--- 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}.