| 77 |
|
\label{introEq:SM2} |
| 78 |
|
\end{equation} |
| 79 |
|
|
| 80 |
< |
The solution to Eq.~\ref{introEq:SM2} maximizes the number of |
| 80 |
> |
The solution of interest to Eq.~\ref{introEq:SM2} maximizes the number of |
| 81 |
|
degenerate configurations in $E$. \cite{Frenkel1996} |
| 82 |
|
This gives |
| 83 |
|
\begin{equation} |
| 101 |
|
degeneracy in Eq.~\ref{introEq:SM3} and the second law of |
| 102 |
|
thermodynamics. Namely, that for a closed system, entropy will |
| 103 |
|
increase for an irreversible process.\cite{chandler:1987} Here the |
| 104 |
< |
process is the partitioning of energy among the two systems. This |
| 104 |
> |
maximization of the degeneracy when partitioning the energy of the system can be likened to the maximization of the entropy for this process. This |
| 105 |
|
allows the following definition of entropy: |
| 106 |
|
\begin{equation} |
| 107 |
|
S = k_B \ln \Omega(E), |
| 383 |
|
Eq.~\ref{introEq:MCmarkovEquil} is solved such that |
| 384 |
|
$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzmann distribution |
| 385 |
|
of states. The method accomplishes this by imposing the strong |
| 386 |
< |
condition of microscopic reversibility on the equilibrium |
| 387 |
< |
distribution. This means that at equilibrium, the probability of going |
| 386 |
> |
condition of detailed balance on the equilibrium |
| 387 |
> |
distribution. This means that the probability of going |
| 388 |
|
from $m$ to $n$ is the same as going from $n$ to $m$, |
| 389 |
|
\begin{equation} |
| 390 |
|
\rho_m\pi_{mn} = \rho_n\pi_{nm}. |
| 401 |
|
For a Boltzmann limiting distribution, |
| 402 |
|
\begin{equation} |
| 403 |
|
\frac{\rho_n}{\rho_m} = e^{-\beta[\mathcal{U}(n) - \mathcal{U}(m)]} |
| 404 |
< |
= e^{-\beta \Delta \mathcal{U}}. |
| 404 |
> |
= e^{-\beta \Delta \mathcal{U}}, |
| 405 |
|
\label{introEq:MCmicro3} |
| 406 |
|
\end{equation} |
| 407 |
< |
This allows for the following set of acceptance rules be defined: |
| 407 |
> |
where $\Delta\mathcal{U}$ is the change in the total energy of the system. This allows for the following set of acceptance rules be defined: |
| 408 |
|
\begin{equation} |
| 409 |
|
\accMe( m \rightarrow n ) = |
| 410 |
|
\begin{cases} |
| 921 |
|
|
| 922 |
|
The chapter concerning random sequential adsorption simulations is a |
| 923 |
|
study in applying Statistical Mechanics simulation techniques in order |
| 924 |
< |
to obtain a simple model capable of explaining the results. My |
| 924 |
> |
to obtain a simple model capable of explaining experimental observations. My |
| 925 |
|
advisor, Dr. Gezelter, and I were approached by a colleague, |
| 926 |
|
Dr. Lieberman, about possible explanations for the partial coverage of |
| 927 |
|
a gold surface by a particular compound synthesized in her group. We |
