--- trunk/xDissertation/Introduction.tex	2008/03/19 21:05:07	3373
+++ trunk/xDissertation/Introduction.tex	2008/03/20 22:21:43	3374
@@ -3,25 +3,63 @@ 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. Lipids, when dispersed in water, self assemble into a
-mumber of topologically distinct bilayer structures. The phase
-behavior of lipid bilayers has been explored
-experimentally~\cite{Cevc87}, however, a complete understanding of the
-mechanism and driving forces behind the various phases has not been
-achieved.
+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}.
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{./figures/inLipid.pdf}
+\caption{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.
+\begin{table*}
+\begin{minipage}{\linewidth}
+\begin{center}
+\caption{A number types of phosphatidycholine.}
+\begin{tabular}{lll}
+\hline 
+  & Fatty acid & Generic Name \\
+\hline 
+\textcolor{red}{DMPC} & Myristic: CH$_3$(CH$_2$)$_{12}$COOH &
+\textcolor{red}{D}i\textcolor{red}{M}yristoyl\textcolor{red}{P}hosphatidyl\textcolor{red}{C}holine \\
+\textcolor{red}{DPPC} & Palmitic: CH$_3$(CH$_2$)$_{14}$COOH & \textcolor{red}{D}i\textcolor{red}{P}almtoyl\textcolor{red}{P}hosphatidyl\textcolor{red}{C}holine
+\\
+\textcolor{red}{DSPC} & Stearic: CH$_3$(CH$_2$)$_{16}$COOH & \textcolor{red}{D}i\textcolor{red}{S}tearoyl\textcolor{red}{P}hosphatidyl\textcolor{red}{C}holine \\
+\end{tabular}
+\label{Intab:pc}
+\end{center}
+\end{minipage}
+\end{table*}
+When dispersed in water, lipids self assemble into a mumber 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}}
 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
-figure~\ref{Infig:phaseDiagram}.~\cite{Cevc87}
+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.}
+($P_{\beta'}$) to fluid ($L_\alpha$) phase transition.~\cite{Cevc80}}
 \label{Infig:phaseDiagram}
 \end{figure}
 Most structural information of the ripple phase has been obtained by
@@ -37,15 +75,16 @@ mica.~\cite{Kaasgaard03}
 \centering
 \includegraphics[width=\linewidth]{./figures/inRipple.pdf}
 \caption{The experimental observed ripple phase. The top image is
-obtained by X-ray diffraction~\cite{Sun96}, and the bottom one is
-observed by AFM.~\cite{Kaasgaard03}}
+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}}
 \label{Infig:ripple}
 \end{figure}
 Figure~\ref{Infig:ripple} shows the ripple phase oberved by 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{Cevc87} although Tenchov
+lipid packing within the gel phase,~\cite{Cevc80} although Tenchov
 {\it et al.} have recently observed near-hexagonal packing in some
 phosphatidylcholine (PC) gel phases.~\cite{Tenchov2001} However, the
 physical mechanism for the formation of the ripple phase has never
@@ -413,77 +452,80 @@ C_{AA}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^
 function in a classical mechanics 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), q^N(t+\tau)] \rho(q, p)
+A[q^N(t+\tau), p^N(t+\tau)] \rho(q, p)
 \label{Ineq:autocorrelationFunctionCM}
 \end{equation}
 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), q^N(t+\tau)] \rho(q, p)
+B[q^N(t+\tau), p^N(t+\tau)] \rho(q, p)
 \label{Ineq:crosscorrelationFunctionCM}
 \end{equation}
-as autocorrelation function and cross correlation function
+as the autocorrelation function and cross correlation functions
 respectively. $\rho(q, p)$ is the density distribution at equillibrium
-in phase space. In many cases, the correlation function decay is a
-single exponential
+in phase space. In many cases, correlation functions decay as a
+single exponential in time
 \begin{equation}
 C(t) \sim e^{-t / \tau_r},
 \label{Ineq:relaxation}
 \end{equation}
-where $\tau_r$ is known as relaxation time which discribes the rate of
+where $\tau_r$ is known as relaxation time which describes the rate of
 the decay.
 
-\section{Methodolody\label{In:sec:method}}
-The simulations performed in this dissertation are branched into two
-main catalog, Monte Carlo and Molecular Dynamics. There are two main
-difference between Monte Carlo and Molecular Dynamics simulations. One
-is that the Monte Carlo simulation is time independent, and Molecular
-Dynamics simulation is time involved. Another dissimilar is that the
-Monte Carlo is a stochastic process, the configuration of the system
-is not determinated by its past, however, using Moleuclar Dynamics,
-the system is propagated by Newton's equation of motion, the
-trajectory of the system evolved in the phase space is determined. A
-brief introduction of the two algorithms are given in
-section~\ref{In:ssec:mc} and~\ref{In:ssec:md}. An extension of the
-Molecular Dynamics, Langevin Dynamics, is introduced by
+\section{Methodology\label{In:sec:method}}
+The simulations performed in this dissertation branch into two main
+categories, Monte Carlo and Molecular Dynamics. There are two main
+differences between Monte Carlo and Molecular Dynamics
+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
+by its past. However, in Molecular Dynamics, the system is propagated
+by Newton's equation 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
+Dynamics that includes implicit solvent effects, is introduced by
 section~\ref{In:ssec:ld}.
 
 \subsection{Monte Carlo\label{In:ssec:mc}}
-Monte Carlo algorithm was first introduced by Metropolis {\it et
-al.}.~\cite{Metropolis53} Basic 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. The average energy is given by substituding
-Eq.~\ref{Ineq:canonicalEnsemble} into Eq.~\ref{Ineq:statAverage2},
+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
 \begin{equation}
-\langle E \rangle = \frac{1}{Z_N} \int d \vec q~^N \int d \vec p~^N E e^{-H(q^N, p^N) / k_B T}.
+\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}.
 \label{Ineq:energyofCanonicalEnsemble}
 \end{equation}
-So are the other properties of the system. The Hamiltonian is the
-summation of Kinetic energy $K(p^N)$ as a function of momenta and
-Potential energy $U(q^N)$ as a function of positions,
+The Hamiltonian is the sum of the kinetic energy, $K(p^N)$, a function
+of momenta and the potential energy, $U(q^N)$, a function of
+positions,
 \begin{equation}
 H(q^N, p^N) = K(p^N) + U(q^N).
 \label{Ineq:hamiltonian}
 \end{equation}
-If the property $A$ is only a function of position ($ A = A(q^N)$),
-the mean value of $A$ is reduced to
+If the property $A$ is a function only of position ($ A = A(q^N)$),
+the mean value of $A$ can be reduced to
 \begin{equation}
-\langle A \rangle = \frac{\int d \vec q~^N \int d \vec p~^N A e^{-U(q^N) / k_B T}}{\int d \vec q~^N \int d \vec p~^N e^{-U(q^N) / k_B T}},
+\langle A \rangle = \frac{\int d \vec q~^N A e^{-U(q^N) / k_B T}}{\int d \vec q~^N e^{-U(q^N) / k_B T}},
 \label{Ineq:configurationIntegral}
 \end{equation}
 The kinetic energy $K(p^N)$ is factored out in
 Eq.~\ref{Ineq:configurationIntegral}. $\langle A
-\rangle$ is a configuration integral now, and the
+\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).
 \label{Ineq:configurationAve}
 \end{equation}
 
-In a Monte Carlo simulation of canonical ensemble, the probability of
-the system being in a state $s$ is $\rho_s$, the change of this
-probability with time is given by
+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
+of this probability with time is given by
 \begin{equation}
 \frac{d \rho_s}{dt} = \sum_{s'} [ -w_{ss'}\rho_s + w_{s's}\rho_{s'} ],
 \label{Ineq:timeChangeofProb}
@@ -500,104 +542,95 @@ for all $s'$. So
 \frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = \frac{w_{s's}}{w_{ss'}}.
 \label{Ineq:relationshipofRhoandW}
 \end{equation}
-If
+If the ratio of state populations
 \begin{equation}
-\frac{w_{s's}}{w_{ss'}} = e^{-(U_s - U_{s'}) / k_B T},
-\label{Ineq:conditionforBoltzmannStatistics}
-\end{equation}
-then
-\begin{equation}
 \frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = e^{-(U_s - U_{s'}) / k_B T}.
 \label{Ineq:satisfyofBoltzmannStatistics}
 \end{equation}
-Eq.~\ref{Ineq:satisfyofBoltzmannStatistics} implies that
-$\rho^{equilibrium}$ satisfies Boltzmann statistics. An algorithm,
-shows how Monte Carlo simulation generates a transition probability
-governed by \ref{Ineq:conditionforBoltzmannStatistics}, is schemed as
+then the ratio of transition probabilities,
+\begin{equation}
+\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
+transition probability governed by
+\ref{Ineq:conditionforBoltzmannStatistics}, is given schematically as,
 \begin{enumerate}
-\item\label{Initm:oldEnergy} Choose an particle randomly, calculate the energy.
-\item\label{Initm:newEnergy} Make a random displacement for particle,
-calculate the new energy.
+\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.
+\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 Keep the new configuration and return to step~\ref{Initm:oldEnergy} if energy
-goes down.
-     \item Pick a random number between $[0,1]$ if energy goes up.
+     \item If the energy goes down, keep the new configuration and return to
+step~\ref{Initm:oldEnergy}.
+     \item If the energy goes up, pick a random number between $[0,1]$.
         \begin{itemize}
-           \item Keep the new configuration and return to
-step~\ref{Initm:oldEnergy} if the random number smaller than
-$e^{-(U_{new} - U_{old})} / k_B T$.
-           \item Keep the old configuration and return to
-step~\ref{Initm:oldEnergy} if the random number larger than
-$e^{-(U_{new} - U_{old})} / k_B T$.
+           \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}.
+           \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}.
         \end{itemize}
   \end{itemize}
-\item\label{Initm:accumulateAvg} Accumulate the average after it converges.
+\item\label{Initm:accumulateAvg} Accumulate the averages based on the
+current configuartion.
+\item Go to step~\ref{Initm:oldEnergy}.
 \end{enumerate}
-It is important to notice that the old configuration has to be sampled
-again if it is kept.
+It is important for sampling accurately that the old configuration is
+sampled again if it is kept.
 
 \subsection{Molecular Dynamics\label{In:ssec:md}}
 Although some of properites of the system can be calculated from the
-ensemble average in Monte Carlo simulations, due to the nature of
-lacking in the time dependence, it is impossible to gain information
-of those dynamic properties from Monte Carlo simulations. Molecular
-Dynamics is a measurement of the evolution of the positions and
-momenta of the particles in the system. The evolution of the system
-obeys laws of classical mechanics, in most situations, there is no
-need for the count of the quantum effects. For a real experiment, the
-instantaneous positions and momenta of the particles in the system are
-neither important nor measurable, the observable quantities are
-usually a average value for a finite time interval. These quantities
-are expressed as a function of positions and momenta in Melecular
-Dynamics simulations. Like the thermal temperature of the system is
-defined by
+ensemble average in Monte Carlo simulations, due to the absence of the
+time dependence, it is impossible to gain information on dynamic
+properties from Monte Carlo simulations. Molecular Dynamics evolves
+the positions and momenta of the particles in the system. The
+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,
+temperature of the system is defined by
 \begin{equation}
-\frac{1}{2} k_B T = \langle \frac{1}{2} m v_\alpha \rangle,
+\frac{3}{2} N k_B T = \langle \sum_{i=1}^N \frac{1}{2} m_i v_i \rangle,
 \label{Ineq:temperature}
 \end{equation}
-here $m$ is the mass of the particle and $v_\alpha$ is the $\alpha$
-component of the velocity of the particle. The right side of
-Eq.~\ref{Ineq:temperature} is the average kinetic energy of the
-system. A simple Molecular Dynamics simulation scheme
-is:~\cite{Frenkel1996}
-\begin{enumerate}
-\item\label{Initm:initialize} Assign the initial positions and momenta
-for the particles in the system.
-\item\label{Initm:calcForce} Calculate the forces.
-\item\label{Initm:equationofMotion} Integrate the equation of motion.
-  \begin{itemize}
-     \item Return to step~\ref{Initm:calcForce} if the equillibrium is
-not achieved.
-     \item Go to step~\ref{Initm:calcAvg} if the equillibrium is
-achieved.
-  \end{itemize}
-\item\label{Initm:calcAvg} Compute the quantities we are interested in.
-\end{enumerate}
-The initial positions of the particles are chosen as that there is no
-overlap for the particles. The initial velocities at first are
-distributed randomly to the particles, and then shifted to make the
-momentum of the system $0$, at last scaled to the desired temperature
-of the simulation according Eq.~\ref{Ineq:temperature}.
+here $m_i$ is the mass of particle $i$ and $v_i$ is the velocity of
+particle $i$. The right side of Eq.~\ref{Ineq:temperature} is the
+average kinetic energy of the system.
 
-The core of Molecular Dynamics simulations is step~\ref{Initm:calcForce}
-and~\ref{Initm:equationofMotion}. The calculation of the forces are
-often involved numerous effort, this is the most time consuming step
-in the Molecular Dynamics scheme. The evaluation of the forces is
-followed by
+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 $0$.
+
+The core of Molecular Dynamics simulations is the calculation of
+forces and the integration algorithm. Calculation of the forces often
+involves enormous effort. This is the most time consuming step in the
+Molecular Dynamics scheme. Evaluation of the forces is mathematically
+simple,
 \begin{equation}
 f(q) = - \frac{\partial U(q)}{\partial q},
 \label{Ineq:force}
 \end{equation}
-$U(q)$ is the potential of the system. Once the forces computed, are
-the positions and velocities updated by integrating Newton's equation
-of motion,
-\begin{equation}
-f(q) = \frac{dp}{dt} = \frac{m dv}{dt}.
+where $U(q)$ is the potential of the system. However, the numerical
+details of this computation are often quite complex. Once the forces
+computed, the positions and velocities are updated by integrating
+Hamilton's equations of motion,
+\begin{eqnarray}
+\dot p_i & = & -\frac{\partial H}{\partial q_i} = -\frac{\partial
+U(q_i)}{\partial q_i} = f(q_i) \\
+\dot q_i & = & p_i 
 \label{Ineq:newton}
-\end{equation}
-Here is an example of integrating algorithms, Verlet algorithm, which
-is one of the best algorithms to integrate Newton's equation of
-motion. The Taylor expension of the position at time $t$ is
+\end{eqnarray}
+The classic example of an integrating algorithm is the Verlet
+algorithm, which is one of the simplest algorithms for integrating the
+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 +
 	\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} +
@@ -611,8 +644,8 @@ q(t-\Delta t)= q(t) - v(t) \Delta t + \frac{f(t)}{2m}\
 	\mathcal{O}(\Delta t^4) ,
 \label{Ineq:verletPrevious}
 \end{equation}
-for a previous time $t-\Delta t$. The summation of the
-Eq.~\ref{Ineq:verletFuture} and~\ref{Ineq:verletPrevious} gives
+for a previous time $t-\Delta t$. Adding Eq.~\ref{Ineq:verletFuture}
+and~\ref{Ineq:verletPrevious} gives
 \begin{equation}
 q(t+\Delta t)+q(t-\Delta t) = 
 	2q(t) + \frac{f(t)}{m}\Delta t^2 + \mathcal{O}(\Delta t^4),
@@ -624,49 +657,53 @@ q(t+\Delta t) \approx 
 	2q(t) - q(t-\Delta t) + \frac{f(t)}{m}\Delta t^2.
 \label{Ineq:newPosition}
 \end{equation}
-The higher order of the $\Delta t$ is omitted.
+The higher order terms in $\Delta t$ are omitted.
 
-Numerous technics and tricks are applied to Molecular Dynamics
-simulation to gain more efficiency or more accuracy. The simulation
-engine used in this dissertation for the Molecular Dynamics
-simulations is {\sc oopse}, more details of the algorithms and
-technics can be found in~\cite{Meineke2005}.
+Numerous techniques and tricks have been applied to Molecular Dynamics
+simulations to gain greater efficiency or accuracy. The engine used in
+this dissertation for the Molecular Dynamics simulations is {\sc
+oopse}. More details of the algorithms and techniques used in this
+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
-lipid-water system studied in this dissertation, are less important to
-the researchers. However, the major computational expense is spent on
-the solvent in the Molecular Dynamics simulations because of the large
-number of the solvent molecules compared to that of solute molecules,
-the ratio of the number of lipid molecules to the number of water
+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.
+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 simulations, the Langevin
-Dynamics seeks a way to avoid integrating equation of motion for
-solvent particles without losing the Brownian properites of solute
-particles. A common approximation is that the coupling of the solute
-and solvent is expressed as a set of harmonic oscillators. So the
-Hamiltonian of such a system is written as
+As an extension of the Molecular Dynamics methodologies, Langevin
+Dynamics seeks a way to avoid integrating the equations of motion for
+solvent particles without losing the solvent effects on the solute
+particles. One common approximation is to express the coupling of the
+solute and solvent as a set of harmonic oscillators. The Hamiltonian
+of such a system is written as
 \begin{equation}
 H = \frac{p^2}{2m} + U(q) + H_B + \Delta U(q),
 \label{Ineq:hamiltonianofCoupling}
 \end{equation}
-where $H_B$ is the Hamiltonian of the bath which equals to
+where $H_B$ is the Hamiltonian of the bath which  is a set of N
+harmonic oscillators
 \begin{equation}
 H_B = \sum_{\alpha = 1}^{N} \left\{ \frac{p_\alpha^2}{2m_\alpha} +
 \frac{1}{2} m_\alpha \omega_\alpha^2 q_\alpha^2\right\},
 \label{Ineq:hamiltonianofBath}
 \end{equation}
-$\alpha$ is all the degree of freedoms of the bath, $\omega$ is the
-bath frequency, and $\Delta U(q)$ is the bilinear coupling given by
+$\alpha$ runs over all the degree of freedoms of the bath,
+$\omega_\alpha$ is the bath frequency of oscillator $\alpha$, and
+$\Delta U(q)$ is the bilinear coupling given by
 \begin{equation}
 \Delta U = -\sum_{\alpha = 1}^{N} g_\alpha q_\alpha q,
 \label{Ineq:systemBathCoupling}
 \end{equation}
-where $g$ is the coupling constant. By solving the Hamilton's equation
-of motion, the {\it Generalized Langevin Equation} for this system is
-derived to
+where $g_\alpha$ is the coupling constant for oscillator $\alpha$. By
+solving the Hamilton's equations of motion, the {\it Generalized
+Langevin Equation} for this system is derived as
 \begin{equation}
 m \ddot q = -\frac{\partial W(q)}{\partial q} - \int_0^t \xi(t) \dot q(t-t')dt' + R(t),
 \label{Ineq:gle}
@@ -674,41 +711,42 @@ W(q) = U(q) - \sum_{\alpha = 1}^N \frac{g_\alpha^2}{2m
 with mean force, 
 \begin{equation}
 W(q) = U(q) - \sum_{\alpha = 1}^N \frac{g_\alpha^2}{2m_\alpha
-\omega_\alpha^2}q^2,
+\omega_\alpha^2}q^2.
 \label{Ineq:meanForce}
 \end{equation}
-being only a dependence of coordinates of the solute particles, {\it
-friction kernel},
+The {\it friction kernel}, $\xi(t)$, depends only on the coordinates
+of the solute particles,
 \begin{equation}
 \xi(t) = \sum_{\alpha = 1}^N \frac{-g_\alpha}{m_\alpha
 \omega_\alpha} \cos(\omega_\alpha t),
 \label{Ineq:xiforGLE}
 \end{equation}
-and the random force,
+and a ``random'' force,
 \begin{equation}
 R(t) = \sum_{\alpha = 1}^N \left( g_\alpha q_\alpha(0)-\frac{g_\alpha}{m_\alpha
 \omega_\alpha^2}q(0)\right) \cos(\omega_\alpha t) + \frac{\dot
 q_\alpha(0)}{\omega_\alpha} \sin(\omega_\alpha t),
 \label{Ineq:randomForceforGLE}
 \end{equation}
-as only a dependence of the initial conditions. The relationship of
-friction kernel $\xi(t)$ and random force $R(t)$ is given by
+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
+\xi(t) = \frac{1}{k_B T} \langle R(t)R(0) \rangle.
 \label{Ineq:relationshipofXiandR}
 \end{equation}
-from their definitions. In Langevin limit, the friction is treated
-static, which means
+In the harmonic bath this relation is exact and provable from the
+definitions of these quantities. In the limit of static friction,
 \begin{equation}
 \xi(t) = 2 \xi_0 \delta(t).
 \label{Ineq:xiofStaticFriction}
 \end{equation}
-After substitude $\xi(t)$ into Eq.~\ref{Ineq:gle} with
-Eq.~\ref{Ineq:xiofStaticFriction}, {\it Langevin Equation} is extracted
-to
+After substituting $\xi(t)$ into Eq.~\ref{Ineq:gle} with
+Eq.~\ref{Ineq:xiofStaticFriction}, the {\it Langevin Equation} is
+extracted,
 \begin{equation}
 m \ddot q = -\frac{\partial U(q)}{\partial q} - \xi \dot q(t) + R(t).
 \label{Ineq:langevinEquation}
 \end{equation}
-The applying of Langevin Equation to dynamic simulations is discussed
-in Ch.~\ref{chap:ld}.
+Application of the Langevin Equation to dynamic simulations is
+discussed in Ch.~\ref{chap:ld}.
