Kris's Research Notes

October 6, 2010

Comparing Marginal Distributions – Part 3

Filed under: Stat. Mech for SOS Model — Kris Reyes @ 4:53 pm

This is a follow up of this post.

Recall we are evolving a height profile \bar h(t) that is initially sinusoidal \bar h_i(0) = 64 \sin\left(\frac{2\pi}{64}(i-1)\right) for i = 1, \hdots, 64 and we have fixed  h_0(t) = 0 for all t. We simulate the evolution with atom hopping rates

\displaystyle r(i, i\pm1) = \frac{\Omega^\prime}{2}e^{-\beta\gamma n_i}

where n_i is the number of lateral neighbors of the top atom at site i (always counting half bonds with the wall for n_M) and \Omega^\prime = 5\times 10^7. We also set the rates r(1, 0) and r(M, M+1) to zero. We noted two things in the previous post. First, we had observed a discontinuity between h_1(t) and h_0(t) = 0 for small t.  Second, comparing the expected and empirical marginal distributions  \mathbb P_{emp}(t; h_1) and

\displaystyle \mathbb P_{leq} (t;h_1) = \frac{\sinh \theta - \sinh \beta m_1(t)}{\cosh \theta} e^{\theta|h_1(t)| + m_1(t)h_1(t)},

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

We had discussed that this was probably due to the discontinuity between h_0 and h_1.  To avoid this, we decided to use an initial height profile that was very smooth at x=0, e.g. sin^5(x). So we repeat the previous experiment with initial profile

\bar h_i(0) = 64 \sin^5\left(\frac{2\pi}{64}(i-1)\right)

Here is a movie of the simulated evolution of \bar h(t) with rates specified above and total simulated time of 0.1 seconds. The predicted average height profile was solved using Euler method with a time step of \Delta t = 10^{-9} seconds.

As we can see, we still get a discontinuity between h_0 = 0 and h_1. Here are the predicted vs. observed height profiles along with marginals at several times in the interval [0, 0.1].

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1 Comment »

  1. […] reran the trials detailed in this post, but with an initial profile of the […]

    Pingback by Marginal Distributions – Part 4 « Kris's Research Notes — October 8, 2010 @ 3:56 pm


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