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\chapter{\label{chapt:intro}Introduction and Theoretical Background} |
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|
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– |
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\section{\label{introSec:theory}Theoretical Background} |
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The techniques used in the course of this research fall under the two |
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main classes of molecular simulation: Molecular Dynamics and Monte |
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Carlo. Molecular Dynamic simulations integrate the equations of motion |
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thermodynamic properties of the system are being probed, then chose |
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which method best suits that objective. |
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|
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\subsection{\label{introSec:statThermo}Statistical Mechanics} |
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\section{\label{introSec:statThermo}Statistical Mechanics} |
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|
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The following section serves as a brief introduction to some of the |
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Statistical Mechanics concepts present in this dissertation. What |
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\subsection{\label{introSec:boltzman}Boltzman weighted statistics} |
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|
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Consider a system, $\gamma$, with some total energy,, $E_{\gamma}$. |
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Let $\Omega(E_{gamma})$ represent the number of degenerate ways |
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Let $\Omega(E_{\gamma})$ represent the number of degenerate ways |
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$\boldsymbol{\Gamma}$, the collection of positions and conjugate |
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momenta of system $\gamma$, can be configured to give |
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$E_{\gamma}$. Further, if $\gamma$ is in contact with a bath system |
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where energy is exchanged between the two systems, $\Omega(E)$, where |
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$E$ is the total energy of both systems, can be represented as |
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\begin{equation} |
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eq here |
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\Omega(E) = \Omega(E_{\gamma}) \times \Omega(E - E_{\gamma}) |
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\label{introEq:SM1} |
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\end{equation} |
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Or additively as |
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\begin{equation} |
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eq here |
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\ln \Omega(E) = \ln \Omega(E_{\gamma}) + \ln \Omega(E - E_{\gamma}) |
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\label{introEq:SM2} |
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\end{equation} |
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|
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The solution to Eq.~\ref{introEq:SM2} maximizes the number of |
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degenerative configurations in $E$. \cite{fix} |
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degenerative configurations in $E$. \cite{Frenkel1996} |
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This gives |
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\begin{equation} |
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eq here |
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\frac{\partial \ln \Omega(E)}{\partial E_{\gamma}} = 0 = |
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\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
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+ |
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\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
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\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}} |
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\label{introEq:SM3} |
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\end{equation} |
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Where $E_{\text{bath}}$ is $E-E_{\gamma}$, and |
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$\frac{partialE_{\text{bath}}}{\partial E_{\gamma}}$ is |
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$\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}}$ is |
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$-1$. Eq.~\ref{introEq:SM3} becomes |
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\begin{equation} |
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eq here |
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\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} = |
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\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
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\label{introEq:SM4} |
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\end{equation} |
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|
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At this point, one can draw a relationship between the maximization of |
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degeneracy in Eq.~\ref{introEq:SM3} and the second law of |
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thermodynamics. Namely, that for a closed system, entropy wil |
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increase for an irreversible process.\cite{fix} Here the |
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thermodynamics. Namely, that for a closed system, entropy will |
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increase for an irreversible process.\cite{chandler:1987} Here the |
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process is the partitioning of energy among the two systems. This |
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allows the following definition of entropy: |
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\begin{equation} |
| 86 |
< |
eq here |
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> |
S = k_B \ln \Omega(E) |
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\label{introEq:SM5} |
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\end{equation} |
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Where $k_B$ is the Boltzman constant. Having defined entropy, one can |
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also define the temperature of the system using the relation |
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\begin{equation} |
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< |
eq here |
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> |
\frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V} |
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\label{introEq:SM6} |
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\end{equation} |
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The temperature in the system $\gamma$ is then |
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\begin{equation} |
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< |
eq here |
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> |
\beta( E_{\gamma} ) = \frac{1}{k_B T} = |
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\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
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\label{introEq:SM7} |
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\end{equation} |
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Applying this to Eq.~\ref{introEq:SM4} gives the following |
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\begin{equation} |
| 103 |
< |
eq here |
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> |
\beta( E_{\gamma} ) = \beta( E_{\text{bath}} ) |
| 104 |
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\label{introEq:SM8} |
| 105 |
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\end{equation} |
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Showing that the partitioning of energy between the two systems is |
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actually a process of thermal equilibration. \cite{fix} |
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actually a process of thermal equilibration.\cite{Frenkel1996} |
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|
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An application of these results is to formulate the form of an |
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expectation value of an observable, $A$, in the canonical ensemble. In |
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and the temperature, $T$, are all held constant while the energy, $E$, |
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is allowed to fluctuate. Returning to the previous example, the bath |
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system is now an infinitly large thermal bath, whose exchange of |
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energy with the system $\gamma$ holds teh temperature constant. The |
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energy with the system $\gamma$ holds the temperature constant. The |
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partitioning of energy in the bath system is then related to the total |
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energy of both systems and the fluctuations in $E_{\gamma}}$: |
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> |
energy of both systems and the fluctuations in $E_{\gamma}$: |
| 118 |
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\begin{equation} |
| 119 |
< |
eq here |
| 119 |
> |
\Omega( E_{\gamma} ) = \Omega( E - E_{\gamma} ) |
| 120 |
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\label{introEq:SM9} |
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\end{equation} |
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As for the expectation value, it can be defined |
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\begin{equation} |
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< |
eq here |
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> |
\langle A \rangle = |
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> |
\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} |
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P_{\gamma} A(\boldsymbol{\Gamma}) |
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\label{introEq:SM10} |
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\end{eequation} |
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Where $\int_{\boldsymbol{\Gamma}} d\Boldsymbol{\Gamma}$ denotes an |
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integration over all accessable phase space, $P_{\gamma}$ is the |
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\end{equation} |
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Where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma}$ denotes |
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an integration over all accessable phase space, $P_{\gamma}$ is the |
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probability of being in a given phase state and |
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$A(\boldsymbol{\Gamma})$ is some observable that is a function of the |
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phase state. |
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$\gamma$ will be directly proportional to the number of allowable |
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states the coupled system is able to assume. Namely, |
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\begin{equation} |
| 140 |
< |
eq here |
| 140 |
> |
P_{\gamma} \propto \Omega( E_{\text{bath}} ) = |
| 141 |
> |
e^{\ln \Omega( E - E_{\gamma})} |
| 142 |
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\label{introEq:SM11} |
| 143 |
|
\end{equation} |
| 144 |
< |
With $E_{\gamma} \lE$, $\ln \Omega$ can be expanded in a Taylor series: |
| 144 |
> |
With $E_{\gamma} \ll E$, $\ln \Omega$ can be expanded in a Taylor series: |
| 145 |
|
\begin{equation} |
| 146 |
< |
eq here |
| 146 |
> |
\ln \Omega ( E - E_{\gamma}) = |
| 147 |
> |
\ln \Omega (E) - |
| 148 |
> |
E_{\gamma} \frac{\partial \ln \Omega }{\partial E} |
| 149 |
> |
+ \ldots |
| 150 |
|
\label{introEq:SM12} |
| 151 |
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\end{equation} |
| 152 |
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Higher order terms are omitted as $E$ is an infinite thermal |
| 153 |
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bath. Further, using Eq.~\ref{introEq:SM7}, Eq.~\ref{introEq:SM11} can |
| 154 |
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be rewritten: |
| 155 |
|
\begin{equation} |
| 156 |
< |
eq here |
| 156 |
> |
P_{\gamma} \propto e^{-\beta E_{\gamma}} |
| 157 |
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\label{introEq:SM13} |
| 158 |
|
\end{equation} |
| 159 |
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Where $\ln \Omega(E)$ has been factored out of the porpotionality as a |
| 160 |
< |
constant. Normalizing the probability ($\int_{\boldsymbol{\Gamma}} |
| 161 |
< |
d\boldsymbol{\Gamma} P_{\gamma} =1$ gives |
| 160 |
> |
constant. Normalizing the probability ($\int\limits_{\boldsymbol{\Gamma}} |
| 161 |
> |
d\boldsymbol{\Gamma} P_{\gamma} = 1$) gives |
| 162 |
|
\begin{equation} |
| 163 |
< |
eq here |
| 163 |
> |
P_{\gamma} = \frac{e^{-\beta E_{\gamma}}} |
| 164 |
> |
{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}} |
| 165 |
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\label{introEq:SM14} |
| 166 |
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\end{equation} |
| 167 |
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This result is the standard Boltzman statistical distribution. |
| 168 |
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Applying it to Eq.~\ref{introEq:SM10} one can obtain the following relation for ensemble averages: |
| 169 |
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\begin{equation} |
| 170 |
< |
eq here |
| 170 |
> |
\langle A \rangle = |
| 171 |
> |
\frac{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} |
| 172 |
> |
A( \boldsymbol{\Gamma} ) e^{-\beta E_{\gamma}}} |
| 173 |
> |
{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}} |
| 174 |
|
\label{introEq:SM15} |
| 175 |
|
\end{equation} |
| 176 |
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|
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|
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One last important consideration is that of ergodicity. Ergodicity is |
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the assumption that given an infinite amount of time, a system will |
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< |
visit every available point in phase space.\cite{fix} For most |
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> |
visit every available point in phase space.\cite{Frenkel1996} For most |
| 182 |
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systems, this is a valid assumption, except in cases where the system |
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< |
may be trapped in a local feature (\emph{i.~e.~glasses}). When valid, |
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> |
may be trapped in a local feature (\emph{e.g.}~glasses). When valid, |
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ergodicity allows the unification of a time averaged observation and |
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an ensemble averged one. If an observation is averaged over a |
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sufficiently long time, the system is assumed to visit all |
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weighted statistical average. This allows the researcher freedom of |
| 189 |
|
choice when deciding how best to measure a given observable. When an |
| 190 |
|
ensemble averaged approach seems most logical, the Monte Carlo |
| 191 |
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techniques described in Sec.~\ref{introSec:MC} can be utilized. |
| 191 |
> |
techniques described in Sec.~\ref{introSec:monteCarlo} can be utilized. |
| 192 |
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Conversely, if a problem lends itself to a time averaging approach, |
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the Molecular Dynamics techniques in Sec.~\ref{introSec:MD} can be |
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employed. |
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|
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< |
\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations} |
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> |
\section{\label{introSec:monteCarlo}Monte Carlo Simulations} |
| 197 |
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|
| 198 |
|
The Monte Carlo method was developed by Metropolis and Ulam for their |
| 199 |
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work in fissionable material.\cite{metropolis:1949} The method is so |
| 229 |
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\label{eq:mcEnsAvg} |
| 230 |
|
\end{equation} |
| 231 |
|
Where $\mathbf{r}^N$ stands for the coordinates of all $N$ particles |
| 232 |
< |
and $A$ is some observable that is only dependent on |
| 233 |
< |
position. $\langle A \rangle$ is the ensemble average of $A$ as |
| 234 |
< |
presented in Sec.~\ref{introSec:statThermo}. Because $A$ is |
| 235 |
< |
independent of momentum, the momenta contribution of the integral can |
| 236 |
< |
be factored out, leaving the configurational integral. Application of |
| 237 |
< |
the brute force method to this system would yield highly inefficient |
| 232 |
> |
and $A$ is some observable that is only dependent on position. This is |
| 233 |
> |
the ensemble average of $A$ as presented in |
| 234 |
> |
Sec.~\ref{introSec:statThermo}, except here $A$ is independent of |
| 235 |
> |
momentum. Therefore the momenta contribution of the integral can be |
| 236 |
> |
factored out, leaving the configurational integral. Application of the |
| 237 |
> |
brute force method to this system would yield highly inefficient |
| 238 |
|
results. Due to the Boltzman weighting of this integral, most random |
| 239 |
|
configurations will have a near zero contribution to the ensemble |
| 240 |
< |
average. This is where a importance sampling comes into |
| 240 |
> |
average. This is where importance sampling comes into |
| 241 |
|
play.\cite{allen87:csl} |
| 242 |
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|
| 243 |
|
Importance Sampling is a method where one selects a distribution from |
| 289 |
|
the use of a Markov chain whose limiting distribution was |
| 290 |
|
$\rho_{kT}(\mathbf{r}^N)$. |
| 291 |
|
|
| 292 |
< |
\subsubsection{\label{introSec:markovChains}Markov Chains} |
| 292 |
> |
\subsection{\label{introSec:markovChains}Markov Chains} |
| 293 |
|
|
| 294 |
|
A Markov chain is a chain of states satisfying the following |
| 295 |
|
conditions:\cite{leach01:mm} |
| 339 |
|
\label{introEq:MCmarkovEquil} |
| 340 |
|
\end{equation} |
| 341 |
|
|
| 342 |
< |
\subsubsection{\label{introSec:metropolisMethod}The Metropolis Method} |
| 342 |
> |
\subsection{\label{introSec:metropolisMethod}The Metropolis Method} |
| 343 |
|
|
| 344 |
|
In the Metropolis method\cite{metropolis:1953} |
| 345 |
|
Eq.~\ref{introEq:MCmarkovEquil} is solved such that |
| 368 |
|
\end{equation} |
| 369 |
|
This allows for the following set of acceptance rules be defined: |
| 370 |
|
\begin{equation} |
| 371 |
< |
EQ Here |
| 371 |
> |
\accMe( m \rightarrow n ) = |
| 372 |
> |
\begin{cases} |
| 373 |
> |
1& \text{if $\Delta \mathcal{U} \leq 0$,} \\ |
| 374 |
> |
e^{-\beta \Delta \mathcal{U}}& \text{if $\Delta \mathcal{U} > 0$.} |
| 375 |
> |
\end{cases} |
| 376 |
> |
\label{introEq:accRules} |
| 377 |
|
\end{equation} |
| 378 |
|
|
| 379 |
< |
Using the acceptance criteria from Eq.~\ref{fix} the Metropolis method |
| 380 |
< |
proceeds as follows |
| 381 |
< |
\begin{itemize} |
| 382 |
< |
\item Generate an initial configuration $fix$ which has some finite probability in $fix$. |
| 383 |
< |
\item Modify $fix$, to generate configuratioon $fix$. |
| 384 |
< |
\item If configuration $n$ lowers the energy of the system, accept the move with unity ($fix$ becomes $fix$). Otherwise accept with probability $fix$. |
| 379 |
> |
Using the acceptance criteria from Eq.~\ref{introEq:accRules} the |
| 380 |
> |
Metropolis method proceeds as follows |
| 381 |
> |
\begin{enumerate} |
| 382 |
> |
\item Generate an initial configuration $\mathbf{r}^N$ which has some finite probability in $\rho_{kT}$. |
| 383 |
> |
\item Modify $\mathbf{r}^N$, to generate configuratioon $\mathbf{r^{\prime}}^N$. |
| 384 |
> |
\item If the new configuration lowers the energy of the system, accept the move with unity ($\mathbf{r}^N$ becomes $\mathbf{r^{\prime}}^N$). Otherwise accept with probability $e^{-\beta \Delta \mathcal{U}}$. |
| 385 |
|
\item Accumulate the average for the configurational observable of intereest. |
| 386 |
< |
\item Repeat from step 2 until average converges. |
| 387 |
< |
\end{itemize} |
| 386 |
> |
\item Repeat from step 2 until the average converges. |
| 387 |
> |
\end{enumerate} |
| 388 |
|
One important note is that the average is accumulated whether the move |
| 389 |
|
is accepted or not, this ensures proper weighting of the average. |
| 390 |
< |
Using Eq.~\ref{fix} it becomes clear that the accumulated averages are |
| 391 |
< |
the ensemble averages, as this method ensures that the limiting |
| 392 |
< |
distribution is the Boltzman distribution. |
| 390 |
> |
Using Eq.~\ref{introEq:Importance4} it becomes clear that the |
| 391 |
> |
accumulated averages are the ensemble averages, as this method ensures |
| 392 |
> |
that the limiting distribution is the Boltzman distribution. |
| 393 |
|
|
| 394 |
< |
\subsection{\label{introSec:MD}Molecular Dynamics Simulations} |
| 394 |
> |
\section{\label{introSec:MD}Molecular Dynamics Simulations} |
| 395 |
|
|
| 396 |
|
The main simulation tool used in this research is Molecular Dynamics. |
| 397 |
|
Molecular Dynamics is when the equations of motion for a system are |
| 399 |
|
momentum of a system, allowing the calculation of not only |
| 400 |
|
configurational observables, but momenta dependent ones as well: |
| 401 |
|
diffusion constants, velocity auto correlations, folding/unfolding |
| 402 |
< |
events, etc. Due to the principle of ergodicity, Eq.~\ref{fix}, the |
| 403 |
< |
average of these observables over the time period of the simulation |
| 404 |
< |
are taken to be the ensemble averages for the system. |
| 402 |
> |
events, etc. Due to the principle of ergodicity, |
| 403 |
> |
Sec.~\ref{introSec:ergodic}, the average of these observables over the |
| 404 |
> |
time period of the simulation are taken to be the ensemble averages |
| 405 |
> |
for the system. |
| 406 |
|
|
| 407 |
|
The choice of when to use molecular dynamics over Monte Carlo |
| 408 |
|
techniques, is normally decided by the observables in which the |
| 415 |
|
centered around the dynamic properties of phospholipid bilayers, |
| 416 |
|
making molecular dynamics key in the simulation of those properties. |
| 417 |
|
|
| 418 |
< |
\subsubsection{Molecular dynamics Algorithm} |
| 418 |
> |
\subsection{\label{introSec:mdAlgorithm}The Molecular Dynamics Algorithm} |
| 419 |
|
|
| 420 |
|
To illustrate how the molecular dynamics technique is applied, the |
| 421 |
|
following sections will describe the sequence involved in a |
| 422 |
< |
simulation. Sec.~\ref{fix} deals with the initialization of a |
| 423 |
< |
simulation. Sec.~\ref{fix} discusses issues involved with the |
| 424 |
< |
calculation of the forces. Sec.~\ref{fix} concludes the algorithm |
| 425 |
< |
discussion with the integration of the equations of motion. \cite{fix} |
| 422 |
> |
simulation. Sec.~\ref{introSec:mdInit} deals with the initialization |
| 423 |
> |
of a simulation. Sec.~\ref{introSec:mdForce} discusses issues |
| 424 |
> |
involved with the calculation of the forces. |
| 425 |
> |
Sec.~\ref{introSec:mdIntegrate} concludes the algorithm discussion |
| 426 |
> |
with the integration of the equations of motion.\cite{Frenkel1996} |
| 427 |
|
|
| 428 |
< |
\subsubsection{initialization} |
| 428 |
> |
\subsection{\label{introSec:mdInit}Initialization} |
| 429 |
|
|
| 430 |
|
When selecting the initial configuration for the simulation it is |
| 431 |
|
important to consider what dynamics one is hoping to observe. |
| 432 |
< |
Ch.~\ref{fix} deals with the formation and equilibrium dynamics of |
| 432 |
> |
Ch.~\ref{chapt:lipid} deals with the formation and equilibrium dynamics of |
| 433 |
|
phospholipid membranes. Therefore in these simulations initial |
| 434 |
|
positions were selected that in some cases dispersed the lipids in |
| 435 |
|
water, and in other cases structured the lipids into preformed |
| 448 |
|
whole. This arises from the fact that most simulations are of systems |
| 449 |
|
in equilibrium in the absence of outside forces. Therefore any net |
| 450 |
|
movement would be unphysical and an artifact of the simulation method |
| 451 |
< |
used. The final point addresses teh selection of the magnitude of the |
| 451 |
> |
used. The final point addresses the selection of the magnitude of the |
| 452 |
|
initial velocities. For many simulations it is convienient to use |
| 453 |
|
this opportunity to scale the amount of kinetic energy to reflect the |
| 454 |
|
desired thermal distribution of the system. However, it must be noted |
| 456 |
|
first few initial simulation steps due to either loss or gain of |
| 457 |
|
kinetic energy from energy stored in potential degrees of freedom. |
| 458 |
|
|
| 459 |
< |
\subsubsection{Force Evaluation} |
| 459 |
> |
\subsection{\label{introSec:mdForce}Force Evaluation} |
| 460 |
|
|
| 461 |
|
The evaluation of forces is the most computationally expensive portion |
| 462 |
|
of a given molecular dynamics simulation. This is due entirely to the |
| 463 |
|
evaluation of long range forces in a simulation, typically pair-wise. |
| 464 |
|
These forces are most commonly the Van der Waals force, and sometimes |
| 465 |
< |
Coulombic forces as well. For a pair-wise force, there are $fix$ |
| 466 |
< |
pairs to be evaluated, where $n$ is the number of particles in the |
| 467 |
< |
system. This leads to the calculations scaling as $fix$, making large |
| 465 |
> |
Coulombic forces as well. For a pair-wise force, there are $N(N-1)/ 2$ |
| 466 |
> |
pairs to be evaluated, where $N$ is the number of particles in the |
| 467 |
> |
system. This leads to the calculations scaling as $N^2$, making large |
| 468 |
|
simulations prohibitive in the absence of any computation saving |
| 469 |
|
techniques. |
| 470 |
|
|
| 471 |
|
Another consideration one must resolve, is that in a given simulation |
| 472 |
|
a disproportionate number of the particles will feel the effects of |
| 473 |
< |
the surface.\cite{fix} For a cubic system of 1000 particles arranged |
| 474 |
< |
in a $10x10x10$ cube, 488 particles will be exposed to the surface. |
| 475 |
< |
Unless one is simulating an isolated particle group in a vacuum, the |
| 476 |
< |
behavior of the system will be far from the desired bulk |
| 477 |
< |
charecteristics. To offset this, simulations employ the use of |
| 478 |
< |
periodic boundary images.\cite{fix} |
| 473 |
> |
the surface.\cite{allen87:csl} For a cubic system of 1000 particles |
| 474 |
> |
arranged in a $10 \times 10 \times 10$ cube, 488 particles will be |
| 475 |
> |
exposed to the surface. Unless one is simulating an isolated particle |
| 476 |
> |
group in a vacuum, the behavior of the system will be far from the |
| 477 |
> |
desired bulk charecteristics. To offset this, simulations employ the |
| 478 |
> |
use of periodic boundary images.\cite{born:1912} |
| 479 |
|
|
| 480 |
|
The technique involves the use of an algorithm that replicates the |
| 481 |
|
simulation box on an infinite lattice in cartesian space. Any given |
| 482 |
|
particle leaving the simulation box on one side will have an image of |
| 483 |
< |
itself enter on the opposite side (see Fig.~\ref{fix}). |
| 484 |
< |
\begin{equation} |
| 485 |
< |
EQ Here |
| 486 |
< |
\end{equation} |
| 487 |
< |
In addition, this sets that any given particle pair has an image, real |
| 468 |
< |
or periodic, within $fix$ of each other. A discussion of the method |
| 469 |
< |
used to calculate the periodic image can be found in Sec.\ref{fix}. |
| 483 |
> |
itself enter on the opposite side (see Fig.~\ref{introFig:pbc}). In |
| 484 |
> |
addition, this sets that any given particle pair has an image, real or |
| 485 |
> |
periodic, within $fix$ of each other. A discussion of the method used |
| 486 |
> |
to calculate the periodic image can be found in |
| 487 |
> |
Sec.\ref{oopseSec:pbc}. |
| 488 |
|
|
| 489 |
+ |
\begin{figure} |
| 490 |
+ |
\centering |
| 491 |
+ |
\includegraphics[width=\linewidth]{pbcFig.eps} |
| 492 |
+ |
\caption[An illustration of periodic boundry conditions]{A 2-D illustration of periodic boundry conditions. As one particle leaves the right of the simulation box, an image of it enters the left.} |
| 493 |
+ |
\label{introFig:pbc} |
| 494 |
+ |
\end{figure} |
| 495 |
+ |
|
| 496 |
|
Returning to the topic of the computational scale of the force |
| 497 |
|
evaluation, the use of periodic boundary conditions requires that a |
| 498 |
|
cutoff radius be employed. Using a cutoff radius improves the |
| 499 |
|
efficiency of the force evaluation, as particles farther than a |
| 500 |
< |
predetermined distance, $fix$, are not included in the |
| 501 |
< |
calculation.\cite{fix} In a simultation with periodic images, $fix$ |
| 502 |
< |
has a maximum value of $fix$. Fig.~\ref{fix} illustrates how using an |
| 503 |
< |
$fix$ larger than this value, or in the extreme limit of no $fix$ at |
| 504 |
< |
all, the corners of the simulation box are unequally weighted due to |
| 505 |
< |
the lack of particle images in the $x$, $y$, or $z$ directions past a |
| 506 |
< |
disance of $fix$. |
| 500 |
> |
predetermined distance, $r_{\text{cut}}$, are not included in the |
| 501 |
> |
calculation.\cite{Frenkel1996} In a simultation with periodic images, |
| 502 |
> |
$r_{\text{cut}}$ has a maximum value of $\text{box}/2$. |
| 503 |
> |
Fig.~\ref{introFig:rMax} illustrates how when using an |
| 504 |
> |
$r_{\text{cut}}$ larger than this value, or in the extreme limit of no |
| 505 |
> |
$r_{\text{cut}}$ at all, the corners of the simulation box are |
| 506 |
> |
unequally weighted due to the lack of particle images in the $x$, $y$, |
| 507 |
> |
or $z$ directions past a disance of $\text{box} / 2$. |
| 508 |
|
|
| 509 |
< |
With the use of an $fix$, however, comes a discontinuity in the |
| 510 |
< |
potential energy curve (Fig.~\ref{fix}). To fix this discontinuity, |
| 511 |
< |
one calculates the potential energy at the $r_{\text{cut}}$, and add |
| 512 |
< |
that value to the potential. This causes the function to go smoothly |
| 513 |
< |
to zero at the cutoff radius. This ensures conservation of energy |
| 514 |
< |
when integrating the Newtonian equations of motion. |
| 509 |
> |
\begin{figure} |
| 510 |
> |
\centering |
| 511 |
> |
\includegraphics[width=\linewidth]{rCutMaxFig.eps} |
| 512 |
> |
\caption[An explanation of $r_{\text{cut}}$]{The yellow atom has all other images wrapped to itself as the center. If $r_{\text{cut}}=\text{box}/2$, then the distribution is uniform (blue atoms). However, when $r_{\text{cut}}>\text{box}/2$ the corners are disproportionately weighted (green atoms) vs the axial directions (shaded regions).} |
| 513 |
> |
\label{introFig:rMax} |
| 514 |
> |
\end{figure} |
| 515 |
> |
|
| 516 |
> |
With the use of an $r_{\text{cut}}$, however, comes a discontinuity in |
| 517 |
> |
the potential energy curve (Fig.~\ref{introFig:shiftPot}). To fix this |
| 518 |
> |
discontinuity, one calculates the potential energy at the |
| 519 |
> |
$r_{\text{cut}}$, and adds that value to the potential, causing |
| 520 |
> |
the function to go smoothly to zero at the cutoff radius. This |
| 521 |
> |
shifted potential ensures conservation of energy when integrating the |
| 522 |
> |
Newtonian equations of motion. |
| 523 |
|
|
| 524 |
+ |
\begin{figure} |
| 525 |
+ |
\centering |
| 526 |
+ |
\includegraphics[width=\linewidth]{shiftedPot.eps} |
| 527 |
+ |
\caption[Shifting the Lennard-Jones Potential]{The Lennard-Jones potential is shifted to remove the discontiuity at $r_{\text{cut}}$.} |
| 528 |
+ |
\label{introFig:shiftPot} |
| 529 |
+ |
\end{figure} |
| 530 |
+ |
|
| 531 |
|
The second main simplification used in this research is the Verlet |
| 532 |
|
neighbor list. \cite{allen87:csl} In the Verlet method, one generates |
| 533 |
|
a list of all neighbor atoms, $j$, surrounding atom $i$ within some |
| 539 |
|
giving rise to the possibility that a particle has left or joined a |
| 540 |
|
neighbor list. |
| 541 |
|
|
| 542 |
< |
\subsection{\label{introSec:MDintegrate} Integration of the equations of motion} |
| 542 |
> |
\subsection{\label{introSec:mdIntegrate} Integration of the equations of motion} |
| 543 |
|
|
| 544 |
|
A starting point for the discussion of molecular dynamics integrators |
| 545 |
|
is the Verlet algorithm. \cite{Frenkel1996} It begins with a Taylor |
