| 5 |
|
|
| 6 |
|
The techniques used in the course of this research fall under the two |
| 7 |
|
main classes of molecular simulation: Molecular Dynamics and Monte |
| 8 |
< |
Carlo. Molecular Dynamic simulations integrate the equations of motion |
| 9 |
< |
for a given system of particles, allowing the researcher to gain |
| 10 |
< |
insight into the time dependent evolution of a system. Diffusion |
| 8 |
> |
Carlo. Molecular Dynamics simulations integrate the equations of |
| 9 |
> |
motion for a given system of particles, allowing the researcher to |
| 10 |
> |
gain insight into the time dependent evolution of a system. Diffusion |
| 11 |
|
phenomena are readily studied with this simulation technique, making |
| 12 |
|
Molecular Dynamics the main simulation technique used in this |
| 13 |
|
research. Other aspects of the research fall under the Monte Carlo |
| 14 |
|
class of simulations. In Monte Carlo, the configuration space |
| 15 |
< |
available to the collection of particles is sampled stochastically, |
| 16 |
< |
or randomly. Each configuration is chosen with a given probability |
| 17 |
< |
based on the Maxwell Boltzmann distribution. These types of simulations |
| 18 |
< |
are best used to probe properties of a system that are only dependent |
| 19 |
< |
only on the state of the system. Structural information about a system |
| 20 |
< |
is most readily obtained through these types of methods. |
| 15 |
> |
available to the collection of particles is sampled stochastically, or |
| 16 |
> |
randomly. Each configuration is chosen with a given probability based |
| 17 |
> |
on the Maxwell Boltzmann distribution. These types of simulations are |
| 18 |
> |
best used to probe properties of a system that are dependent only on |
| 19 |
> |
the state of the system. Structural information about a system is most |
| 20 |
> |
readily obtained through these types of methods. |
| 21 |
|
|
| 22 |
|
Although the two techniques employed seem dissimilar, they are both |
| 23 |
|
linked by the overarching principles of Statistical |
| 24 |
< |
Thermodynamics. Statistical Thermodynamics governs the behavior of |
| 24 |
> |
Mechanics. Statistical Meachanics governs the behavior of |
| 25 |
|
both classes of simulations and dictates what each method can and |
| 26 |
|
cannot do. When investigating a system, one most first analyze what |
| 27 |
|
thermodynamic properties of the system are being probed, then chose |
| 31 |
|
|
| 32 |
|
The following section serves as a brief introduction to some of the |
| 33 |
|
Statistical Mechanics concepts present in this dissertation. What |
| 34 |
< |
follows is a brief derivation of Boltzmann weighted statistics, and an |
| 34 |
> |
follows is a brief derivation of Boltzmann weighted statistics and an |
| 35 |
|
explanation of how one can use the information to calculate an |
| 36 |
|
observable for a system. This section then concludes with a brief |
| 37 |
|
discussion of the ergodic hypothesis and its relevance to this |
| 39 |
|
|
| 40 |
|
\subsection{\label{introSec:boltzman}Boltzmann weighted statistics} |
| 41 |
|
|
| 42 |
< |
Consider a system, $\gamma$, with some total energy,, $E_{\gamma}$. |
| 43 |
< |
Let $\Omega(E_{\gamma})$ represent the number of degenerate ways |
| 44 |
< |
$\boldsymbol{\Gamma}$, the collection of positions and conjugate |
| 45 |
< |
momenta of system $\gamma$, can be configured to give |
| 46 |
< |
$E_{\gamma}$. Further, if $\gamma$ is in contact with a bath system |
| 47 |
< |
where energy is exchanged between the two systems, $\Omega(E)$, where |
| 48 |
< |
$E$ is the total energy of both systems, can be represented as |
| 42 |
> |
Consider a system, $\gamma$, with total energy $E_{\gamma}$. Let |
| 43 |
> |
$\Omega(E_{\gamma})$ represent the number of degenerate ways |
| 44 |
> |
$\boldsymbol{\Gamma}\{r_1,r_2,\ldots r_n,p_1,p_2,\ldots p_n\}$, the |
| 45 |
> |
collection of positions and conjugate momenta of system $\gamma$, can |
| 46 |
> |
be configured to give $E_{\gamma}$. Further, if $\gamma$ is a subset |
| 47 |
> |
of a larger system, $\boldsymbol{\Lambda}\{E_1,E_2,\ldots |
| 48 |
> |
E_{\gamma},\ldots E_n\}$, the total degeneracy of the system can be |
| 49 |
> |
expressed as, |
| 50 |
|
\begin{equation} |
| 51 |
+ |
\Omega(\boldsymbol{\Lambda}) = \Omega(E_1) \times \Omega(E_2) \times \ldots |
| 52 |
+ |
\Omega(E_{\gamma}) \times \ldots \Omega(E_n) |
| 53 |
+ |
\label{introEq:SM0.1} |
| 54 |
+ |
\end{equation} |
| 55 |
+ |
This multiplicative combination of degeneracies is illustrated in |
| 56 |
+ |
Fig.~\ref{introFig:degenProd}. |
| 57 |
+ |
|
| 58 |
+ |
\begin{figure} |
| 59 |
+ |
\centering |
| 60 |
+ |
\includegraphics[width=\linewidth]{omegaFig.eps} |
| 61 |
+ |
\caption[An explanation of the combination of degeneracies]{Systems A and B both have three energy levels and two indistinguishable particles. When the total energy is 2, there are two ways for each system to disperse the energy. However, for system C, the superset of A and B, the total degeneracy is the product of the degeneracy of each system. In this case $\Omega(\text{C})$ is 4.} |
| 62 |
+ |
\label{introFig:degenProd} |
| 63 |
+ |
\end{figure} |
| 64 |
+ |
|
| 65 |
+ |
Next, consider the specific case of $\gamma$ in contact with a |
| 66 |
+ |
bath. Exchange of energy is allowed between the bath and the system, |
| 67 |
+ |
subject to the constraint that the total energy |
| 68 |
+ |
($E_{\text{bath}}+E_{\gamma}$) remain constant. $\Omega(E)$, where $E$ |
| 69 |
+ |
is the total energy of both systems, can be represented as |
| 70 |
+ |
\begin{equation} |
| 71 |
|
\Omega(E) = \Omega(E_{\gamma}) \times \Omega(E - E_{\gamma}) |
| 72 |
|
\label{introEq:SM1} |
| 73 |
|
\end{equation} |
| 665 |
|
\begin{figure} |
| 666 |
|
\centering |
| 667 |
|
\includegraphics[width=\linewidth]{eulerRotFig.eps} |
| 668 |
< |
\caption[Euler rotation of Cartesian coordinates]{The rotation scheme for Euler angles. First is a rotation of $\phi$ about the $z$ axis (blue rotation). Next is a rotation of $\theta$ about the new $x\prime$ axis (green rotation). Lastly is a final rotation of $\psi$ about the new $z\prime$ axis (red rotation).} |
| 668 |
> |
\caption[Euler rotation of Cartesian coordinates]{The rotation scheme for Euler angles. First is a rotation of $\phi$ about the $z$ axis (blue rotation). Next is a rotation of $\theta$ about the new $x$ axis (green rotation). Lastly is a final rotation of $\psi$ about the new $z$ axis (red rotation).} |
| 669 |
|
\label{introFig:eulerAngles} |
| 670 |
|
\end{figure} |
| 671 |
|
|
