| 40 |
|
|
| 41 |
|
The Monte Carlo method was developed by Metropolis and Ulam for their |
| 42 |
|
work in fissionable material.\cite{metropolis:1949} The method is so |
| 43 |
< |
named, because it heavily uses random numbers in the solution of the |
| 44 |
< |
problem. |
| 43 |
> |
named, because it heavily uses random numbers in its |
| 44 |
> |
solution.\cite{allen87:csl} The Monte Carlo method allows for the |
| 45 |
> |
solution of integrals through the stochastic sampling of the values |
| 46 |
> |
within the integral. In the simplest case, the evaluation of an |
| 47 |
> |
integral would follow a brute force method of |
| 48 |
> |
sampling.\cite{Frenkel1996} Consider the following single dimensional |
| 49 |
> |
integral: |
| 50 |
> |
\begin{equation} |
| 51 |
> |
I = f(x)dx |
| 52 |
> |
\label{eq:MCex1} |
| 53 |
> |
\end{equation} |
| 54 |
> |
The equation can be recast as: |
| 55 |
> |
\begin{equation} |
| 56 |
> |
I = (b-a)<f(x)> |
| 57 |
> |
\label{eq:MCex2} |
| 58 |
> |
\end{equation} |
| 59 |
> |
Where $<f(x)>$ is the unweighted average over the interval |
| 60 |
> |
$[a,b]$. The calculation of the integral could then be solved by |
| 61 |
> |
randomly choosing points along the interval $[a,b]$ and calculating |
| 62 |
> |
the value of $f(x)$ at each point. The accumulated average would then |
| 63 |
> |
approach $I$ in the limit where the number of trials is infintely |
| 64 |
> |
large. |
| 65 |
|
|
| 66 |
+ |
However, in Statistical Mechanics, one is typically interested in |
| 67 |
+ |
integrals of the form: |
| 68 |
+ |
\begin{equation} |
| 69 |
+ |
<A> = \frac{A}{exp^{-\beta}} |
| 70 |
+ |
\label{eq:mcEnsAvg} |
| 71 |
+ |
\end{equation} |
| 72 |
+ |
Where $r^N$ stands for the coordinates of all $N$ particles and $A$ is |
| 73 |
+ |
some observable that is only dependent on position. $<A>$ is the |
| 74 |
+ |
ensemble average of $A$ as presented in |
| 75 |
+ |
Sec.~\ref{introSec:statThermo}. Because $A$ is independent of |
| 76 |
+ |
momentum, the momenta contribution of the integral can be factored |
| 77 |
+ |
out, leaving the configurational integral. Application of the brute |
| 78 |
+ |
force method to this system would yield highly inefficient |
| 79 |
+ |
results. Due to the Boltzman weighting of this integral, most random |
| 80 |
+ |
configurations will have a near zero contribution to the ensemble |
| 81 |
+ |
average. This is where a importance sampling comes into |
| 82 |
+ |
play.\cite{allen87:csl} |
| 83 |
|
|
| 84 |
+ |
Importance Sampling is a method where one selects a distribution from |
| 85 |
+ |
which the random configurations are chosen in order to more |
| 86 |
+ |
efficiently calculate the integral.\cite{Frenkel1996} Consider again |
| 87 |
+ |
Eq.~\ref{eq:MCex1} rewritten to be: |
| 88 |
+ |
|
| 89 |
+ |
|
| 90 |
+ |
|
| 91 |
|
\subsection{\label{introSec:md}Molecular Dynamics Simulations} |
| 92 |
|
|
| 93 |
|
time averages |
