Kris's Research Notes

September 24, 2010

Comparing Marginal Distributions

Filed under: Stat. Mech for SOS Model — Kris Reyes @ 2:49 pm

Recall: We wish to compare the distributions \pi_{leq}(t) and P(t), where

\pi_{leq}(t;h) = \frac{1}{\Xi} e^{-\beta H_{leq}(t;h)},

and

P(t;h) = \frac{1}{Z} e^{\beta H(t;h)},

for energy functions H, H_{leq} and partition functions Z, \Xi. Because of the form of H_{leq} and H, we may instead (and perhaps more naturally) write these probabilities in terms of the height differences latex g_i = h_i - h_{i-1}. We do so below and write \pi^\Delta_{leq}(t; g) for this probability, where g = (g_1, \hdots, g_n) . For example

\displaystyle \pi^\Delta_{leq}(t;g) = \frac{1}{\Xi}e^{-\beta H_{leq}(t;g)} = \frac{1}{\Xi} \exp\left[-\beta(\sum_{i=1}^N |g_i| + m_ig_i)\right]

One way to visualize this is to consider the marginals

\displaystyle \pi^{\Delta_i}_{leq} (t; g_i) = \sum_{g_1, \hdots, \hat g_i, \hdots g_N} \pi^\Delta_{leq}(t; (g_1,\hdots,g_N)),

and similarly for P^{\Delta_i}(t;g_i). The marginal distribution \pi{^\Delta_i}_{leq}(t;g_i)  may be simplified as

\displaystyle \pi^{\Delta_i}_{leq}(t; g_i) = e^{\theta|g_i| + \beta m_i(t)g_i}\frac{\cosh\theta - \cosh \beta m_i(t)}{\sinh \theta}

where \theta = \beta\gamma/2 and m_i is a function of the average height profile at time t. We compare this distribution with the empirical estimation for P^{\Delta_i}(t, g_i), which we obtained by KMC simulation.

In the simulation we used parameters T = 1000K, \gamma = 0.25eV, N = 64 and prefactor \Omega = 10^9. The initial profile was a sinusoidal profile about h=0 with an amplitude of 64. We let the simulation run until the simulation time reached 1 second. We wrote the height profile out every 0.01 second of simulation time. We simulated 4000 trials and computed the empirical distribution P^{\Delta_i}(t, g_i) and the average height profile \bar h(t). Using this empirical average height profile,  we also calculated \pi^{\Delta_i}_{leq}(t).  Here are some results where we vary

t \in \left\{0.01, 0.25, 0.5, 0.75, 0.99\right\},

and

i \in \left\{1, 16, 32, 48, 64\right\}.

The following table is ordered as follows. Each row corresponds to a particular time. The first column in each row shows the average height profile at that time, and the subsequent columns show the two marginal distributions at particular values of i. Here, \pi^{\Delta_i}_{leq} is plotted in red and P^{\Delta_i} is in blue.

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1 Comment »

  1. […] 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 […]

    Pingback by Comparing Marginal Height Profiles – Part 2 « Kris's Research Notes — October 4, 2010 @ 3:58 pm


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