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\chapter{\label{chap:intro}INTRODUCTION AND BACKGROUND} |
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\section{Background on the Problem\label{In:sec:pro}} |
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Phospholipid molecules are chosen to be studied in this dissertation |
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because of their critical role as a foundation of the bio-membrane |
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construction. The self assembled bilayer of the lipids when dispersed |
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in water is the micro structure of the membrane. The phase behavior of |
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lipid bilayer is explored experimentally~\cite{Cevc87}, however, fully |
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understanding on the mechanism is far beyond accomplished. |
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\subsection{Ripple Phase\label{In:ssec:ripple}} |
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The {\it ripple phase} $P_{\beta'}$ of lipid bilayers, named from the |
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periodic buckling of the membrane, is an intermediate phase which is |
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developed either from heating the gel phase $L_{\beta'}$ or cooling |
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the fluid phase $L_\alpha$. Although the ripple phase is observed in |
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different |
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experiments~\cite{Sun96,Katsaras00,Copeland80,Meyer96,Kaasgaard03}, |
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the mechanism of the formation of the ripple phase has never been |
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explained and the microscopic structure of the ripple phase has never |
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been elucidated by experiments. Computational simulation is a perfect |
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tool to study the microscopic properties for a system, however, the |
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long range dimension of the ripple structure and the long time scale |
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of the formation of the ripples are crucial obstacles to performing |
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the actual work. The idea to break through this dilemma forks into: |
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\begin{itemize} |
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\item Simplify the lipid model. |
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\item Improve the integrating algorithm. |
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\end{itemize} |
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In Ch.~\ref{chap:mc} and~\ref{chap:md}, we use a simple point dipole |
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model and a coarse-grained model to perform the Monte Carlo and |
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Molecular Dynamics simulations respectively, and in Ch.~\ref{chap:ld}, |
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we implement a Langevin Dynamics algorithm to exclude the explicit |
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solvent to improve the efficiency of the simulations. |
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\subsection{Lattice Model\label{In:ssec:model}} |
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The gel like characteristic of the ripple phase ensures the feasiblity |
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of applying the lattice model to study the system. It is claimed that |
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the packing of the lipid molecules in ripple phase is |
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hexagonal~\cite{Cevc87}. The popular $2$ dimensional lattice models, |
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{\it i.e.}, Ising model, Heisenberg model and $X-Y$ model, show |
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{\it frustration} on triangular lattice. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{./figures/inFrustration.pdf} |
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\caption{Sketch to illustrate the frustration on triangular |
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lattice. Spins are represented by arrows, no matter which direction |
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the spin on the top of triangle points to, the Hamiltonian of the |
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system is the same, hence there are infinite possibilities for the |
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packing of the spins.} |
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\label{Infig:frustration} |
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\end{figure} |
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Figure~\ref{Infig:frustration} shows an illustration of the frustration |
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on a triangular lattice. The direction of the spin on top of the |
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triangle has no effects on the Hamiltonian of the system, therefore |
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infinite possibilities for the packing of spins induce the frustration |
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of the lattice. |
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The lack of translational degree of freedom in lattice models prevents |
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their utilization on investigating the emergence of the surface |
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buckling which is the imposition of the ripple formation. In this |
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dissertation, a modified lattice model is introduced to this specific |
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situation in Ch.~\ref{chap:mc}. |
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\section{Overview of Classical Statistical Mechanics\label{In:sec:SM}} |
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Statistical mechanics provides a way to calculate the macroscopic |
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properties of a system from the molecular interactions used in |
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computational simulations. This section serves as a brief introduction |
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to key concepts of classical statistical mechanics that we used in |
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this dissertation. Tolman gives an excellent introduction to the |
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principles of statistical mechanics~\cite{Tolman1979}. A large part of |
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section~\ref{In:sec:SM} will follow Tolman's notation. |
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\subsection{Ensembles\label{In:ssec:ensemble}} |
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In classical mechanics, the state of the system is completely |
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described by the positions and momenta of all particles. If we have an |
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$N$ particle system, there are $6N$ coordinates ($3N$ position $(q_1, |
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q_2, \ldots, q_{3N})$ and $3N$ momenta $(p_1, p_2, \ldots, p_{3N})$) |
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to define the instantaneous state of the system. Each single set of |
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the $6N$ coordinates can be considered as a unique point in a $6N$ |
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dimensional space where each perpendicular axis is one of |
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$q_{i\alpha}$ or $p_{i\alpha}$ ($i$ is the particle and $\alpha$ is |
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the spatial axis). This $6N$ dimensional space is known as phase |
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space. The instantaneous state of the system is a single point in |
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phase space. A trajectory is a connected path of points in phase |
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space. An ensemble is a collection of systems described by the same |
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macroscopic observables but which have microscopic state |
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distributions. In phase space an ensemble is a collection of a set of |
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representive points. A density distribution $\rho(q^N, p^N)$ of the |
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representive points in phase space describes the condition of an |
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ensemble of identical systems. Since this density may change with |
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time, it is also a function of time. $\rho(q^N, p^N, t)$ describes the |
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ensemble at a time $t$, and $\rho(q^N, p^N, t')$ describes the |
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ensemble at a later time $t'$. For convenience, we will use $\rho$ |
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instead of $\rho(q^N, p^N, t)$ in the following disccusion. If we |
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normalize $\rho$ to unity, |
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\begin{equation} |
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1 = \int d \vec q~^N \int d \vec p~^N \rho, |
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\label{Ineq:normalized} |
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\end{equation} |
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then the value of $\rho$ gives the probability of finding the system |
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in a unit volume in the phase space. |
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Liouville's theorem describes the change in density $\rho$ with |
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time. The number of representive points at a given volume in the phase |
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space at any instant can be written as: |
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\begin{equation} |
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\label{Ineq:deltaN} |
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\delta N = \rho~\delta q_1 \delta q_2 \ldots \delta q_N \delta p_1 \delta p_2 \ldots \delta p_N. |
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\end{equation} |
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To calculate the change in the number of representive points in this |
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volume, let us consider a simple condition: the change in the number |
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of representive points in $q_1$ axis. The rate of the number of the |
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representive points entering the volume at $q_1$ per unit time is: |
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\begin{equation} |
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\label{Ineq:deltaNatq1} |
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\rho~\dot q_1 \delta q_2 \ldots \delta q_N \delta p_1 \delta p_2 \ldots \delta p_N, |
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\end{equation} |
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and the rate of the number of representive points leaving the volume |
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at another position $q_1 + \delta q_1$ is: |
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\begin{equation} |
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\label{Ineq:deltaNatq1plusdeltaq1} |
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\left( \rho + \frac{\partial \rho}{\partial q_1} \delta q_1 \right)\left(\dot q_1 + |
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\frac{\partial \dot q_1}{\partial q_1} \delta q_1 \right)\delta q_2 \ldots \delta q_N \delta p_1 \delta p_2 \ldots \delta p_N. |
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\end{equation} |
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Here the higher order differentials are neglected. So the change of |
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the number of the representive points is the difference of |
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eq.~\ref{Ineq:deltaNatq1} and eq.~\ref{Ineq:deltaNatq1plusdeltaq1}, which |
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gives us: |
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\begin{equation} |
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\label{Ineq:deltaNatq1axis} |
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-\left(\rho \frac{\partial {\dot q_1}}{\partial q_1} + \frac{\partial {\rho}}{\partial q_1} \dot q_1 \right)\delta q_1 \delta q_2 \ldots \delta q_N \delta p_1 \delta p_2 \ldots \delta p_N, |
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\end{equation} |
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where, higher order differetials are neglected. If we sum over all the |
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axes in the phase space, we can get the change of the number of |
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representive points in a given volume with time: |
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\begin{equation} |
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\label{Ineq:deltaNatGivenVolume} |
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\frac{d(\delta N)}{dt} = -\sum_{i=1}^N \left[\rho \left(\frac{\partial |
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{\dot q_i}}{\partial q_i} + \frac{\partial |
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{\dot p_i}}{\partial p_i}\right) + \left( \frac{\partial {\rho}}{\partial |
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q_i} \dot q_i + \frac{\partial {\rho}}{\partial p_i} \dot p_i\right)\right]\delta q_1 \delta q_2 \ldots \delta q_N \delta p_1 \delta p_2 \ldots \delta p_N. |
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\end{equation} |
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From Hamilton's equation of motion, |
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\begin{equation} |
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\frac{\partial {\dot q_i}}{\partial q_i} = - \frac{\partial |
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{\dot p_i}}{\partial p_i}, |
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\label{Ineq:canonicalFormOfEquationOfMotion} |
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\end{equation} |
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this cancels out the first term on the right side of |
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eq.~\ref{Ineq:deltaNatGivenVolume}. If both sides of |
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eq.~\ref{Ineq:deltaNatGivenVolume} are divided by $\delta q_1 \delta q_2 |
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\ldots \delta q_N \delta p_1 \delta p_2 \ldots \delta p_N$, then we |
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can derive Liouville's theorem: |
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\begin{equation} |
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\left( \frac{\partial \rho}{\partial t} \right)_{q, p} = -\sum_{i} \left( |
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\frac{\partial {\rho}}{\partial |
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q_i} \dot q_i + \frac{\partial {\rho}}{\partial p_i} \dot p_i \right). |
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\label{Ineq:simpleFormofLiouville} |
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\end{equation} |
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This is the basis of statistical mechanics. If we move the right |
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side of equation~\ref{Ineq:simpleFormofLiouville} to the left, we |
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will obtain |
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\begin{equation} |
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\left( \frac{\partial \rho}{\partial t} \right)_{q, p} + \sum_{i} \left( |
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\frac{\partial {\rho}}{\partial |
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q_i} \dot q_i + \frac{\partial {\rho}}{\partial p_i} \dot p_i \right) |
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= 0. |
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\label{Ineq:anotherFormofLiouville} |
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\end{equation} |
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It is easy to note that the left side of |
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equation~\ref{Ineq:anotherFormofLiouville} is the total derivative of |
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$\rho$ with respect of $t$, which means |
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\begin{equation} |
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\frac{d \rho}{dt} = 0, |
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\label{Ineq:conservationofRho} |
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\end{equation} |
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and the rate of density change is zero in the neighborhood of any |
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selected moving representive points in the phase space. |
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The condition of the ensemble is determined by the density |
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distribution. If we consider the density distribution as only a |
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function of $q$ and $p$, which means the rate of change of the phase |
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space density in the neighborhood of all representive points in the |
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phase space is zero, |
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\begin{equation} |
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\left( \frac{\partial \rho}{\partial t} \right)_{q, p} = 0. |
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\label{Ineq:statEquilibrium} |
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\end{equation} |
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We may conclude the ensemble is in {\it statistical equilibrium}. An |
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ensemble in statistical equilibrium often means the system is also in |
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macroscopic equilibrium. If $\left( \frac{\partial \rho}{\partial t} |
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\right)_{q, p} = 0$, then |
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\begin{equation} |
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\sum_{i} \left( |
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\frac{\partial {\rho}}{\partial |
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q_i} \dot q_i + \frac{\partial {\rho}}{\partial p_i} \dot p_i \right) |
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= 0. |
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\label{Ineq:constantofMotion} |
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\end{equation} |
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If $\rho$ is a function only of some constant of the motion, $\rho$ is |
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independent of time. For a conservative system, the energy of the |
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system is one of the constants of the motion. Here are several |
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examples: when the density distribution is constant everywhere in the |
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phase space, |
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\begin{equation} |
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\rho = \mathrm{const.} |
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\label{Ineq:uniformEnsemble} |
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\end{equation} |
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the ensemble is called {\it uniform ensemble}. Another useful |
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ensemble is called {\it microcanonical ensemble}, for which: |
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\begin{equation} |
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\rho = \delta \left( H(q^N, p^N) - E \right) \frac{1}{\Sigma (N, V, E)} |
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\label{Ineq:microcanonicalEnsemble} |
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\end{equation} |
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where $\Sigma(N, V, E)$ is a normalization constant parameterized by |
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$N$, the total number of particles, $V$, the total physical volume and |
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$E$, the total energy. The physical meaning of $\Sigma(N, V, E)$ is |
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the phase space volume accessible to a microcanonical system with |
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energy $E$ evolving under Hamilton's equations. $H(q^N, p^N)$ is the |
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Hamiltonian of the system. The Gibbs entropy is defined as |
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\begin{equation} |
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S = - k_B \int d \vec q~^N \int d \vec p~^N \rho \ln [C^N \rho], |
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\label{Ineq:gibbsEntropy} |
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\end{equation} |
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where $k_B$ is the Boltzmann constant and $C^N$ is a number which |
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makes the argument of $\ln$ dimensionless, in this case, it is the |
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total phase space volume of one state. The entropy in microcanonical |
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ensemble is given by |
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\begin{equation} |
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S = k_B \ln \left(\frac{\Sigma(N, V, E)}{C^N}\right). |
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\label{Ineq:entropy} |
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\end{equation} |
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If the density distribution $\rho$ is given by |
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\begin{equation} |
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\rho = \frac{1}{Z_N}e^{-H(q^N, p^N) / k_B T}, |
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\label{Ineq:canonicalEnsemble} |
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\end{equation} |
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the ensemble is known as the {\it canonical ensemble}. Here, |
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\begin{equation} |
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Z_N = \int d \vec q~^N \int_\Gamma d \vec p~^N e^{-H(q^N, p^N) / k_B T}, |
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\label{Ineq:partitionFunction} |
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\end{equation} |
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which is also known as {\it partition function}. $\Gamma$ indicates |
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that the integral is over all the phase space. In the canonical |
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ensemble, $N$, the total number of particles, $V$, total volume, and |
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$T$, the temperature are constants. The systems with the lowest |
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energies hold the largest population. According to maximum principle, |
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the thermodynamics maximizes the entropy $S$, |
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\begin{equation} |
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\begin{array}{ccc} |
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\delta S = 0 & \mathrm{and} & \delta^2 S < 0. |
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\end{array} |
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\label{Ineq:maximumPrinciple} |
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\end{equation} |
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From Eq.~\ref{Ineq:maximumPrinciple} and two constrains of the canonical |
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ensemble, {\it i.e.}, total probability and average energy conserved, |
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the partition function is calculated as |
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\begin{equation} |
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Z_N = e^{-A/k_B T}, |
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\label{Ineq:partitionFunctionWithFreeEnergy} |
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\end{equation} |
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where $A$ is the Helmholtz free energy. The significance of |
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Eq.~\ref{Ineq:entropy} and~\ref{Ineq:partitionFunctionWithFreeEnergy} is |
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that they serve as a connection between macroscopic properties of the |
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system and the distribution of the microscopic states. |
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|
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There is an implicit assumption that our arguments are based on so |
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far. A representive point in the phase space is equally to be found in |
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any same extent of the regions. In other words, all energetically |
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accessible states are represented equally, the probabilities to find |
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the system in any of the accessible states is equal. This is called |
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equal a {\it priori} probabilities. |
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|
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\subsection{Statistical Average\label{In:ssec:average}} |
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Given the density distribution $\rho$ in the phase space, the average |
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of any quantity ($F(q^N, p^N$)) which depends on the coordinates |
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($q^N$) and the momenta ($p^N$) for all the systems in the ensemble |
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can be calculated based on the definition shown by |
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Eq.~\ref{Ineq:statAverage1} |
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\begin{equation} |
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\langle F(q^N, p^N, t) \rangle = \frac{\int d \vec q~^N \int d \vec p~^N |
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F(q^N, p^N, t) \rho}{\int d \vec q~^N \int d \vec p~^N \rho}. |
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\label{Ineq:statAverage1} |
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\end{equation} |
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Since the density distribution $\rho$ is normalized to unity, the mean |
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value of $F(q^N, p^N)$ is simplified to |
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\begin{equation} |
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\langle F(q^N, p^N, t) \rangle = \int d \vec q~^N \int d \vec p~^N F(q^N, |
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p^N, t) \rho, |
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\label{Ineq:statAverage2} |
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\end{equation} |
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called {\it ensemble average}. However, the quantity is often averaged |
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for a finite time in real experiments, |
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\begin{equation} |
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\langle F(q^N, p^N) \rangle_t = \lim_{T \rightarrow \infty} |
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\frac{1}{T} \int_{t_0}^{t_0+T} F(q^N, p^N, t) dt. |
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\label{Ineq:timeAverage1} |
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\end{equation} |
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Usually this time average is independent of $t_0$ in statistical |
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mechanics, so Eq.~\ref{Ineq:timeAverage1} becomes |
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3347 |
\begin{equation} |
302 |
|
|
\langle F(q^N, p^N) \rangle_t = \lim_{T \rightarrow \infty} |
303 |
|
|
\frac{1}{T} \int_{0}^{T} F(q^N, p^N, t) dt |
304 |
xsun |
3354 |
\label{Ineq:timeAverage2} |
305 |
xsun |
3347 |
\end{equation} |
306 |
|
|
for an infinite time interval. |
307 |
|
|
|
308 |
|
|
{\it ergodic hypothesis}, an important hypothesis from the statistical |
309 |
|
|
mechanics point of view, states that the system will eventually pass |
310 |
|
|
arbitrarily close to any point that is energetically accessible in |
311 |
|
|
phase space. Mathematically, this leads to |
312 |
|
|
\begin{equation} |
313 |
|
|
\langle F(q^N, p^N, t) \rangle = \langle F(q^N, p^N) \rangle_t. |
314 |
xsun |
3354 |
\label{Ineq:ergodicity} |
315 |
xsun |
3347 |
\end{equation} |
316 |
xsun |
3354 |
Eq.~\ref{Ineq:ergodicity} validates the Monte Carlo method which we will |
317 |
xsun |
3347 |
discuss in section~\ref{In:ssec:mc}. An ensemble average of a quantity |
318 |
|
|
can be related to the time average measured in the experiments. |
319 |
|
|
|
320 |
|
|
\subsection{Correlation Function\label{In:ssec:corr}} |
321 |
|
|
Thermodynamic properties can be computed by equillibrium statistical |
322 |
|
|
mechanics. On the other hand, {\it Time correlation function} is a |
323 |
|
|
powerful method to understand the evolution of a dynamic system in |
324 |
|
|
non-equillibrium statistical mechanics. Imagine a property $A(q^N, |
325 |
|
|
p^N, t)$ as a function of coordinates $q^N$ and momenta $p^N$ has an |
326 |
|
|
intial value at $t_0$, at a later time $t_0 + \tau$ this value is |
327 |
|
|
changed. If $\tau$ is very small, the change of the value is minor, |
328 |
|
|
and the later value of $A(q^N, p^N, t_0 + |
329 |
|
|
\tau)$ is correlated to its initial value. Howere, when $\tau$ is large, |
330 |
|
|
this correlation is lost. The correlation function is a measurement of |
331 |
|
|
this relationship and is defined by~\cite{Berne90} |
332 |
|
|
\begin{equation} |
333 |
|
|
C(t) = \langle A(0)A(\tau) \rangle = \lim_{T \rightarrow \infty} |
334 |
|
|
\frac{1}{T} \int_{0}^{T} dt A(t) A(t + \tau). |
335 |
xsun |
3354 |
\label{Ineq:autocorrelationFunction} |
336 |
xsun |
3347 |
\end{equation} |
337 |
xsun |
3354 |
Eq.~\ref{Ineq:autocorrelationFunction} is the correlation function of a |
338 |
xsun |
3347 |
single variable, called {\it autocorrelation function}. The defination |
339 |
|
|
of the correlation function for two different variables is similar to |
340 |
|
|
that of autocorrelation function, which is |
341 |
|
|
\begin{equation} |
342 |
|
|
C(t) = \langle A(0)B(\tau) \rangle = \lim_{T \rightarrow \infty} |
343 |
|
|
\frac{1}{T} \int_{0}^{T} dt A(t) B(t + \tau), |
344 |
xsun |
3354 |
\label{Ineq:crosscorrelationFunction} |
345 |
xsun |
3347 |
\end{equation} |
346 |
|
|
and called {\it cross correlation function}. |
347 |
|
|
|
348 |
xsun |
3354 |
In section~\ref{In:ssec:average} we know from Eq.~\ref{Ineq:ergodicity} |
349 |
xsun |
3347 |
the relationship between time average and ensemble average. We can put |
350 |
|
|
the correlation function in a classical mechanics form, |
351 |
|
|
\begin{equation} |
352 |
|
|
C(t) = \langle A(0)A(\tau) \rangle = \int d \vec q~^N \int d \vec p~^N A(t) A(t + \tau) \rho(q, p) |
353 |
xsun |
3354 |
\label{Ineq:autocorrelationFunctionCM} |
354 |
xsun |
3347 |
\end{equation} |
355 |
|
|
and |
356 |
|
|
\begin{equation} |
357 |
|
|
C(t) = \langle A(0)B(\tau) \rangle = \int d \vec q~^N \int d \vec p~^N A(t) B(t + \tau) |
358 |
|
|
\rho(q, p) |
359 |
xsun |
3354 |
\label{Ineq:crosscorrelationFunctionCM} |
360 |
xsun |
3347 |
\end{equation} |
361 |
|
|
as autocorrelation function and cross correlation function |
362 |
|
|
respectively. $\rho(q, p)$ is the density distribution at equillibrium |
363 |
|
|
in phase space. In many cases, the correlation function decay is a |
364 |
|
|
single exponential |
365 |
|
|
\begin{equation} |
366 |
|
|
C(t) \sim e^{-t / \tau_r}, |
367 |
xsun |
3354 |
\label{Ineq:relaxation} |
368 |
xsun |
3347 |
\end{equation} |
369 |
|
|
where $\tau_r$ is known as relaxation time which discribes the rate of |
370 |
|
|
the decay. |
371 |
|
|
|
372 |
|
|
\section{Methodolody\label{In:sec:method}} |
373 |
|
|
The simulations performed in this dissertation are branched into two |
374 |
|
|
main catalog, Monte Carlo and Molecular Dynamics. There are two main |
375 |
|
|
difference between Monte Carlo and Molecular Dynamics simulations. One |
376 |
|
|
is that the Monte Carlo simulation is time independent, and Molecular |
377 |
|
|
Dynamics simulation is time involved. Another dissimilar is that the |
378 |
|
|
Monte Carlo is a stochastic process, the configuration of the system |
379 |
|
|
is not determinated by its past, however, using Moleuclar Dynamics, |
380 |
|
|
the system is propagated by Newton's equation of motion, the |
381 |
|
|
trajectory of the system evolved in the phase space is determined. A |
382 |
|
|
brief introduction of the two algorithms are given in |
383 |
|
|
section~\ref{In:ssec:mc} and~\ref{In:ssec:md}. An extension of the |
384 |
|
|
Molecular Dynamics, Langevin Dynamics, is introduced by |
385 |
|
|
section~\ref{In:ssec:ld}. |
386 |
|
|
|
387 |
|
|
\subsection{Monte Carlo\label{In:ssec:mc}} |
388 |
|
|
Monte Carlo algorithm was first introduced by Metropolis {\it et |
389 |
|
|
al.}.~\cite{Metropolis53} Basic Monte Carlo algorithm is usually |
390 |
|
|
applied to the canonical ensemble, a Boltzmann-weighted ensemble, in |
391 |
|
|
which the $N$, the total number of particles, $V$, total volume, $T$, |
392 |
|
|
temperature are constants. The average energy is given by substituding |
393 |
xsun |
3354 |
Eq.~\ref{Ineq:canonicalEnsemble} into Eq.~\ref{Ineq:statAverage2}, |
394 |
xsun |
3347 |
\begin{equation} |
395 |
|
|
\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}. |
396 |
xsun |
3354 |
\label{Ineq:energyofCanonicalEnsemble} |
397 |
xsun |
3347 |
\end{equation} |
398 |
|
|
So are the other properties of the system. The Hamiltonian is the |
399 |
|
|
summation of Kinetic energy $K(p^N)$ as a function of momenta and |
400 |
|
|
Potential energy $U(q^N)$ as a function of positions, |
401 |
|
|
\begin{equation} |
402 |
|
|
H(q^N, p^N) = K(p^N) + U(q^N). |
403 |
xsun |
3354 |
\label{Ineq:hamiltonian} |
404 |
xsun |
3347 |
\end{equation} |
405 |
|
|
If the property $A$ is only a function of position ($ A = A(q^N)$), |
406 |
|
|
the mean value of $A$ is reduced to |
407 |
|
|
\begin{equation} |
408 |
|
|
\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}}, |
409 |
xsun |
3354 |
\label{Ineq:configurationIntegral} |
410 |
xsun |
3347 |
\end{equation} |
411 |
|
|
The kinetic energy $K(p^N)$ is factored out in |
412 |
xsun |
3354 |
Eq.~\ref{Ineq:configurationIntegral}. $\langle A |
413 |
xsun |
3347 |
\rangle$ is a configuration integral now, and the |
414 |
xsun |
3354 |
Eq.~\ref{Ineq:configurationIntegral} is equivalent to |
415 |
xsun |
3347 |
\begin{equation} |
416 |
|
|
\langle A \rangle = \int d \vec q~^N A \rho(q^N). |
417 |
xsun |
3354 |
\label{Ineq:configurationAve} |
418 |
xsun |
3347 |
\end{equation} |
419 |
|
|
|
420 |
|
|
In a Monte Carlo simulation of canonical ensemble, the probability of |
421 |
|
|
the system being in a state $s$ is $\rho_s$, the change of this |
422 |
|
|
probability with time is given by |
423 |
|
|
\begin{equation} |
424 |
|
|
\frac{d \rho_s}{dt} = \sum_{s'} [ -w_{ss'}\rho_s + w_{s's}\rho_{s'} ], |
425 |
xsun |
3354 |
\label{Ineq:timeChangeofProb} |
426 |
xsun |
3347 |
\end{equation} |
427 |
|
|
where $w_{ss'}$ is the tansition probability of going from state $s$ |
428 |
|
|
to state $s'$. Since $\rho_s$ is independent of time at equilibrium, |
429 |
|
|
\begin{equation} |
430 |
|
|
\frac{d \rho_{s}^{equilibrium}}{dt} = 0, |
431 |
xsun |
3354 |
\label{Ineq:equiProb} |
432 |
xsun |
3347 |
\end{equation} |
433 |
|
|
which means $\sum_{s'} [ -w_{ss'}\rho_s + w_{s's}\rho_{s'} ]$ is $0$ |
434 |
|
|
for all $s'$. So |
435 |
|
|
\begin{equation} |
436 |
|
|
\frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = \frac{w_{s's}}{w_{ss'}}. |
437 |
xsun |
3354 |
\label{Ineq:relationshipofRhoandW} |
438 |
xsun |
3347 |
\end{equation} |
439 |
|
|
If |
440 |
|
|
\begin{equation} |
441 |
|
|
\frac{w_{s's}}{w_{ss'}} = e^{-(U_s - U_{s'}) / k_B T}, |
442 |
xsun |
3354 |
\label{Ineq:conditionforBoltzmannStatistics} |
443 |
xsun |
3347 |
\end{equation} |
444 |
|
|
then |
445 |
|
|
\begin{equation} |
446 |
|
|
\frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = e^{-(U_s - U_{s'}) / k_B T}. |
447 |
xsun |
3354 |
\label{Ineq:satisfyofBoltzmannStatistics} |
448 |
xsun |
3347 |
\end{equation} |
449 |
xsun |
3354 |
Eq.~\ref{Ineq:satisfyofBoltzmannStatistics} implies that |
450 |
xsun |
3347 |
$\rho^{equilibrium}$ satisfies Boltzmann statistics. An algorithm, |
451 |
|
|
shows how Monte Carlo simulation generates a transition probability |
452 |
xsun |
3354 |
governed by \ref{Ineq:conditionforBoltzmannStatistics}, is schemed as |
453 |
xsun |
3347 |
\begin{enumerate} |
454 |
xsun |
3354 |
\item\label{Initm:oldEnergy} Choose an particle randomly, calculate the energy. |
455 |
|
|
\item\label{Initm:newEnergy} Make a random displacement for particle, |
456 |
xsun |
3347 |
calculate the new energy. |
457 |
|
|
\begin{itemize} |
458 |
xsun |
3354 |
\item Keep the new configuration and return to step~\ref{Initm:oldEnergy} if energy |
459 |
xsun |
3347 |
goes down. |
460 |
|
|
\item Pick a random number between $[0,1]$ if energy goes up. |
461 |
|
|
\begin{itemize} |
462 |
|
|
\item Keep the new configuration and return to |
463 |
xsun |
3354 |
step~\ref{Initm:oldEnergy} if the random number smaller than |
464 |
xsun |
3347 |
$e^{-(U_{new} - U_{old})} / k_B T$. |
465 |
|
|
\item Keep the old configuration and return to |
466 |
xsun |
3354 |
step~\ref{Initm:oldEnergy} if the random number larger than |
467 |
xsun |
3347 |
$e^{-(U_{new} - U_{old})} / k_B T$. |
468 |
|
|
\end{itemize} |
469 |
|
|
\end{itemize} |
470 |
xsun |
3354 |
\item\label{Initm:accumulateAvg} Accumulate the average after it converges. |
471 |
xsun |
3347 |
\end{enumerate} |
472 |
|
|
It is important to notice that the old configuration has to be sampled |
473 |
|
|
again if it is kept. |
474 |
|
|
|
475 |
|
|
\subsection{Molecular Dynamics\label{In:ssec:md}} |
476 |
|
|
Although some of properites of the system can be calculated from the |
477 |
|
|
ensemble average in Monte Carlo simulations, due to the nature of |
478 |
|
|
lacking in the time dependence, it is impossible to gain information |
479 |
|
|
of those dynamic properties from Monte Carlo simulations. Molecular |
480 |
|
|
Dynamics is a measurement of the evolution of the positions and |
481 |
|
|
momenta of the particles in the system. The evolution of the system |
482 |
|
|
obeys laws of classical mechanics, in most situations, there is no |
483 |
|
|
need for the count of the quantum effects. For a real experiment, the |
484 |
|
|
instantaneous positions and momenta of the particles in the system are |
485 |
|
|
neither important nor measurable, the observable quantities are |
486 |
|
|
usually a average value for a finite time interval. These quantities |
487 |
|
|
are expressed as a function of positions and momenta in Melecular |
488 |
|
|
Dynamics simulations. Like the thermal temperature of the system is |
489 |
|
|
defined by |
490 |
|
|
\begin{equation} |
491 |
|
|
\frac{1}{2} k_B T = \langle \frac{1}{2} m v_\alpha \rangle, |
492 |
xsun |
3354 |
\label{Ineq:temperature} |
493 |
xsun |
3347 |
\end{equation} |
494 |
|
|
here $m$ is the mass of the particle and $v_\alpha$ is the $\alpha$ |
495 |
|
|
component of the velocity of the particle. The right side of |
496 |
xsun |
3354 |
Eq.~\ref{Ineq:temperature} is the average kinetic energy of the |
497 |
xsun |
3347 |
system. A simple Molecular Dynamics simulation scheme |
498 |
|
|
is:~\cite{Frenkel1996} |
499 |
|
|
\begin{enumerate} |
500 |
xsun |
3354 |
\item\label{Initm:initialize} Assign the initial positions and momenta |
501 |
xsun |
3347 |
for the particles in the system. |
502 |
xsun |
3354 |
\item\label{Initm:calcForce} Calculate the forces. |
503 |
|
|
\item\label{Initm:equationofMotion} Integrate the equation of motion. |
504 |
xsun |
3347 |
\begin{itemize} |
505 |
xsun |
3354 |
\item Return to step~\ref{Initm:calcForce} if the equillibrium is |
506 |
xsun |
3347 |
not achieved. |
507 |
xsun |
3354 |
\item Go to step~\ref{Initm:calcAvg} if the equillibrium is |
508 |
xsun |
3347 |
achieved. |
509 |
|
|
\end{itemize} |
510 |
xsun |
3354 |
\item\label{Initm:calcAvg} Compute the quantities we are interested in. |
511 |
xsun |
3347 |
\end{enumerate} |
512 |
|
|
The initial positions of the particles are chosen as that there is no |
513 |
|
|
overlap for the particles. The initial velocities at first are |
514 |
|
|
distributed randomly to the particles, and then shifted to make the |
515 |
|
|
momentum of the system $0$, at last scaled to the desired temperature |
516 |
xsun |
3354 |
of the simulation according Eq.~\ref{Ineq:temperature}. |
517 |
xsun |
3347 |
|
518 |
xsun |
3354 |
The core of Molecular Dynamics simulations is step~\ref{Initm:calcForce} |
519 |
|
|
and~\ref{Initm:equationofMotion}. The calculation of the forces are |
520 |
xsun |
3347 |
often involved numerous effort, this is the most time consuming step |
521 |
|
|
in the Molecular Dynamics scheme. The evaluation of the forces is |
522 |
|
|
followed by |
523 |
|
|
\begin{equation} |
524 |
|
|
f(q) = - \frac{\partial U(q)}{\partial q}, |
525 |
xsun |
3354 |
\label{Ineq:force} |
526 |
xsun |
3347 |
\end{equation} |
527 |
|
|
$U(q)$ is the potential of the system. Once the forces computed, are |
528 |
|
|
the positions and velocities updated by integrating Newton's equation |
529 |
|
|
of motion, |
530 |
|
|
\begin{equation} |
531 |
|
|
f(q) = \frac{dp}{dt} = \frac{m dv}{dt}. |
532 |
xsun |
3354 |
\label{Ineq:newton} |
533 |
xsun |
3347 |
\end{equation} |
534 |
|
|
Here is an example of integrating algorithms, Verlet algorithm, which |
535 |
|
|
is one of the best algorithms to integrate Newton's equation of |
536 |
|
|
motion. The Taylor expension of the position at time $t$ is |
537 |
|
|
\begin{equation} |
538 |
|
|
q(t+\Delta t)= q(t) + v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 + |
539 |
|
|
\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} + |
540 |
|
|
\mathcal{O}(\Delta t^4) |
541 |
xsun |
3354 |
\label{Ineq:verletFuture} |
542 |
xsun |
3347 |
\end{equation} |
543 |
|
|
for a later time $t+\Delta t$, and |
544 |
|
|
\begin{equation} |
545 |
|
|
q(t-\Delta t)= q(t) - v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 - |
546 |
|
|
\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} + |
547 |
|
|
\mathcal{O}(\Delta t^4) , |
548 |
xsun |
3354 |
\label{Ineq:verletPrevious} |
549 |
xsun |
3347 |
\end{equation} |
550 |
|
|
for a previous time $t-\Delta t$. The summation of the |
551 |
xsun |
3354 |
Eq.~\ref{Ineq:verletFuture} and~\ref{Ineq:verletPrevious} gives |
552 |
xsun |
3347 |
\begin{equation} |
553 |
|
|
q(t+\Delta t)+q(t-\Delta t) = |
554 |
|
|
2q(t) + \frac{f(t)}{m}\Delta t^2 + \mathcal{O}(\Delta t^4), |
555 |
xsun |
3354 |
\label{Ineq:verletSum} |
556 |
xsun |
3347 |
\end{equation} |
557 |
|
|
so, the new position can be expressed as |
558 |
|
|
\begin{equation} |
559 |
|
|
q(t+\Delta t) \approx |
560 |
|
|
2q(t) - q(t-\Delta t) + \frac{f(t)}{m}\Delta t^2. |
561 |
xsun |
3354 |
\label{Ineq:newPosition} |
562 |
xsun |
3347 |
\end{equation} |
563 |
|
|
The higher order of the $\Delta t$ is omitted. |
564 |
|
|
|
565 |
|
|
Numerous technics and tricks are applied to Molecular Dynamics |
566 |
|
|
simulation to gain more efficiency or more accuracy. The simulation |
567 |
|
|
engine used in this dissertation for the Molecular Dynamics |
568 |
|
|
simulations is {\sc oopse}, more details of the algorithms and |
569 |
|
|
technics can be found in~\cite{Meineke2005}. |
570 |
|
|
|
571 |
|
|
\subsection{Langevin Dynamics\label{In:ssec:ld}} |
572 |
|
|
In many cases, the properites of the solvent in a system, like the |
573 |
|
|
lipid-water system studied in this dissertation, are less important to |
574 |
|
|
the researchers. However, the major computational expense is spent on |
575 |
|
|
the solvent in the Molecular Dynamics simulations because of the large |
576 |
|
|
number of the solvent molecules compared to that of solute molecules, |
577 |
|
|
the ratio of the number of lipid molecules to the number of water |
578 |
|
|
molecules is $1:25$ in our lipid-water system. The efficiency of the |
579 |
|
|
Molecular Dynamics simulations is greatly reduced. |
580 |
|
|
|
581 |
|
|
As an extension of the Molecular Dynamics simulations, the Langevin |
582 |
|
|
Dynamics seeks a way to avoid integrating equation of motion for |
583 |
|
|
solvent particles without losing the Brownian properites of solute |
584 |
|
|
particles. A common approximation is that the coupling of the solute |
585 |
|
|
and solvent is expressed as a set of harmonic oscillators. So the |
586 |
|
|
Hamiltonian of such a system is written as |
587 |
|
|
\begin{equation} |
588 |
|
|
H = \frac{p^2}{2m} + U(q) + H_B + \Delta U(q), |
589 |
xsun |
3354 |
\label{Ineq:hamiltonianofCoupling} |
590 |
xsun |
3347 |
\end{equation} |
591 |
|
|
where $H_B$ is the Hamiltonian of the bath which equals to |
592 |
|
|
\begin{equation} |
593 |
|
|
H_B = \sum_{\alpha = 1}^{N} \left\{ \frac{p_\alpha^2}{2m_\alpha} + |
594 |
|
|
\frac{1}{2} m_\alpha \omega_\alpha^2 q_\alpha^2\right\}, |
595 |
xsun |
3354 |
\label{Ineq:hamiltonianofBath} |
596 |
xsun |
3347 |
\end{equation} |
597 |
|
|
$\alpha$ is all the degree of freedoms of the bath, $\omega$ is the |
598 |
|
|
bath frequency, and $\Delta U(q)$ is the bilinear coupling given by |
599 |
|
|
\begin{equation} |
600 |
|
|
\Delta U = -\sum_{\alpha = 1}^{N} g_\alpha q_\alpha q, |
601 |
xsun |
3354 |
\label{Ineq:systemBathCoupling} |
602 |
xsun |
3347 |
\end{equation} |
603 |
|
|
where $g$ is the coupling constant. By solving the Hamilton's equation |
604 |
|
|
of motion, the {\it Generalized Langevin Equation} for this system is |
605 |
|
|
derived to |
606 |
|
|
\begin{equation} |
607 |
|
|
m \ddot q = -\frac{\partial W(q)}{\partial q} - \int_0^t \xi(t) \dot q(t-t')dt' + R(t), |
608 |
xsun |
3354 |
\label{Ineq:gle} |
609 |
xsun |
3347 |
\end{equation} |
610 |
|
|
with mean force, |
611 |
|
|
\begin{equation} |
612 |
|
|
W(q) = U(q) - \sum_{\alpha = 1}^N \frac{g_\alpha^2}{2m_\alpha |
613 |
|
|
\omega_\alpha^2}q^2, |
614 |
xsun |
3354 |
\label{Ineq:meanForce} |
615 |
xsun |
3347 |
\end{equation} |
616 |
|
|
being only a dependence of coordinates of the solute particles, {\it |
617 |
|
|
friction kernel}, |
618 |
|
|
\begin{equation} |
619 |
|
|
\xi(t) = \sum_{\alpha = 1}^N \frac{-g_\alpha}{m_\alpha |
620 |
|
|
\omega_\alpha} \cos(\omega_\alpha t), |
621 |
xsun |
3354 |
\label{Ineq:xiforGLE} |
622 |
xsun |
3347 |
\end{equation} |
623 |
|
|
and the random force, |
624 |
|
|
\begin{equation} |
625 |
|
|
R(t) = \sum_{\alpha = 1}^N \left( g_\alpha q_\alpha(0)-\frac{g_\alpha}{m_\alpha |
626 |
|
|
\omega_\alpha^2}q(0)\right) \cos(\omega_\alpha t) + \frac{\dot |
627 |
|
|
q_\alpha(0)}{\omega_\alpha} \sin(\omega_\alpha t), |
628 |
xsun |
3354 |
\label{Ineq:randomForceforGLE} |
629 |
xsun |
3347 |
\end{equation} |
630 |
|
|
as only a dependence of the initial conditions. The relationship of |
631 |
|
|
friction kernel $\xi(t)$ and random force $R(t)$ is given by |
632 |
|
|
\begin{equation} |
633 |
|
|
\xi(t) = \frac{1}{k_B T} \langle R(t)R(0) \rangle |
634 |
xsun |
3354 |
\label{Ineq:relationshipofXiandR} |
635 |
xsun |
3347 |
\end{equation} |
636 |
|
|
from their definitions. In Langevin limit, the friction is treated |
637 |
|
|
static, which means |
638 |
|
|
\begin{equation} |
639 |
|
|
\xi(t) = 2 \xi_0 \delta(t). |
640 |
xsun |
3354 |
\label{Ineq:xiofStaticFriction} |
641 |
xsun |
3347 |
\end{equation} |
642 |
xsun |
3354 |
After substitude $\xi(t)$ into Eq.~\ref{Ineq:gle} with |
643 |
|
|
Eq.~\ref{Ineq:xiofStaticFriction}, {\it Langevin Equation} is extracted |
644 |
xsun |
3347 |
to |
645 |
|
|
\begin{equation} |
646 |
|
|
m \ddot q = -\frac{\partial U(q)}{\partial q} - \xi \dot q(t) + R(t). |
647 |
xsun |
3354 |
\label{Ineq:langevinEquation} |
648 |
xsun |
3347 |
\end{equation} |
649 |
|
|
The applying of Langevin Equation to dynamic simulations is discussed |
650 |
|
|
in Ch.~\ref{chap:ld}. |