mu_1} + e^{\beta\mu_2}\right) & k = 1 \\ \frac{\Omega}{2}\left(e^{\beta\mu_{i-1}} - 2e^{\beta\mu_i} e^{\beta\mu_{i+1}}\right) & k = 2, \hdots, N-1 \\ \frac{\Omega}{2}\left(e^{\beta\mu_{N-1}} - e^{\beta\mu_N}\right) & k =N \\ \end{cases},$

where $\Omega = \Omega^\prime e^{-\beta\gamma}$.

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 $h_{emp}(t)$ and the empirical distributions for the height differences. We compare this to the expected height profile $\bar h$, which we compute using the evolution equation above and initial conditions $\bar h(0.01) = \bar h_{emp}(0.01)$. We also compare empirical vs. predicted marginal distributions.

We plot the empirical data along with the predicted data below. Each row corresponds to $t \in \left\{0.01, 0.04, 0.09, 0.14, 0.19, 0.24\right\}$. The first plot in a row is the plot of $\bar h$ and $\bar h_{emp}$. Subsequent plots in a row are expected and observed plots of the marginal distribution for $h_i(t) - h_{i-1}(t)$ for $i \in \left\{1, 16, 32, 48, 64\right\}$. 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 $i=1$, we get a good match in the marginal distributions as well. When $t = 1, 4$ or $9$, the marginal distribution for $h_1$ is not good. Note these correspond to the case when $h_1$ is much larger than $h_0 =0$.