# 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 $T = 1000K, \gamma = 0.25$eV, $N = 64$ 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.