--- trunk/xDissertation/Introduction.tex	2008/03/20 22:21:43	3374
+++ trunk/xDissertation/Introduction.tex	2008/04/16 21:56:34	3383
@@ -3,31 +3,35 @@ because of their critical role as the foundation of bi
 \section{Background on the Problem\label{In:sec:pro}}
 Phospholipid molecules are the primary topic of this dissertation
 because of their critical role as the foundation of biological
-membranes. The chemical structure of phospholipids includes the polar
-head group which is due to a large charge separation between phosphate
-and amino alcohol, and the nonpolar tails that contains fatty acid
-chains. Depending on the alcohol which phosphate and fatty acid chains
-are esterified to, the phospholipids are divided into
-glycerophospholipids and sphingophospholipids.~\cite{Cevc80} The
-chemical structures are shown in figure~\ref{Infig:lipid}.
+membranes. The chemical structure of phospholipids includes a head
+group with a large dipole moment which is due to the large charge
+separation between phosphate and amino alcohol, and a nonpolar tail
+that contains fatty acid chains. Depending on the specific alcohol
+which the phosphate and fatty acid chains are esterified to, the
+phospholipids are divided into glycerophospholipids and
+sphingophospholipids.~\cite{Cevc80} The chemical structures are shown
+in figure~\ref{Infig:lipid}.
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{./figures/inLipid.pdf}
-\caption{The chemical structure of glycerophospholipids (left) and
-sphingophospholipids (right).\cite{Cevc80}}
+\caption[The chemical structure of lipids]{The chemical structure of
+glycerophospholipids (left) and sphingophospholipids
+(right).\cite{Cevc80}}
 \label{Infig:lipid}
 \end{figure}
-The glycerophospholipid is the dominant phospholipid in membranes. The
-types of glycerophospholipids depend on the X group, and the
-chains. For example, if X is choline, the lipids are known as
-phosphatidylcholine (PC), or if X is ethanolamine, the lipids are
-known as phosphatidyethanolamine (PE). Table~\ref{Intab:pc} listed a
-number types of phosphatidycholine with different fatty acids as the
-lipid chains.
+Glycerophospholipids are the dominant phospholipids in biological
+membranes. The type of glycerophospholipid depends on the identity of
+the X group, and the chains. For example, if X is choline
+[(CH$_3$)$_3$N$^+$CH$_2$CH$_2$OH], the lipids are known as
+phosphatidylcholine (PC), or if X is ethanolamine
+[H$_3$N$^+$CH$_2$CH$_2$OH], the lipids are known as
+phosphatidyethanolamine (PE). Table~\ref{Intab:pc} listed a number
+types of phosphatidycholine with different fatty acids as the lipid
+chains.
 \begin{table*}
 \begin{minipage}{\linewidth}
 \begin{center}
-\caption{A number types of phosphatidycholine.}
+\caption{A NUMBER TYPES OF PHOSPHATIDYCHOLINE}
 \begin{tabular}{lll}
 \hline 
   & Fatty acid & Generic Name \\
@@ -42,28 +46,30 @@ lipid chains.
 \end{center}
 \end{minipage}
 \end{table*}
-When dispersed in water, lipids self assemble into a mumber of
+When dispersed in water, lipids self assemble into a number of
 topologically distinct bilayer structures. The phase behavior of lipid
 bilayers has been explored experimentally~\cite{Cevc80}, however, a
 complete understanding of the mechanism and driving forces behind the
 various phases has not been achieved.
 
-\subsection{Ripple Phase\label{In:ssec:ripple}}
+\subsection{The Ripple Phase\label{In:ssec:ripple}}
 The $P_{\beta'}$ {\it ripple phase} of lipid bilayers, named from the
 periodic buckling of the membrane, is an intermediate phase which is
 developed either from heating the gel phase $L_{\beta'}$ or cooling
-the fluid phase $L_\alpha$. A Sketch is shown in
+the fluid phase $L_\alpha$. A sketch of the phases is shown in
 figure~\ref{Infig:phaseDiagram}.~\cite{Cevc80}
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{./figures/inPhaseDiagram.pdf}
-\caption{A phase diagram of lipid bilayer. With increasing the
-temperature, the bilayer can go through a gel ($L_{\beta'}$), ripple
-($P_{\beta'}$) to fluid ($L_\alpha$) phase transition.~\cite{Cevc80}}
+\caption[Phases of PC lipid bilayers]{Phases of PC lipid
+bilayers. With increasing temperature, phosphotidylcholine (PC)
+bilayers can go through $L_{\beta'} \rightarrow P_{\beta'}$ (gel
+$\rightarrow$ ripple) and $P_{\beta'} \rightarrow L_\alpha$ (ripple
+$\rightarrow$ fluid) phase transitions.~\cite{Cevc80}}
 \label{Infig:phaseDiagram}
 \end{figure}
-Most structural information of the ripple phase has been obtained by
-the X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture
+Most structural information about the ripple phase has been obtained
+by X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture
 electron microscopy (FFEM).~\cite{Copeland80,Meyer96} The X-ray
 diffraction work by Katsaras {\it et al.} showed that a rich phase
 diagram exhibiting both {\it asymmetric} and {\it symmetric} ripples
@@ -74,40 +80,43 @@ mica.~\cite{Kaasgaard03}
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{./figures/inRipple.pdf}
-\caption{The experimental observed ripple phase. The top image is
-obtained by Sun {\it et al.} using X-ray diffraction~\cite{Sun96},
-and the bottom one is observed by Kaasgaard {\it et al.} using
-AFM.~\cite{Kaasgaard03}}
+\caption[Experimental observations of the riple phase]{Experimental
+observations of the riple phase. The top image is an electrostatic
+density map obtained by Sun {\it et al.} using X-ray
+diffraction~\cite{Sun96}.  The lower figures are the surface topology
+of various ripple domains in bilayers supported in mica. The AFM
+images were observed by Kaasgaard {\it et al.}.~\cite{Kaasgaard03}}
 \label{Infig:ripple}
 \end{figure}
-Figure~\ref{Infig:ripple} shows the ripple phase oberved by X-ray
+Figure~\ref{Infig:ripple} shows the ripple phase oberved by both X-ray
 diffraction and AFM. The experimental results provide strong support
 for a 2-dimensional triangular packing lattice of the lipid molecules
 within the ripple phase.  This is a notable change from the observed
 lipid packing within the gel phase,~\cite{Cevc80} although Tenchov
-{\it et al.} have recently observed near-hexagonal packing in some
+{\it et al.} have recently observed near-triangular packing in some
 phosphatidylcholine (PC) gel phases.~\cite{Tenchov2001} However, the
 physical mechanism for the formation of the ripple phase has never
 been explained and the microscopic structure of the ripple phase has
 never been elucidated by experiments. Computational simulation is a
-perfect tool to study the microscopic properties for a
-system. However, the large length scale the ripple structure and the
-long time scale of the formation of the ripples are crucial obstacles
-to performing the actual work. The principal ideas explored in this
-dissertation are attempts to break the computational task up by
+very good tool to study the microscopic properties for a
+system. However, the large length scale of the ripple structures and
+the long time required for the formation of the ripples are crucial
+obstacles to performing the actual work. The principal ideas explored
+in this dissertation are attempts to break the computational task up
+by
 \begin{itemize}
 \item Simplifying the lipid model.
-\item Improving algorithm for integrating the equations of motion.
+\item Improving the algorithm for integrating the equations of motion.
 \end{itemize}
 In chapters~\ref{chap:mc} and~\ref{chap:md}, we use a simple point
-dipole spin model and a coarse-grained molecualr scale model to
+dipole spin model and a coarse-grained molecular scale model to
 perform the Monte Carlo and Molecular Dynamics simulations
 respectively, and in chapter~\ref{chap:ld}, we develop a Langevin
 Dynamics algorithm which excludes the explicit solvent to improve the
 efficiency of the simulations.
 
-\subsection{Lattice Model\label{In:ssec:model}}
-The gel-like characteristic (relatively immobile molecules) exhibited
+\subsection{Lattice Models\label{In:ssec:model}}
+The gel-like characteristic (laterally immobile molecules) exhibited
 by the ripple phase makes it feasible to apply a lattice model to
 study the system. The popular $2$ dimensional lattice models, {\it
 i.e.}, the Ising, $X-Y$, and Heisenberg models, show {\it frustration}
@@ -128,24 +137,24 @@ anti-aligned structure.
 \begin{figure}
 \centering
 \includegraphics[width=3in]{./figures/inFrustration.pdf}
-\caption{Frustration on triangular lattice, the spins and dipoles are
-represented by arrows. The multiple local minima of energy states
-induce the frustration for spins and dipoles picking the directions.}
+\caption[Frustration on triangular lattice]{Frustration on triangular
+lattice, the spins and dipoles are represented by arrows. The multiple
+local minima of energy states induce frustration for spins and dipoles
+resulting in disordered low-temperature phases.}
 \label{Infig:frustration}
 \end{figure}
-The spins in figure~\ref{Infig:frustration} shows an illustration of
-the frustration for $J < 0$ on a triangular lattice. There are
-multiple local minima energy states which are independent of the
-direction of the spin on top of the triangle, therefore infinite
-possibilities for the packing of spins which induces what is known as
-``complete regular frustration'' which leads to disordered low
-temperature phases. The similarity goes to the dipoles on a hexagonal
-lattice, which are shown by the dipoles in
-figure~\ref{Infig:frustration}. In this circumstance, the dipoles want
-to be aligned, however, due to the long wave fluctuation, at low
-temperature, the aligned state becomes unstable, vortex is formed and
-results in multiple local minima of energy states. The dipole on the
-center of the hexagonal lattice is frustrated.
+The spins in figure~\ref{Infig:frustration} illustrate frustration for
+$J < 0$ on a triangular lattice. There are multiple local minima
+energy states which are independent of the direction of the spin on
+top of the triangle, therefore infinite possibilities for orienting
+large numbers spins. This induces what is known as ``complete regular
+frustration'' which leads to disordered low temperature phases. This
+behavior extends to dipoles on a triangular lattices, which are shown
+in the lower portion of figure~\ref{Infig:frustration}. In this case,
+dipole-aligned structures are energetically favorable, however, at low
+temperature, vortices are easily formed, and, this results in multiple
+local minima of energy states for a central dipole. The dipole on the
+center of the hexagonal lattice is therefore frustrated.
 
 The lack of translational degrees of freedom in lattice models
 prevents their utilization in models for surface buckling. In
@@ -159,7 +168,7 @@ principles of statistical mechanics.~\cite{Tolman1979}
 to key concepts of classical statistical mechanics that we used in
 this dissertation. Tolman gives an excellent introduction to the
 principles of statistical mechanics.~\cite{Tolman1979} A large part of
-section~\ref{In:sec:SM} will follow Tolman's notation.
+section~\ref{In:sec:SM} follows Tolman's notation.
 
 \subsection{Ensembles\label{In:ssec:ensemble}}
 In classical mechanics, the state of the system is completely
@@ -269,7 +278,7 @@ selected moving representative points in the phase spa
 and the rate of density change is zero in the neighborhood of any
 selected moving representative points in the phase space.
 
-The condition of the ensemble is determined by the density
+The type of thermodynamic ensemble is determined by the density
 distribution. If we consider the density distribution as only a
 function of $q$ and $p$, which means the rate of change of the phase
 space density in the neighborhood of all representative points in the
@@ -298,11 +307,13 @@ constant everywhere in the phase space,
 \rho = \mathrm{const.}
 \label{Ineq:uniformEnsemble}
 \end{equation}
-the ensemble is called {\it uniform ensemble}.
+the ensemble is called {\it uniform ensemble}, but this ensemble has
+little relevance for physical chemistry. It is an ensemble with
+essentially infinite temperature.
 
 \subsubsection{The Microcanonical Ensemble\label{In:sssec:microcanonical}}
-Another useful ensemble is the {\it microcanonical ensemble}, for
-which:
+The most useful ensemble for Molecular Dynamics is the {\it
+microcanonical ensemble}, for which:
 \begin{equation}
 \rho = \delta \left( H(q^N, p^N) - E \right) \frac{1}{\Sigma (N, V, E)}
 \label{Ineq:microcanonicalEnsemble}
@@ -319,8 +330,9 @@ makes the argument of $\ln$ dimensionless. In this cas
 \end{equation}
 where $k_B$ is the Boltzmann constant and $C^N$ is a number which
 makes the argument of $\ln$ dimensionless. In this case, $C^N$ is the
-total phase space volume of one state. The entropy of a microcanonical
-ensemble is given by
+total phase space volume of one state which has the same units as
+$\Sigma(N, V, E)$. The entropy of a microcanonical ensemble is given
+by
 \begin{equation}
 S = k_B \ln \left(\frac{\Sigma(N, V, E)}{C^N}\right).
 \label{Ineq:entropy}
@@ -337,12 +349,12 @@ Z_N = \int d \vec q~^N \int_\Gamma d \vec p~^N  e^{-H(
 Z_N = \int d \vec q~^N \int_\Gamma d \vec p~^N  e^{-H(q^N, p^N) / k_B T},
 \label{Ineq:partitionFunction}
 \end{equation}
-which is also known as the canonical{\it partition function}. $\Gamma$
-indicates that the integral is over all phase space. In the canonical
-ensemble, $N$, the total number of particles, $V$, total volume, and
-$T$, the temperature, are constants. The systems with the lowest
-energies hold the largest population. According to maximum principle,
-thermodynamics maximizes the entropy $S$, implying that
+which is also known as the canonical {\it partition
+function}. $\Gamma$ indicates that the integral is over all phase
+space. In the canonical ensemble, $N$, the total number of particles,
+$V$, total volume, and $T$, the temperature, are constants. The
+systems with the lowest energies hold the largest
+population. Thermodynamics maximizes the entropy, $S$, implying that
 \begin{equation}
 \begin{array}{ccc}
 \delta S = 0 & \mathrm{and} & \delta^2 S < 0.
@@ -363,11 +375,11 @@ There is an implicit assumption that our arguments are
 system and the distribution of microscopic states.
 
 There is an implicit assumption that our arguments are based on so
-far. A representative point in the phase space is equally likely to be
-found in any energetically allowed region. In other words, all
-energetically accessible states are represented equally, the
-probabilities to find the system in any of the accessible states is
-equal. This is called the principle of equal a {\it priori}
+far. Tow representative points in phase space are equally likely to be
+found if they have the same energy. In other words, all energetically
+accessible states are represented , and the probabilities to find the
+system in any of the accessible states is equal to that states
+Boltzmann weight. This is called the principle of equal a {\it priori}
 probabilities.
 
 \subsection{Statistical Averages\label{In:ssec:average}}
@@ -402,7 +414,7 @@ mechanics, so Eq.~\ref{Ineq:timeAverage1} becomes
 \frac{1}{T} \int_{0}^{T} F[q^N(t), p^N(t)] dt
 \label{Ineq:timeAverage2}
 \end{equation}
-for an infinite time interval.
+for an finite time interval, $T$.
 
 \subsubsection{Ergodicity\label{In:sssec:ergodicity}}
 The {\it ergodic hypothesis}, an important hypothesis governing modern
@@ -445,20 +457,22 @@ C_{AB}(\tau) = \langle A(0)B(\tau) \rangle = \lim_{T \
 \frac{1}{T} \int_{0}^{T} dt A(t) B(t + \tau),
 \label{Ineq:crosscorrelationFunction}
 \end{equation}
-and called {\it cross correlation function}.
+and is called a {\it cross correlation function}.
 
 We know from the ergodic hypothesis that there is a relationship
 between time average and ensemble average. We can put the correlation
-function in a classical mechanics form,
+function in a classical mechanical form,
 \begin{equation}
-C_{AA}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N(t), p^N(t)]
-A[q^N(t+\tau), p^N(t+\tau)] \rho(q, p)
+C_{AA}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N, p^N]
+A[q^N(\tau), p^N(\tau)] \rho(q^N, p^N)
 \label{Ineq:autocorrelationFunctionCM}
 \end{equation}
-and
+where $q^N(\tau)$, $p^N(\tau)$ is the phase space point that follows
+classical evolution of the point $q^N$, $p^N$ after a tme $\tau$ has
+elapsed, and
 \begin{equation}
-C_{AB}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N(t), p^N(t)]
-B[q^N(t+\tau), p^N(t+\tau)] \rho(q, p)
+C_{AB}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N, p^N]
+B[q^N(\tau), p^N(\tau)] \rho(q^N, p^N)
 \label{Ineq:crosscorrelationFunctionCM}
 \end{equation}
 as the autocorrelation function and cross correlation functions
@@ -479,10 +493,10 @@ while Molecular Dynamics simulations provide dynamic
 simulations. One is that the Monte Carlo simulations are time
 independent methods of sampling structural features of an ensemble,
 while Molecular Dynamics simulations provide dynamic
-information. Additionally, Monte Carlo methods are a stochastic
-processes, the future configurations of the system are not determined
+information. Additionally, Monte Carlo methods are stochastic
+processes; the future configurations of the system are not determined
 by its past. However, in Molecular Dynamics, the system is propagated
-by Newton's equation of motion, and the trajectory of the system
+by Hamilton's equations of motion, and the trajectory of the system
 evolving in phase space is deterministic. Brief introductions of the
 two algorithms are given in section~\ref{In:ssec:mc}
 and~\ref{In:ssec:md}. Langevin Dynamics, an extension of the Molecular
@@ -493,9 +507,9 @@ algorithm is usually applied to the canonical ensemble
 A Monte Carlo integration algorithm was first introduced by Metropolis
 {\it et al.}~\cite{Metropolis53} The basic Metropolis Monte Carlo
 algorithm is usually applied to the canonical ensemble, a
-Boltzmann-weighted ensemble, in which the $N$, the total number of
-particles, $V$, total volume, $T$, temperature are constants. An
-average in this ensemble is given
+Boltzmann-weighted ensemble, in which $N$, the total number of
+particles, $V$, the total volume, and $T$, the temperature are
+constants. An average in this ensemble is given
 \begin{equation}
 \langle A \rangle = \frac{1}{Z_N} \int d \vec q~^N \int d \vec p~^N
 A(q^N, p^N) e^{-H(q^N, p^N) / k_B T}.
@@ -519,9 +533,14 @@ Eq.~\ref{Ineq:configurationIntegral} is equivalent to
 \rangle$ is now a configuration integral, and
 Eq.~\ref{Ineq:configurationIntegral} is equivalent to
 \begin{equation}
-\langle A \rangle = \int d \vec q~^N A \rho(q^N).
+\langle A \rangle = \int d \vec q~^N A \rho(q^N),
 \label{Ineq:configurationAve}
 \end{equation}
+where $\rho(q^N)$ is a configurational probability
+\begin{equation}
+\rho(q^N) = \frac{e^{-U(q^N) / k_B T}}{\int d \vec q~^N e^{-U(q^N) / k_B T}}.
+\label{Ineq:configurationProb}
+\end{equation}
 
 In a Monte Carlo simulation of a system in the canonical ensemble, the
 probability of the system being in a state $s$ is $\rho_s$, the change
@@ -536,8 +555,8 @@ to state $s'$. Since $\rho_s$ is independent of time a
 \frac{d \rho_{s}^{equilibrium}}{dt} = 0,
 \label{Ineq:equiProb}
 \end{equation}
-which means $\sum_{s'} [ -w_{ss'}\rho_s + w_{s's}\rho_{s'} ]$ is $0$
-for all $s'$. So
+the sum of transition probabilities $\sum_{s'} [ -w_{ss'}\rho_s +
+w_{s's}\rho_{s'} ]$ is $0$ for all $s'$. So
 \begin{equation}
 \frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = \frac{w_{s's}}{w_{ss'}}.
 \label{Ineq:relationshipofRhoandW}
@@ -552,32 +571,30 @@ then the ratio of transition probabilities,
 \frac{w_{s's}}{w_{ss'}} = e^{-(U_s - U_{s'}) / k_B T},
 \label{Ineq:conditionforBoltzmannStatistics}
 \end{equation}
-An algorithm that shows how Monte Carlo simulation generates a
+An algorithm that indicates how a Monte Carlo simulation generates a
 transition probability governed by
 \ref{Ineq:conditionforBoltzmannStatistics}, is given schematically as,
 \begin{enumerate}
 \item\label{Initm:oldEnergy} Choose a particle randomly, and calculate
-the energy of the rest of the system due to the location of the particle.
+the energy of the rest of the system due to the current location of
+the particle.
 \item\label{Initm:newEnergy} Make a random displacement of the particle,
 calculate the new energy taking the movement of the particle into account.
   \begin{itemize}
-     \item If the energy goes down, keep the new configuration and return to
-step~\ref{Initm:oldEnergy}.
+     \item If the energy goes down, keep the new configuration.
      \item If the energy goes up, pick a random number between $[0,1]$.
         \begin{itemize}
            \item If the random number smaller than
-$e^{-(U_{new} - U_{old})} / k_B T$, keep the new configuration and return to
-step~\ref{Initm:oldEnergy}.
+$e^{-(U_{new} - U_{old})} / k_B T$, keep the new configuration.
            \item If the random number is larger than
-$e^{-(U_{new} - U_{old})} / k_B T$, keep the old configuration and return to
-step~\ref{Initm:oldEnergy}.
+$e^{-(U_{new} - U_{old})} / k_B T$, keep the old configuration.
         \end{itemize}
   \end{itemize}
 \item\label{Initm:accumulateAvg} Accumulate the averages based on the
-current configuartion.
+current configuration.
 \item Go to step~\ref{Initm:oldEnergy}.
 \end{enumerate}
-It is important for sampling accurately that the old configuration is
+It is important for sampling accuracy that the old configuration is
 sampled again if it is kept.
 
 \subsection{Molecular Dynamics\label{In:ssec:md}}
@@ -589,10 +606,10 @@ real experiment, the instantaneous positions and momen
 evolution of the system obeys the laws of classical mechanics, and in
 most situations, there is no need to account for quantum effects. In a
 real experiment, the instantaneous positions and momenta of the
-particles in the system are neither important nor measurable, the
-observable quantities are usually an average value for a finite time
-interval. These quantities are expressed as a function of positions
-and momenta in Molecular Dynamics simulations. For example,
+particles in the system are ofter neither important nor measurable,
+the observable quantities are usually an average value for a finite
+time interval. These quantities are expressed as a function of
+positions and momenta in Molecular Dynamics simulations. For example,
 temperature of the system is defined by
 \begin{equation}
 \frac{3}{2} N k_B T = \langle \sum_{i=1}^N \frac{1}{2} m_i v_i \rangle,
@@ -605,8 +622,8 @@ distributed randomly to the particles using a Maxwell-
 The initial positions of the particles are chosen so that there is no
 overlap of the particles. The initial velocities at first are
 distributed randomly to the particles using a Maxwell-Boltzmann
-ditribution, and then shifted to make the total linear momentum of the
-system $0$.
+distribution, and then shifted to make the total linear momentum of
+the system $0$.
 
 The core of Molecular Dynamics simulations is the calculation of
 forces and the integration algorithm. Calculation of the forces often
@@ -632,14 +649,14 @@ is
 equations of motion. The Taylor expansion of the position at time $t$
 is
 \begin{equation}
-q(t+\Delta t)= q(t) + v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 +
+q(t+\Delta t)= q(t) + \frac{p(t)}{m} \Delta t + \frac{f(t)}{2m}\Delta t^2 +
 	\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} +
 	\mathcal{O}(\Delta t^4)
 \label{Ineq:verletFuture}
 \end{equation}
 for a later time $t+\Delta t$, and 
 \begin{equation}
-q(t-\Delta t)= q(t) - v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 -
+q(t-\Delta t)= q(t) - \frac{p(t)}{m} \Delta t + \frac{f(t)}{2m}\Delta t^2 -
 	\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} +
 	\mathcal{O}(\Delta t^4) ,
 \label{Ineq:verletPrevious}
@@ -666,16 +683,16 @@ code can be found in Ref.~\cite{Meineke2005}.
 code can be found in Ref.~\cite{Meineke2005}.
 
 \subsection{Langevin Dynamics\label{In:ssec:ld}}
-In many cases, the properites of the solvent in a system, like the
-water in the lipid-water system studied in this dissertation, are less
-interesting to the researchers than the behavior of the
-solute. However, the major computational expense is ofter the
-solvent-solvent interation, this is due to the large number of the
-solvent molecules when compared to the number of solute molecules, the
-ratio of the number of lipid molecules to the number of water
-molecules is $1:25$ in our lipid-water system. The efficiency of the
-Molecular Dynamics simulations is greatly reduced, with up to 85\% of
-CPU time spent calculating only water-water interactions.
+In many cases, the properites of the solvent (like the water in the
+lipid-water system studied in this dissertation) are less interesting
+to the researchers than the behavior of the solute. However, the major
+computational expense is ofter the solvent-solvent interactions, this
+is due to the large number of the solvent molecules when compared to
+the number of solute molecules. The ratio of the number of lipid
+molecules to the number of water molecules is $1:25$ in our
+lipid-water system. The efficiency of the Molecular Dynamics
+simulations is greatly reduced, with up to 85\% of CPU time spent
+calculating only water-water interactions.
 
 As an extension of the Molecular Dynamics methodologies, Langevin
 Dynamics seeks a way to avoid integrating the equations of motion for
@@ -728,9 +745,9 @@ q_\alpha(0)}{\omega_\alpha} \sin(\omega_\alpha t),
 q_\alpha(0)}{\omega_\alpha} \sin(\omega_\alpha t),
 \label{Ineq:randomForceforGLE}
 \end{equation}
-depends only on the initial conditions. The relationship of friction
-kernel $\xi(t)$ and random force $R(t)$ is given by the second
-fluctuation dissipation theorem,
+that depends only on the initial conditions. The relationship of
+friction kernel $\xi(t)$ and random force $R(t)$ is given by the
+second fluctuation dissipation theorem,
 \begin{equation}
 \xi(t) = \frac{1}{k_B T} \langle R(t)R(0) \rangle.
 \label{Ineq:relationshipofXiandR}
