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.

 

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1 Comment »

  1. […] Comparing Marginal Distributions – Part 3 Filed under: Uncategorized — Kris Reyes @ 4:53 pm This is a follow up of this post. […]

    Pingback by Comparing Marginal Distributions – Part 3 « Kris's Research Notes — October 6, 2010 @ 4:53 pm


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