Recall: We wish to compare the distributions and , where

and

for energy functions and partition functions . Because of the form of and , we may instead (and perhaps more naturally) write these probabilities in terms of the height differences latex . We do so below and write for this probability, where . For example

One way to visualize this is to consider the marginals

and similarly for . The marginal distribution may be simplified as

where and is a function of the average height profile at time . We compare this distribution with the empirical estimation for , which we obtained by KMC simulation.

In the simulation we used parameters eV, and prefactor . The initial profile was a sinusoidal profile about 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 and the average height profile . Using this empirical average height profile, we also calculated . Here are some results where we vary

and

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 . Here, is plotted in red and is in blue.

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