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\chapter{\label{chapt:intro}Introduction and Theoretical Background} |
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\chapter{\label{chapt:intro}INTRODUCTION AND THEORETICAL BACKGROUND} |
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\section{\label{introSec:theory}Theoretical Background} |
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The techniques used in the course of this research fall under the two |
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main classes of molecular simulation: Molecular Dynamics and Monte |
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Carlo. Molecular Dynamic simulations integrate the equations of motion |
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for a given system of particles, allowing the researher to gain |
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insight into the time dependent evolution of a system. Diffusion |
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Carlo. Molecular Dynamics simulations integrate the equations of |
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motion for a given system of particles, allowing the researcher to |
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gain insight into the time dependent evolution of a system. Diffusion |
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phenomena are readily studied with this simulation technique, making |
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Molecular Dynamics the main simulation technique used in this |
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research. Other aspects of the research fall under the Monte Carlo |
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class of simulations. In Monte Carlo, the configuration space |
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available to the collection of particles is sampled stochastichally, |
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or randomly. Each configuration is chosen with a given probability |
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based on the Maxwell Boltzman distribution. These types of simulations |
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are best used to probe properties of a system that are only dependent |
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only on the state of the system. Structural information about a system |
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is most readily obtained through these types of methods. |
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available to the collection of particles is sampled stochastically, or |
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randomly. Each configuration is chosen with a given probability based |
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on the Maxwell Boltzmann distribution. These types of simulations are |
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best used to probe properties of a system that are dependent only on |
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the state of the system. Structural information about a system is most |
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readily obtained through these types of methods. |
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Although the two techniques employed seem dissimilar, they are both |
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linked by the overarching principles of Statistical |
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Thermodynamics. Statistical Thermodynamics governs the behavior of |
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Mechanics. Statistical Meachanics governs the behavior of |
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both classes of simulations and dictates what each method can and |
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cannot do. When investigating a system, one most first analyze what |
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thermodynamic properties of the system are being probed, then chose |
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which method best suits that objective. |
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\subsection{\label{introSec:statThermo}Statistical Thermodynamics} |
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\section{\label{introSec:statThermo}Statistical Mechanics} |
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ergodic hypothesis |
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The following section serves as a brief introduction to some of the |
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Statistical Mechanics concepts present in this dissertation. What |
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follows is a brief derivation of Boltzmann weighted statistics and an |
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explanation of how one can use the information to calculate an |
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observable for a system. This section then concludes with a brief |
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discussion of the ergodic hypothesis and its relevance to this |
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research. |
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enesemble averages |
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\subsection{\label{introSec:boltzman}Boltzmann weighted statistics} |
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\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations} |
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Consider a system, $\gamma$, with total energy $E_{\gamma}$. Let |
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$\Omega(E_{\gamma})$ represent the number of degenerate ways |
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$\boldsymbol{\Gamma}\{r_1,r_2,\ldots r_n,p_1,p_2,\ldots p_n\}$, the |
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collection of positions and conjugate momenta of system $\gamma$, can |
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be configured to give $E_{\gamma}$. Further, if $\gamma$ is a subset |
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of a larger system, $\boldsymbol{\Lambda}\{E_1,E_2,\ldots |
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E_{\gamma},\ldots E_n\}$, the total degeneracy of the system can be |
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expressed as, |
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\begin{equation} |
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\Omega(\boldsymbol{\Lambda}) = \Omega(E_1) \times \Omega(E_2) \times \ldots |
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\Omega(E_{\gamma}) \times \ldots \Omega(E_n) |
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\label{introEq:SM0.1} |
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\end{equation} |
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This multiplicative combination of degeneracies is illustrated in |
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Fig.~\ref{introFig:degenProd}. |
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The Monte Carlo method was developed by Metropolis and Ulam for their |
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work in fissionable material.\cite{metropolis:1949} The method is so |
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named, because it heavily uses random numbers in the solution of the |
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problem. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{omegaFig.eps} |
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\caption[An explanation of the combination of degeneracies]{Systems A and B both have three energy levels and two indistinguishable particles. When the total energy is 2, there are two ways for each system to disperse the energy. However, for system C, the superset of A and B, the total degeneracy is the product of the degeneracy of each system. In this case $\Omega(\text{C})$ is 4.} |
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\label{introFig:degenProd} |
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\end{figure} |
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Next, consider the specific case of $\gamma$ in contact with a |
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bath. Exchange of energy is allowed between the bath and the system, |
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subject to the constraint that the total energy |
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($E_{\text{bath}}+E_{\gamma}$) remain constant. $\Omega(E)$, where $E$ |
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is the total energy of both systems, can be represented as |
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\begin{equation} |
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\Omega(E) = \Omega(E_{\gamma}) \times \Omega(E - E_{\gamma}) |
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\label{introEq:SM1} |
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\end{equation} |
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Or additively as |
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\begin{equation} |
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\ln \Omega(E) = \ln \Omega(E_{\gamma}) + \ln \Omega(E - E_{\gamma}) |
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\label{introEq:SM2} |
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\end{equation} |
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\subsection{\label{introSec:md}Molecular Dynamics Simulations} |
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The solution to Eq.~\ref{introEq:SM2} maximizes the number of |
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degenerate configurations in $E$. \cite{Frenkel1996} |
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This gives |
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\begin{equation} |
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\frac{\partial \ln \Omega(E)}{\partial E_{\gamma}} = 0 = |
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\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
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+ |
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\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
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\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}} |
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\label{introEq:SM3} |
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\end{equation} |
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Where $E_{\text{bath}}$ is $E-E_{\gamma}$, and |
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$\frac{\partial E_{\text{bath}}}{\partial E_{\gamma}}$ is |
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$-1$. Eq.~\ref{introEq:SM3} becomes |
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\begin{equation} |
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\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} = |
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\frac{\partial \ln \Omega(E_{\text{bath}})}{\partial E_{\text{bath}}} |
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\label{introEq:SM4} |
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\end{equation} |
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time averages |
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At this point, one can draw a relationship between the maximization of |
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degeneracy in Eq.~\ref{introEq:SM3} and the second law of |
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thermodynamics. Namely, that for a closed system, entropy will |
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increase for an irreversible process.\cite{chandler:1987} Here the |
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process is the partitioning of energy among the two systems. This |
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allows the following definition of entropy: |
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\begin{equation} |
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S = k_B \ln \Omega(E) |
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\label{introEq:SM5} |
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\end{equation} |
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Where $k_B$ is the Boltzmann constant. Having defined entropy, one can |
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also define the temperature of the system using the relation |
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\begin{equation} |
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\frac{1}{T} = \biggl ( \frac{\partial S}{\partial E} \biggr )_{N,V} |
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\label{introEq:SM6} |
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\end{equation} |
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The temperature in the system $\gamma$ is then |
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\begin{equation} |
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\beta( E_{\gamma} ) = \frac{1}{k_B T} = |
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\frac{\partial \ln \Omega(E_{\gamma})}{\partial E_{\gamma}} |
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\label{introEq:SM7} |
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\end{equation} |
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Applying this to Eq.~\ref{introEq:SM4} gives the following |
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\begin{equation} |
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\beta( E_{\gamma} ) = \beta( E_{\text{bath}} ) |
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\label{introEq:SM8} |
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\end{equation} |
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Showing that the partitioning of energy between the two systems is |
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actually a process of thermal equilibration.\cite{Frenkel1996} |
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time integrating schemes |
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An application of these results is to formulate the form of an |
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expectation value of an observable, $A$, in the canonical ensemble. In |
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the canonical ensemble, the number of particles, $N$, the volume, $V$, |
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and the temperature, $T$, are all held constant while the energy, $E$, |
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is allowed to fluctuate. Returning to the previous example, the bath |
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system is now an infinitely large thermal bath, whose exchange of |
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energy with the system $\gamma$ holds the temperature constant. The |
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partitioning of energy in the bath system is then related to the total |
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energy of both systems and the fluctuations in $E_{\gamma}$: |
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\begin{equation} |
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\Omega( E_{\gamma} ) = \Omega( E - E_{\gamma} ) |
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\label{introEq:SM9} |
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\end{equation} |
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As for the expectation value, it can be defined |
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\begin{equation} |
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\langle A \rangle = |
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\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} |
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P_{\gamma} A(\boldsymbol{\Gamma}) |
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\label{introEq:SM10} |
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\end{equation} |
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Where $\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma}$ denotes |
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an integration over all accessible phase space, $P_{\gamma}$ is the |
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probability of being in a given phase state and |
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$A(\boldsymbol{\Gamma})$ is some observable that is a function of the |
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phase state. |
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time reversible |
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Because entropy seeks to maximize the number of degenerate states at a |
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given energy, the probability of being in a particular state in |
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$\gamma$ will be directly proportional to the number of allowable |
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states the coupled system is able to assume. Namely, |
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\begin{equation} |
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P_{\gamma} \propto \Omega( E_{\text{bath}} ) = |
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e^{\ln \Omega( E - E_{\gamma})} |
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\label{introEq:SM11} |
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\end{equation} |
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With $E_{\gamma} \ll E$, $\ln \Omega$ can be expanded in a Taylor series: |
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\begin{equation} |
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\ln \Omega ( E - E_{\gamma}) = |
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\ln \Omega (E) - |
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E_{\gamma} \frac{\partial \ln \Omega }{\partial E} |
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+ \ldots |
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\label{introEq:SM12} |
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\end{equation} |
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Higher order terms are omitted as $E$ is an infinite thermal |
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bath. Further, using Eq.~\ref{introEq:SM7}, Eq.~\ref{introEq:SM11} can |
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be rewritten: |
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\begin{equation} |
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P_{\gamma} \propto e^{-\beta E_{\gamma}} |
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\label{introEq:SM13} |
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\end{equation} |
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Where $\ln \Omega(E)$ has been factored out of the proportionality as a |
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constant. Normalizing the probability ($\int\limits_{\boldsymbol{\Gamma}} |
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d\boldsymbol{\Gamma} P_{\gamma} = 1$) gives |
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\begin{equation} |
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P_{\gamma} = \frac{e^{-\beta E_{\gamma}}} |
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{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}} |
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\label{introEq:SM14} |
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\end{equation} |
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This result is the standard Boltzmann statistical distribution. |
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Applying it to Eq.~\ref{introEq:SM10} one can obtain the following relation for ensemble averages: |
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\begin{equation} |
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\langle A \rangle = |
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\frac{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} |
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A( \boldsymbol{\Gamma} ) e^{-\beta E_{\gamma}}} |
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{\int\limits_{\boldsymbol{\Gamma}} d\boldsymbol{\Gamma} e^{-\beta E_{\gamma}}} |
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\label{introEq:SM15} |
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\end{equation} |
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symplectic methods |
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\subsection{\label{introSec:ergodic}The Ergodic Hypothesis} |
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Extended ensembles (NVT NPT) |
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In the case of a Molecular Dynamics simulation, rather than |
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calculating an ensemble average integral over phase space as in |
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Eq.~\ref{introEq:SM15}, it becomes easier to caclulate the time |
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average of an observable. Namely, |
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\begin{equation} |
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\langle A \rangle_t = \frac{1}{\tau} |
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\int_0^{\tau} A[\boldsymbol{\Gamma}(t)]\,dt |
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\label{introEq:SM16} |
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\end{equation} |
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Where the value of an observable is averaged over the length of time |
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that the simulation is run. This type of measurement mirrors the |
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experimental measurement of an observable. In an experiment, the |
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instrument analyzing the system must average its observation of the |
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finite time of the measurement. What is required then, is a principle |
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to relate the time average to the ensemble average. This is the |
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ergodic hypothesis. |
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constrained dynamics |
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Ergodicity is the assumption that given an infinite amount of time, a |
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system will visit every available point in phase |
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space.\cite{Frenkel1996} For most systems, this is a valid assumption, |
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except in cases where the system may be trapped in a local feature |
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(\emph{e.g.}~glasses). When valid, ergodicity allows the unification |
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of a time averaged observation and an ensemble averaged one. If an |
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observation is averaged over a sufficiently long time, the system is |
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assumed to visit all appropriately available points in phase space, |
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giving a properly weighted statistical average. This allows the |
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researcher freedom of choice when deciding how best to measure a given |
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observable. When an ensemble averaged approach seems most logical, |
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the Monte Carlo techniques described in Sec.~\ref{introSec:monteCarlo} |
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can be utilized. Conversely, if a problem lends itself to a time |
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averaging approach, the Molecular Dynamics techniques in |
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Sec.~\ref{introSec:MD} can be employed. |
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\section{\label{introSec:chapterLayout}Chapter Layout} |
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\section{\label{introSec:monteCarlo}Monte Carlo Simulations} |
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\subsection{\label{introSec:RSA}Random Sequential Adsorption} |
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The Monte Carlo method was developed by Metropolis and Ulam for their |
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work in fissionable material.\cite{metropolis:1949} The method is so |
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named, because it heavily uses random numbers in its |
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solution.\cite{allen87:csl} The Monte Carlo method allows for the |
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solution of integrals through the stochastic sampling of the values |
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within the integral. In the simplest case, the evaluation of an |
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integral would follow a brute force method of |
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sampling.\cite{Frenkel1996} Consider the following single dimensional |
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integral: |
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\begin{equation} |
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I = f(x)dx |
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\label{eq:MCex1} |
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\end{equation} |
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The equation can be recast as: |
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\begin{equation} |
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I = \int^b_a \frac{f(x)}{\rho(x)} \rho(x) dx |
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\label{introEq:Importance1} |
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\end{equation} |
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Where $\rho(x)$ is an arbitrary probability distribution in $x$. If |
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one conducts $\tau$ trials selecting a random number, $\zeta_\tau$, |
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from the distribution $\rho(x)$ on the interval $[a,b]$, then |
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Eq.~\ref{introEq:Importance1} becomes |
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\begin{equation} |
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I= \lim_{\tau \rightarrow \infty}\biggl \langle \frac{f(x)}{\rho(x)} \biggr \rangle_{\text{trials}[a,b]} |
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\label{introEq:Importance2} |
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\end{equation} |
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If $\rho(x)$ is uniformly distributed over the interval $[a,b]$, |
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\begin{equation} |
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\rho(x) = \frac{1}{b-a} |
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\label{introEq:importance2b} |
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\end{equation} |
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then the integral can be rewritten as |
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\begin{equation} |
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I = (b-a)\lim_{\tau \rightarrow \infty} |
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\langle f(x) \rangle_{\text{trials}[a,b]} |
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\label{eq:MCex2} |
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\end{equation} |
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\subsection{\label{introSec:OOPSE}The OOPSE Simulation Package} |
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However, in Statistical Mechanics, one is typically interested in |
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integrals of the form: |
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\begin{equation} |
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\langle A \rangle = \frac{\int d^N \mathbf{r}~A(\mathbf{r}^N)% |
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e^{-\beta V(\mathbf{r}^N)}}% |
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{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}} |
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\label{eq:mcEnsAvg} |
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\end{equation} |
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Where $\mathbf{r}^N$ stands for the coordinates of all $N$ particles |
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and $A$ is some observable that is only dependent on position. This is |
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the ensemble average of $A$ as presented in |
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Sec.~\ref{introSec:statThermo}, except here $A$ is independent of |
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momentum. Therefore the momenta contribution of the integral can be |
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factored out, leaving the configurational integral. Application of the |
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brute force method to this system would yield highly inefficient |
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results. Due to the Boltzmann weighting of this integral, most random |
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configurations will have a near zero contribution to the ensemble |
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average. This is where importance sampling comes into |
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play.\cite{allen87:csl} |
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\subsection{\label{introSec:bilayers}A Mesoscale Model for Phospholipid Bilayers} |
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Importance sampling is a method where the distribution, from which the |
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random configurations are chosen, is selected in such a way as to |
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efficiently sample the integral in question. Looking at |
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Eq.~\ref{eq:mcEnsAvg}, and realizing |
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\begin {equation} |
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\rho_{kT}(\mathbf{r}^N) = |
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\frac{e^{-\beta V(\mathbf{r}^N)}} |
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{\int d^N \mathbf{r}~e^{-\beta V(\mathbf{r}^N)}} |
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\label{introEq:MCboltzman} |
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\end{equation} |
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Where $\rho_{kT}$ is the Boltzmann distribution. The ensemble average |
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can be rewritten as |
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\begin{equation} |
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\langle A \rangle = \int d^N \mathbf{r}~A(\mathbf{r}^N) |
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\rho_{kT}(\mathbf{r}^N) |
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\label{introEq:Importance3} |
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\end{equation} |
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Applying Eq.~\ref{introEq:Importance1} one obtains |
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\begin{equation} |
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\langle A \rangle = \biggl \langle |
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\frac{ A \rho_{kT}(\mathbf{r}^N) } |
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{\rho(\mathbf{r}^N)} \biggr \rangle_{\text{trials}} |
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\label{introEq:Importance4} |
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\end{equation} |
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By selecting $\rho(\mathbf{r}^N)$ to be $\rho_{kT}(\mathbf{r}^N)$ |
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Eq.~\ref{introEq:Importance4} becomes |
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> |
\begin{equation} |
| 320 |
> |
\langle A \rangle = \langle A(\mathbf{r}^N) \rangle_{kT} |
| 321 |
> |
\label{introEq:Importance5} |
| 322 |
> |
\end{equation} |
| 323 |
> |
The difficulty is selecting points $\mathbf{r}^N$ such that they are |
| 324 |
> |
sampled from the distribution $\rho_{kT}(\mathbf{r}^N)$. A solution |
| 325 |
> |
was proposed by Metropolis \emph{et al}.\cite{metropolis:1953} which involved |
| 326 |
> |
the use of a Markov chain whose limiting distribution was |
| 327 |
> |
$\rho_{kT}(\mathbf{r}^N)$. |
| 328 |
> |
|
| 329 |
> |
\subsection{\label{introSec:markovChains}Markov Chains} |
| 330 |
> |
|
| 331 |
> |
A Markov chain is a chain of states satisfying the following |
| 332 |
> |
conditions:\cite{leach01:mm} |
| 333 |
> |
\begin{enumerate} |
| 334 |
> |
\item The outcome of each trial depends only on the outcome of the previous trial. |
| 335 |
> |
\item Each trial belongs to a finite set of outcomes called the state space. |
| 336 |
> |
\end{enumerate} |
| 337 |
> |
If given two configurations, $\mathbf{r}^N_m$ and $\mathbf{r}^N_n$, |
| 338 |
> |
$\rho_m$ and $\rho_n$ are the probabilities of being in state |
| 339 |
> |
$\mathbf{r}^N_m$ and $\mathbf{r}^N_n$ respectively. Further, the two |
| 340 |
> |
states are linked by a transition probability, $\pi_{mn}$, which is the |
| 341 |
> |
probability of going from state $m$ to state $n$. |
| 342 |
> |
|
| 343 |
> |
\newcommand{\accMe}{\operatorname{acc}} |
| 344 |
> |
|
| 345 |
> |
The transition probability is given by the following: |
| 346 |
> |
\begin{equation} |
| 347 |
> |
\pi_{mn} = \alpha_{mn} \times \accMe(m \rightarrow n) |
| 348 |
> |
\label{introEq:MCpi} |
| 349 |
> |
\end{equation} |
| 350 |
> |
Where $\alpha_{mn}$ is the probability of attempting the move $m |
| 351 |
> |
\rightarrow n$, and $\accMe$ is the probability of accepting the move |
| 352 |
> |
$m \rightarrow n$. Defining a probability vector, |
| 353 |
> |
$\boldsymbol{\rho}$, such that |
| 354 |
> |
\begin{equation} |
| 355 |
> |
\boldsymbol{\rho} = \{\rho_1, \rho_2, \ldots \rho_m, \rho_n, |
| 356 |
> |
\ldots \rho_N \} |
| 357 |
> |
\label{introEq:MCrhoVector} |
| 358 |
> |
\end{equation} |
| 359 |
> |
a transition matrix $\boldsymbol{\Pi}$ can be defined, |
| 360 |
> |
whose elements are $\pi_{mn}$, for each given transition. The |
| 361 |
> |
limiting distribution of the Markov chain can then be found by |
| 362 |
> |
applying the transition matrix an infinite number of times to the |
| 363 |
> |
distribution vector. |
| 364 |
> |
\begin{equation} |
| 365 |
> |
\boldsymbol{\rho}_{\text{limit}} = |
| 366 |
> |
\lim_{N \rightarrow \infty} \boldsymbol{\rho}_{\text{initial}} |
| 367 |
> |
\boldsymbol{\Pi}^N |
| 368 |
> |
\label{introEq:MCmarkovLimit} |
| 369 |
> |
\end{equation} |
| 370 |
> |
The limiting distribution of the chain is independent of the starting |
| 371 |
> |
distribution, and successive applications of the transition matrix |
| 372 |
> |
will only yield the limiting distribution again. |
| 373 |
> |
\begin{equation} |
| 374 |
> |
\boldsymbol{\rho}_{\text{limit}} = \boldsymbol{\rho}_{\text{initial}} |
| 375 |
> |
\boldsymbol{\Pi} |
| 376 |
> |
\label{introEq:MCmarkovEquil} |
| 377 |
> |
\end{equation} |
| 378 |
> |
|
| 379 |
> |
\subsection{\label{introSec:metropolisMethod}The Metropolis Method} |
| 380 |
> |
|
| 381 |
> |
In the Metropolis method\cite{metropolis:1953} |
| 382 |
> |
Eq.~\ref{introEq:MCmarkovEquil} is solved such that |
| 383 |
> |
$\boldsymbol{\rho}_{\text{limit}}$ matches the Boltzmann distribution |
| 384 |
> |
of states. The method accomplishes this by imposing the strong |
| 385 |
> |
condition of microscopic reversibility on the equilibrium |
| 386 |
> |
distribution. Meaning, that at equilibrium the probability of going |
| 387 |
> |
from $m$ to $n$ is the same as going from $n$ to $m$. |
| 388 |
> |
\begin{equation} |
| 389 |
> |
\rho_m\pi_{mn} = \rho_n\pi_{nm} |
| 390 |
> |
\label{introEq:MCmicroReverse} |
| 391 |
> |
\end{equation} |
| 392 |
> |
Further, $\boldsymbol{\alpha}$ is chosen to be a symmetric matrix in |
| 393 |
> |
the Metropolis method. Using Eq.~\ref{introEq:MCpi}, |
| 394 |
> |
Eq.~\ref{introEq:MCmicroReverse} becomes |
| 395 |
> |
\begin{equation} |
| 396 |
> |
\frac{\accMe(m \rightarrow n)}{\accMe(n \rightarrow m)} = |
| 397 |
> |
\frac{\rho_n}{\rho_m} |
| 398 |
> |
\label{introEq:MCmicro2} |
| 399 |
> |
\end{equation} |
| 400 |
> |
For a Boltzmann limiting distribution, |
| 401 |
> |
\begin{equation} |
| 402 |
> |
\frac{\rho_n}{\rho_m} = e^{-\beta[\mathcal{U}(n) - \mathcal{U}(m)]} |
| 403 |
> |
= e^{-\beta \Delta \mathcal{U}} |
| 404 |
> |
\label{introEq:MCmicro3} |
| 405 |
> |
\end{equation} |
| 406 |
> |
This allows for the following set of acceptance rules be defined: |
| 407 |
> |
\begin{equation} |
| 408 |
> |
\accMe( m \rightarrow n ) = |
| 409 |
> |
\begin{cases} |
| 410 |
> |
1& \text{if $\Delta \mathcal{U} \leq 0$,} \\ |
| 411 |
> |
e^{-\beta \Delta \mathcal{U}}& \text{if $\Delta \mathcal{U} > 0$.} |
| 412 |
> |
\end{cases} |
| 413 |
> |
\label{introEq:accRules} |
| 414 |
> |
\end{equation} |
| 415 |
> |
|
| 416 |
> |
Using the acceptance criteria from Eq.~\ref{introEq:accRules} the |
| 417 |
> |
Metropolis method proceeds as follows |
| 418 |
> |
\begin{enumerate} |
| 419 |
> |
\item Generate an initial configuration $\mathbf{r}^N$ which has some finite probability in $\rho_{kT}$. |
| 420 |
> |
\item Modify $\mathbf{r}^N$, to generate configuration $\mathbf{r^{\prime}}^N$. |
| 421 |
> |
\item If the new configuration lowers the energy of the system, accept the move with unity ($\mathbf{r}^N$ becomes $\mathbf{r^{\prime}}^N$). Otherwise accept with probability $e^{-\beta \Delta \mathcal{U}}$. |
| 422 |
> |
\item Accumulate the average for the configurational observable of interest. |
| 423 |
> |
\item Repeat from step 2 until the average converges. |
| 424 |
> |
\end{enumerate} |
| 425 |
> |
One important note is that the average is accumulated whether the move |
| 426 |
> |
is accepted or not, this ensures proper weighting of the average. |
| 427 |
> |
Using Eq.~\ref{introEq:Importance4} it becomes clear that the |
| 428 |
> |
accumulated averages are the ensemble averages, as this method ensures |
| 429 |
> |
that the limiting distribution is the Boltzmann distribution. |
| 430 |
> |
|
| 431 |
> |
\section{\label{introSec:MD}Molecular Dynamics Simulations} |
| 432 |
> |
|
| 433 |
> |
The main simulation tool used in this research is Molecular Dynamics. |
| 434 |
> |
Molecular Dynamics is when the equations of motion for a system are |
| 435 |
> |
integrated in order to obtain information about both the positions and |
| 436 |
> |
momentum of a system, allowing the calculation of not only |
| 437 |
> |
configurational observables, but momenta dependent ones as well: |
| 438 |
> |
diffusion constants, relaxation events, folding/unfolding |
| 439 |
> |
events, etc. With the time dependent information gained from a |
| 440 |
> |
Molecular Dynamics simulation, one can also calculate time correlation |
| 441 |
> |
functions of the form\cite{Hansen86} |
| 442 |
> |
\begin{equation} |
| 443 |
> |
\langle A(t)\,A(0)\rangle = \lim_{\tau\rightarrow\infty} \frac{1}{\tau} |
| 444 |
> |
\int_0^{\tau} A(t+t^{\prime})\,A(t^{\prime})\,dt^{\prime} |
| 445 |
> |
\label{introEq:timeCorr} |
| 446 |
> |
\end{equation} |
| 447 |
> |
These correlations can be used to measure fundamental time constants |
| 448 |
> |
of a system, such as diffusion constants from the velocity |
| 449 |
> |
autocorrelation or dipole relaxation times from the dipole |
| 450 |
> |
autocorrelation. Due to the principle of ergodicity, |
| 451 |
> |
Sec.~\ref{introSec:ergodic}, the average of these observables over the |
| 452 |
> |
time period of the simulation are taken to be the ensemble averages |
| 453 |
> |
for the system. |
| 454 |
> |
|
| 455 |
> |
The choice of when to use molecular dynamics over Monte Carlo |
| 456 |
> |
techniques, is normally decided by the observables in which the |
| 457 |
> |
researcher is interested. If the observables depend on time in |
| 458 |
> |
any fashion, then the only choice is molecular dynamics in some form. |
| 459 |
> |
However, when the observable is dependent only on the configuration, |
| 460 |
> |
then for most small systems, Monte Carlo techniques will be more efficient. |
| 461 |
> |
|
| 462 |
> |
The focus of research in the second half of this dissertation is |
| 463 |
> |
centered around the dynamic properties of phospholipid bilayers, |
| 464 |
> |
making molecular dynamics key in the simulation of those properties. |
| 465 |
> |
|
| 466 |
> |
\subsection{\label{introSec:mdAlgorithm}The Molecular Dynamics Algorithm} |
| 467 |
> |
|
| 468 |
> |
To illustrate how the molecular dynamics technique is applied, the |
| 469 |
> |
following sections will describe the sequence involved in a |
| 470 |
> |
simulation. Sec.~\ref{introSec:mdInit} deals with the initialization |
| 471 |
> |
of a simulation. Sec.~\ref{introSec:mdForce} discusses issues |
| 472 |
> |
involved with the calculation of the forces. |
| 473 |
> |
Sec.~\ref{introSec:mdIntegrate} concludes the algorithm discussion |
| 474 |
> |
with the integration of the equations of motion.\cite{Frenkel1996} |
| 475 |
> |
|
| 476 |
> |
\subsection{\label{introSec:mdInit}Initialization} |
| 477 |
> |
|
| 478 |
> |
When selecting the initial configuration for the simulation it is |
| 479 |
> |
important to consider what dynamics one is hoping to observe. |
| 480 |
> |
Ch.~\ref{chapt:lipid} deals with the formation and equilibrium dynamics of |
| 481 |
> |
phospholipid membranes. Therefore in these simulations initial |
| 482 |
> |
positions were selected that in some cases dispersed the lipids in |
| 483 |
> |
water, and in other cases structured the lipids into performed |
| 484 |
> |
bilayers. Important considerations at this stage of the simulation are: |
| 485 |
> |
\begin{itemize} |
| 486 |
> |
\item There are no major overlaps of molecular or atomic orbitals |
| 487 |
> |
\item Velocities are chosen in such a way as to not give the system a non=zero total momentum or angular momentum. |
| 488 |
> |
\item It is also sometimes desirable to select the velocities to correctly sample the target temperature. |
| 489 |
> |
\end{itemize} |
| 490 |
> |
|
| 491 |
> |
The first point is important due to the amount of potential energy |
| 492 |
> |
generated by having two particles too close together. If overlap |
| 493 |
> |
occurs, the first evaluation of forces will return numbers so large as |
| 494 |
> |
to render the numerical integration of the motion meaningless. The |
| 495 |
> |
second consideration keeps the system from drifting or rotating as a |
| 496 |
> |
whole. This arises from the fact that most simulations are of systems |
| 497 |
> |
in equilibrium in the absence of outside forces. Therefore any net |
| 498 |
> |
movement would be unphysical and an artifact of the simulation method |
| 499 |
> |
used. The final point addresses the selection of the magnitude of the |
| 500 |
> |
initial velocities. For many simulations it is convenient to use |
| 501 |
> |
this opportunity to scale the amount of kinetic energy to reflect the |
| 502 |
> |
desired thermal distribution of the system. However, it must be noted |
| 503 |
> |
that most systems will require further velocity rescaling after the |
| 504 |
> |
first few initial simulation steps due to either loss or gain of |
| 505 |
> |
kinetic energy from energy stored in potential degrees of freedom. |
| 506 |
> |
|
| 507 |
> |
\subsection{\label{introSec:mdForce}Force Evaluation} |
| 508 |
> |
|
| 509 |
> |
The evaluation of forces is the most computationally expensive portion |
| 510 |
> |
of a given molecular dynamics simulation. This is due entirely to the |
| 511 |
> |
evaluation of long range forces in a simulation, typically pair-wise. |
| 512 |
> |
These forces are most commonly the Van der Waals force, and sometimes |
| 513 |
> |
Coulombic forces as well. For a pair-wise force, there are $N(N-1)/ 2$ |
| 514 |
> |
pairs to be evaluated, where $N$ is the number of particles in the |
| 515 |
> |
system. This leads to the calculations scaling as $N^2$, making large |
| 516 |
> |
simulations prohibitive in the absence of any computation saving |
| 517 |
> |
techniques. |
| 518 |
> |
|
| 519 |
> |
Another consideration one must resolve, is that in a given simulation |
| 520 |
> |
a disproportionate number of the particles will feel the effects of |
| 521 |
> |
the surface.\cite{allen87:csl} For a cubic system of 1000 particles |
| 522 |
> |
arranged in a $10 \times 10 \times 10$ cube, 488 particles will be |
| 523 |
> |
exposed to the surface. Unless one is simulating an isolated particle |
| 524 |
> |
group in a vacuum, the behavior of the system will be far from the |
| 525 |
> |
desired bulk characteristics. To offset this, simulations employ the |
| 526 |
> |
use of periodic boundary images.\cite{born:1912} |
| 527 |
> |
|
| 528 |
> |
The technique involves the use of an algorithm that replicates the |
| 529 |
> |
simulation box on an infinite lattice in Cartesian space. Any given |
| 530 |
> |
particle leaving the simulation box on one side will have an image of |
| 531 |
> |
itself enter on the opposite side (see Fig.~\ref{introFig:pbc}). In |
| 532 |
> |
addition, this sets that any two particles have an image, real or |
| 533 |
> |
periodic, within $\text{box}/2$ of each other. A discussion of the |
| 534 |
> |
method used to calculate the periodic image can be found in |
| 535 |
> |
Sec.\ref{oopseSec:pbc}. |
| 536 |
> |
|
| 537 |
> |
\begin{figure} |
| 538 |
> |
\centering |
| 539 |
> |
\includegraphics[width=\linewidth]{pbcFig.eps} |
| 540 |
> |
\caption[An illustration of periodic boundary conditions]{A 2-D illustration of periodic boundary conditions. As one particle leaves the right of the simulation box, an image of it enters the left.} |
| 541 |
> |
\label{introFig:pbc} |
| 542 |
> |
\end{figure} |
| 543 |
> |
|
| 544 |
> |
Returning to the topic of the computational scale of the force |
| 545 |
> |
evaluation, the use of periodic boundary conditions requires that a |
| 546 |
> |
cutoff radius be employed. Using a cutoff radius improves the |
| 547 |
> |
efficiency of the force evaluation, as particles farther than a |
| 548 |
> |
predetermined distance, $r_{\text{cut}}$, are not included in the |
| 549 |
> |
calculation.\cite{Frenkel1996} In a simulation with periodic images, |
| 550 |
> |
$r_{\text{cut}}$ has a maximum value of $\text{box}/2$. |
| 551 |
> |
Fig.~\ref{introFig:rMax} illustrates how when using an |
| 552 |
> |
$r_{\text{cut}}$ larger than this value, or in the extreme limit of no |
| 553 |
> |
$r_{\text{cut}}$ at all, the corners of the simulation box are |
| 554 |
> |
unequally weighted due to the lack of particle images in the $x$, $y$, |
| 555 |
> |
or $z$ directions past a distance of $\text{box} / 2$. |
| 556 |
> |
|
| 557 |
> |
\begin{figure} |
| 558 |
> |
\centering |
| 559 |
> |
\includegraphics[width=\linewidth]{rCutMaxFig.eps} |
| 560 |
> |
\caption[An explanation of $r_{\text{cut}}$]{The yellow atom has all other images wrapped to itself as the center. If $r_{\text{cut}}=\text{box}/2$, then the distribution is uniform (blue atoms). However, when $r_{\text{cut}}>\text{box}/2$ the corners are disproportionately weighted (green atoms) vs the axial directions (shaded regions).} |
| 561 |
> |
\label{introFig:rMax} |
| 562 |
> |
\end{figure} |
| 563 |
> |
|
| 564 |
> |
With the use of an $r_{\text{cut}}$, however, comes a discontinuity in |
| 565 |
> |
the potential energy curve (Fig.~\ref{introFig:shiftPot}). To fix this |
| 566 |
> |
discontinuity, one calculates the potential energy at the |
| 567 |
> |
$r_{\text{cut}}$, and adds that value to the potential, causing |
| 568 |
> |
the function to go smoothly to zero at the cutoff radius. This |
| 569 |
> |
shifted potential ensures conservation of energy when integrating the |
| 570 |
> |
Newtonian equations of motion. |
| 571 |
> |
|
| 572 |
> |
\begin{figure} |
| 573 |
> |
\centering |
| 574 |
> |
\includegraphics[width=\linewidth]{shiftedPot.eps} |
| 575 |
> |
\caption[Shifting the Lennard-Jones Potential]{The Lennard-Jones potential (blue line) is shifted (red line) to remove the discontinuity at $r_{\text{cut}}$.} |
| 576 |
> |
\label{introFig:shiftPot} |
| 577 |
> |
\end{figure} |
| 578 |
> |
|
| 579 |
> |
The second main simplification used in this research is the Verlet |
| 580 |
> |
neighbor list. \cite{allen87:csl} In the Verlet method, one generates |
| 581 |
> |
a list of all neighbor atoms, $j$, surrounding atom $i$ within some |
| 582 |
> |
cutoff $r_{\text{list}}$, where $r_{\text{list}}>r_{\text{cut}}$. |
| 583 |
> |
This list is created the first time forces are evaluated, then on |
| 584 |
> |
subsequent force evaluations, pair calculations are only calculated |
| 585 |
> |
from the neighbor lists. The lists are updated if any given particle |
| 586 |
> |
in the system moves farther than $r_{\text{list}}-r_{\text{cut}}$, |
| 587 |
> |
giving rise to the possibility that a particle has left or joined a |
| 588 |
> |
neighbor list. |
| 589 |
> |
|
| 590 |
> |
\subsection{\label{introSec:mdIntegrate} Integration of the equations of motion} |
| 591 |
> |
|
| 592 |
> |
A starting point for the discussion of molecular dynamics integrators |
| 593 |
> |
is the Verlet algorithm.\cite{Frenkel1996} It begins with a Taylor |
| 594 |
> |
expansion of position in time: |
| 595 |
> |
\begin{equation} |
| 596 |
> |
q(t+\Delta t)= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 + |
| 597 |
> |
\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} + |
| 598 |
> |
\mathcal{O}(\Delta t^4) |
| 599 |
> |
\label{introEq:verletForward} |
| 600 |
> |
\end{equation} |
| 601 |
> |
As well as, |
| 602 |
> |
\begin{equation} |
| 603 |
> |
q(t-\Delta t)= q(t) - v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 - |
| 604 |
> |
\frac{\Delta t^3}{3!}\frac{\partial q(t)}{\partial t} + |
| 605 |
> |
\mathcal{O}(\Delta t^4) |
| 606 |
> |
\label{introEq:verletBack} |
| 607 |
> |
\end{equation} |
| 608 |
> |
Where $m$ is the mass of the particle, $q(t)$ is the position at time |
| 609 |
> |
$t$, $v(t)$ the velocity, and $F(t)$ the force acting on the |
| 610 |
> |
particle. Adding together Eq.~\ref{introEq:verletForward} and |
| 611 |
> |
Eq.~\ref{introEq:verletBack} results in, |
| 612 |
> |
\begin{equation} |
| 613 |
> |
q(t+\Delta t)+q(t-\Delta t) = |
| 614 |
> |
2q(t) + \frac{F(t)}{m}\Delta t^2 + \mathcal{O}(\Delta t^4) |
| 615 |
> |
\label{introEq:verletSum} |
| 616 |
> |
\end{equation} |
| 617 |
> |
Or equivalently, |
| 618 |
> |
\begin{equation} |
| 619 |
> |
q(t+\Delta t) \approx |
| 620 |
> |
2q(t) - q(t-\Delta t) + \frac{F(t)}{m}\Delta t^2 |
| 621 |
> |
\label{introEq:verletFinal} |
| 622 |
> |
\end{equation} |
| 623 |
> |
Which contains an error in the estimate of the new positions on the |
| 624 |
> |
order of $\Delta t^4$. |
| 625 |
> |
|
| 626 |
> |
In practice, however, the simulations in this research were integrated |
| 627 |
> |
with a velocity reformulation of the Verlet method.\cite{allen87:csl} |
| 628 |
> |
\begin{align} |
| 629 |
> |
q(t+\Delta t) &= q(t) + v(t)\Delta t + \frac{F(t)}{2m}\Delta t^2 % |
| 630 |
> |
\label{introEq:MDvelVerletPos} \\% |
| 631 |
> |
% |
| 632 |
> |
v(t+\Delta t) &= v(t) + \frac{\Delta t}{2m}[F(t) + F(t+\Delta t)] % |
| 633 |
> |
\label{introEq:MDvelVerletVel} |
| 634 |
> |
\end{align} |
| 635 |
> |
The original Verlet algorithm can be regained by substituting the |
| 636 |
> |
velocity back into Eq.~\ref{introEq:MDvelVerletPos}. The Verlet |
| 637 |
> |
formulations are chosen in this research because the algorithms have |
| 638 |
> |
very little long term drift in energy conservation. Energy |
| 639 |
> |
conservation in a molecular dynamics simulation is of extreme |
| 640 |
> |
importance, as it is a measure of how closely one is following the |
| 641 |
> |
``true'' trajectory with the finite integration scheme. An exact |
| 642 |
> |
solution to the integration will conserve area in phase space, as well |
| 643 |
> |
as be reversible in time, that is, the trajectory integrated forward |
| 644 |
> |
or backwards will exactly match itself. Having a finite algorithm |
| 645 |
> |
that both conserves area in phase space and is time reversible, |
| 646 |
> |
therefore increases, but does not guarantee the ``correctness'' or the |
| 647 |
> |
integrated trajectory. |
| 648 |
> |
|
| 649 |
> |
It can be shown,\cite{Frenkel1996} that although the Verlet algorithm |
| 650 |
> |
does not rigorously preserve the actual Hamiltonian, it does preserve |
| 651 |
> |
a pseudo-Hamiltonian which shadows the real one in phase space. This |
| 652 |
> |
pseudo-Hamiltonian is provably area-conserving as well as time |
| 653 |
> |
reversible. The fact that it shadows the true Hamiltonian in phase |
| 654 |
> |
space is acceptable in actual simulations as one is interested in the |
| 655 |
> |
ensemble average of the observable being measured. From the ergodic |
| 656 |
> |
hypothesis (Sec.~\ref{introSec:statThermo}), it is known that the time |
| 657 |
> |
average will match the ensemble average, therefore two similar |
| 658 |
> |
trajectories in phase space should give matching statistical averages. |
| 659 |
> |
|
| 660 |
> |
\subsection{\label{introSec:MDfurther}Further Considerations} |
| 661 |
> |
|
| 662 |
> |
In the simulations presented in this research, a few additional |
| 663 |
> |
parameters are needed to describe the motions. The simulations |
| 664 |
> |
involving water and phospholipids in Ch.~\ref{chapt:lipid} are |
| 665 |
> |
required to integrate the equations of motions for dipoles on atoms. |
| 666 |
> |
This involves an additional three parameters be specified for each |
| 667 |
> |
dipole atom: $\phi$, $\theta$, and $\psi$. These three angles are |
| 668 |
> |
taken to be the Euler angles, where $\phi$ is a rotation about the |
| 669 |
> |
$z$-axis, and $\theta$ is a rotation about the new $x$-axis, and |
| 670 |
> |
$\psi$ is a final rotation about the new $z$-axis (see |
| 671 |
> |
Fig.~\ref{introFig:eulerAngles}). This sequence of rotations can be |
| 672 |
> |
accumulated into a single $3 \times 3$ matrix, $\mathbf{A}$, |
| 673 |
> |
defined as follows: |
| 674 |
> |
\begin{equation} |
| 675 |
> |
\mathbf{A} = |
| 676 |
> |
\begin{bmatrix} |
| 677 |
> |
\cos\phi\cos\psi-\sin\phi\cos\theta\sin\psi &% |
| 678 |
> |
\sin\phi\cos\psi+\cos\phi\cos\theta\sin\psi &% |
| 679 |
> |
\sin\theta\sin\psi \\% |
| 680 |
> |
% |
| 681 |
> |
-\cos\phi\sin\psi-\sin\phi\cos\theta\cos\psi &% |
| 682 |
> |
-\sin\phi\sin\psi+\cos\phi\cos\theta\cos\psi &% |
| 683 |
> |
\sin\theta\cos\psi \\% |
| 684 |
> |
% |
| 685 |
> |
\sin\phi\sin\theta &% |
| 686 |
> |
-\cos\phi\sin\theta &% |
| 687 |
> |
\cos\theta |
| 688 |
> |
\end{bmatrix} |
| 689 |
> |
\label{introEq:EulerRotMat} |
| 690 |
> |
\end{equation} |
| 691 |
> |
|
| 692 |
> |
\begin{figure} |
| 693 |
> |
\centering |
| 694 |
> |
\includegraphics[width=\linewidth]{eulerRotFig.eps} |
| 695 |
> |
\caption[Euler rotation of Cartesian coordinates]{The rotation scheme for Euler angles. First is a rotation of $\phi$ about the $z$ axis (blue rotation). Next is a rotation of $\theta$ about the new $x$ axis (green rotation). Lastly is a final rotation of $\psi$ about the new $z$ axis (red rotation).} |
| 696 |
> |
\label{introFig:eulerAngles} |
| 697 |
> |
\end{figure} |
| 698 |
> |
|
| 699 |
> |
The equations of motion for Euler angles can be written down |
| 700 |
> |
as\cite{allen87:csl} |
| 701 |
> |
\begin{align} |
| 702 |
> |
\dot{\phi} &= -\omega^s_x \frac{\sin\phi\cos\theta}{\sin\theta} + |
| 703 |
> |
\omega^s_y \frac{\cos\phi\cos\theta}{\sin\theta} + |
| 704 |
> |
\omega^s_z |
| 705 |
> |
\label{introEq:MDeulerPhi} \\% |
| 706 |
> |
% |
| 707 |
> |
\dot{\theta} &= \omega^s_x \cos\phi + \omega^s_y \sin\phi |
| 708 |
> |
\label{introEq:MDeulerTheta} \\% |
| 709 |
> |
% |
| 710 |
> |
\dot{\psi} &= \omega^s_x \frac{\sin\phi}{\sin\theta} - |
| 711 |
> |
\omega^s_y \frac{\cos\phi}{\sin\theta} |
| 712 |
> |
\label{introEq:MDeulerPsi} |
| 713 |
> |
\end{align} |
| 714 |
> |
Where $\omega^s_i$ is the angular velocity in the lab space frame |
| 715 |
> |
along Cartesian coordinate $i$. However, a difficulty arises when |
| 716 |
> |
attempting to integrate Eq.~\ref{introEq:MDeulerPhi} and |
| 717 |
> |
Eq.~\ref{introEq:MDeulerPsi}. The $\frac{1}{\sin \theta}$ present in |
| 718 |
> |
both equations means there is a non-physical instability present when |
| 719 |
> |
$\theta$ is 0 or $\pi$. To correct for this, the simulations integrate |
| 720 |
> |
the rotation matrix, $\mathbf{A}$, directly, thus avoiding the |
| 721 |
> |
instability. This method was proposed by Dullweber |
| 722 |
> |
\emph{et. al.}\cite{Dullweber1997}, and is presented in |
| 723 |
> |
Sec.~\ref{introSec:MDsymplecticRot}. |
| 724 |
> |
|
| 725 |
> |
\subsection{\label{introSec:MDliouville}Liouville Propagator} |
| 726 |
> |
|
| 727 |
> |
Before discussing the integration of the rotation matrix, it is |
| 728 |
> |
necessary to understand the construction of a ``good'' integration |
| 729 |
> |
scheme. It has been previously |
| 730 |
> |
discussed(Sec.~\ref{introSec:mdIntegrate}) how it is desirable for an |
| 731 |
> |
integrator to be symplectic, or time reversible. The following is an |
| 732 |
> |
outline of the Trotter factorization of the Liouville Propagator as a |
| 733 |
> |
scheme for generating symplectic integrators.\cite{Tuckerman92} |
| 734 |
> |
|
| 735 |
> |
For a system with $f$ degrees of freedom the Liouville operator can be |
| 736 |
> |
defined as, |
| 737 |
> |
\begin{equation} |
| 738 |
> |
iL=\sum^f_{j=1} \biggl [\dot{q}_j\frac{\partial}{\partial q_j} + |
| 739 |
> |
F_j\frac{\partial}{\partial p_j} \biggr ] |
| 740 |
> |
\label{introEq:LiouvilleOperator} |
| 741 |
> |
\end{equation} |
| 742 |
> |
Here, $q_j$ and $p_j$ are the position and conjugate momenta of a |
| 743 |
> |
degree of freedom, and $F_j$ is the force on that degree of freedom. |
| 744 |
> |
$\Gamma$ is defined as the set of all positions and conjugate momenta, |
| 745 |
> |
$\{q_j,p_j\}$, and the propagator, $U(t)$, is defined |
| 746 |
> |
\begin {equation} |
| 747 |
> |
U(t) = e^{iLt} |
| 748 |
> |
\label{introEq:Lpropagator} |
| 749 |
> |
\end{equation} |
| 750 |
> |
This allows the specification of $\Gamma$ at any time $t$ as |
| 751 |
> |
\begin{equation} |
| 752 |
> |
\Gamma(t) = U(t)\Gamma(0) |
| 753 |
> |
\label{introEq:Lp2} |
| 754 |
> |
\end{equation} |
| 755 |
> |
It is important to note, $U(t)$ is a unitary operator meaning |
| 756 |
> |
\begin{equation} |
| 757 |
> |
U(-t)=U^{-1}(t) |
| 758 |
> |
\label{introEq:Lp3} |
| 759 |
> |
\end{equation} |
| 760 |
> |
|
| 761 |
> |
Decomposing $L$ into two parts, $iL_1$ and $iL_2$, one can use the |
| 762 |
> |
Trotter theorem to yield |
| 763 |
> |
\begin{align} |
| 764 |
> |
e^{iLt} &= e^{i(L_1 + L_2)t} \notag \\% |
| 765 |
> |
% |
| 766 |
> |
&= \biggl [ e^{i(L_1 +L_2)\frac{t}{P}} \biggr]^P \notag \\% |
| 767 |
> |
% |
| 768 |
> |
&= \biggl [ e^{iL_1\frac{\Delta t}{2}}\, e^{iL_2\Delta t}\, |
| 769 |
> |
e^{iL_1\frac{\Delta t}{2}} \biggr ]^P + |
| 770 |
> |
\mathcal{O}\biggl (\frac{t^3}{P^2} \biggr ) \label{introEq:Lp4} |
| 771 |
> |
\end{align} |
| 772 |
> |
Where $\Delta t = t/P$. |
| 773 |
> |
With this, a discrete time operator $G(\Delta t)$ can be defined: |
| 774 |
> |
\begin{align} |
| 775 |
> |
G(\Delta t) &= e^{iL_1\frac{\Delta t}{2}}\, e^{iL_2\Delta t}\, |
| 776 |
> |
e^{iL_1\frac{\Delta t}{2}} \notag \\% |
| 777 |
> |
% |
| 778 |
> |
&= U_1 \biggl ( \frac{\Delta t}{2} \biggr )\, U_2 ( \Delta t )\, |
| 779 |
> |
U_1 \biggl ( \frac{\Delta t}{2} \biggr ) |
| 780 |
> |
\label{introEq:Lp5} |
| 781 |
> |
\end{align} |
| 782 |
> |
Because $U_1(t)$ and $U_2(t)$ are unitary, $G(\Delta t)$ is also |
| 783 |
> |
unitary. Meaning an integrator based on this factorization will be |
| 784 |
> |
reversible in time. |
| 785 |
> |
|
| 786 |
> |
As an example, consider the following decomposition of $L$: |
| 787 |
> |
\begin{align} |
| 788 |
> |
iL_1 &= \dot{q}\frac{\partial}{\partial q}% |
| 789 |
> |
\label{introEq:Lp6a} \\% |
| 790 |
> |
% |
| 791 |
> |
iL_2 &= F(q)\frac{\partial}{\partial p}% |
| 792 |
> |
\label{introEq:Lp6b} |
| 793 |
> |
\end{align} |
| 794 |
> |
This leads to propagator $G( \Delta t )$ as, |
| 795 |
> |
\begin{equation} |
| 796 |
> |
G(\Delta t) = e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} \, |
| 797 |
> |
e^{\Delta t\,\dot{q}\frac{\partial}{\partial q}} \, |
| 798 |
> |
e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} |
| 799 |
> |
\label{introEq:Lp7} |
| 800 |
> |
\end{equation} |
| 801 |
> |
Operating $G(\Delta t)$ on $\Gamma(0)$, and utilizing the operator property |
| 802 |
> |
\begin{equation} |
| 803 |
> |
e^{c\frac{\partial}{\partial x}}\, f(x) = f(x+c) |
| 804 |
> |
\label{introEq:Lp8} |
| 805 |
> |
\end{equation} |
| 806 |
> |
Where $c$ is independent of $x$. One obtains the following: |
| 807 |
> |
\begin{align} |
| 808 |
> |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
| 809 |
> |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)] \label{introEq:Lp9a}\\% |
| 810 |
> |
% |
| 811 |
> |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr )% |
| 812 |
> |
\label{introEq:Lp9b}\\% |
| 813 |
> |
% |
| 814 |
> |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 815 |
> |
\frac{\Delta t}{2m}\, F[q(0)] \label{introEq:Lp9c} |
| 816 |
> |
\end{align} |
| 817 |
> |
Or written another way, |
| 818 |
> |
\begin{align} |
| 819 |
> |
q(t+\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 820 |
> |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2} % |
| 821 |
> |
\label{introEq:Lp10a} \\% |
| 822 |
> |
% |
| 823 |
> |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
| 824 |
> |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr] % |
| 825 |
> |
\label{introEq:Lp10b} |
| 826 |
> |
\end{align} |
| 827 |
> |
This is the velocity Verlet formulation presented in |
| 828 |
> |
Sec.~\ref{introSec:mdIntegrate}. Because this integration scheme is |
| 829 |
> |
comprised of unitary propagators, it is symplectic, and therefore area |
| 830 |
> |
preserving in phase space. From the preceding factorization, one can |
| 831 |
> |
see that the integration of the equations of motion would follow: |
| 832 |
> |
\begin{enumerate} |
| 833 |
> |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
| 834 |
> |
|
| 835 |
> |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 836 |
> |
|
| 837 |
> |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 838 |
> |
|
| 839 |
> |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 840 |
> |
\end{enumerate} |
| 841 |
> |
|
| 842 |
> |
\subsection{\label{introSec:MDsymplecticRot} Symplectic Propagation of the Rotation Matrix} |
| 843 |
> |
|
| 844 |
> |
Based on the factorization from the previous section, |
| 845 |
> |
Dullweber\emph{et al}.\cite{Dullweber1997}~ proposed a scheme for the |
| 846 |
> |
symplectic propagation of the rotation matrix, $\mathbf{A}$, as an |
| 847 |
> |
alternative method for the integration of orientational degrees of |
| 848 |
> |
freedom. The method starts with a straightforward splitting of the |
| 849 |
> |
Liouville operator: |
| 850 |
> |
\begin{align} |
| 851 |
> |
iL_{\text{pos}} &= \dot{q}\frac{\partial}{\partial q} + |
| 852 |
> |
\mathbf{\dot{A}}\frac{\partial}{\partial \mathbf{A}} |
| 853 |
> |
\label{introEq:SR1a} \\% |
| 854 |
> |
% |
| 855 |
> |
iL_F &= F(q)\frac{\partial}{\partial p} |
| 856 |
> |
\label{introEq:SR1b} \\% |
| 857 |
> |
iL_{\tau} &= \tau(\mathbf{A})\frac{\partial}{\partial \pi} |
| 858 |
> |
\label{introEq:SR1b} \\% |
| 859 |
> |
\end{align} |
| 860 |
> |
Where $\tau(\mathbf{A})$ is the torque of the system |
| 861 |
> |
due to the configuration, and $\pi$ is the conjugate |
| 862 |
> |
angular momenta of the system. The propagator, $G(\Delta t)$, becomes |
| 863 |
> |
\begin{equation} |
| 864 |
> |
G(\Delta t) = e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} \, |
| 865 |
> |
e^{\frac{\Delta t}{2} \tau(\mathbf{A})\frac{\partial}{\partial \pi}} \, |
| 866 |
> |
e^{\Delta t\,iL_{\text{pos}}} \, |
| 867 |
> |
e^{\frac{\Delta t}{2} \tau(\mathbf{A})\frac{\partial}{\partial \pi}} \, |
| 868 |
> |
e^{\frac{\Delta t}{2} F(q)\frac{\partial}{\partial p}} |
| 869 |
> |
\label{introEq:SR2} |
| 870 |
> |
\end{equation} |
| 871 |
> |
Propagation of the linear and angular momenta follows as in the Verlet |
| 872 |
> |
scheme. The propagation of positions also follows the Verlet scheme |
| 873 |
> |
with the addition of a further symplectic splitting of the rotation |
| 874 |
> |
matrix propagation, $\mathcal{U}_{\text{rot}}(\Delta t)$, within |
| 875 |
> |
$U_{\text{pos}}(\Delta t)$. |
| 876 |
> |
\begin{equation} |
| 877 |
> |
\mathcal{U}_{\text{rot}}(\Delta t) = |
| 878 |
> |
\mathcal{U}_x \biggl(\frac{\Delta t}{2}\biggr)\, |
| 879 |
> |
\mathcal{U}_y \biggl(\frac{\Delta t}{2}\biggr)\, |
| 880 |
> |
\mathcal{U}_z (\Delta t)\, |
| 881 |
> |
\mathcal{U}_y \biggl(\frac{\Delta t}{2}\biggr)\, |
| 882 |
> |
\mathcal{U}_x \biggl(\frac{\Delta t}{2}\biggr)\, |
| 883 |
> |
\label{introEq:SR3} |
| 884 |
> |
\end{equation} |
| 885 |
> |
Where $\mathcal{U}_j$ is a unitary rotation of $\mathbf{A}$ and |
| 886 |
> |
$\pi$ about each axis $j$. As all propagations are now |
| 887 |
> |
unitary and symplectic, the entire integration scheme is also |
| 888 |
> |
symplectic and time reversible. |
| 889 |
> |
|
| 890 |
> |
\section{\label{introSec:layout}Dissertation Layout} |
| 891 |
> |
|
| 892 |
> |
This dissertation is divided as follows:Ch.~\ref{chapt:RSA} |
| 893 |
> |
presents the random sequential adsorption simulations of related |
| 894 |
> |
pthalocyanines on a gold (111) surface. Ch.~\ref{chapt:OOPSE} |
| 895 |
> |
is about the writing of the molecular dynamics simulation package |
| 896 |
> |
{\sc oopse}. Ch.~\ref{chapt:lipid} regards the simulations of |
| 897 |
> |
phospholipid bilayers using a mesoscale model. And lastly, |
| 898 |
> |
Ch.~\ref{chapt:conclusion} concludes this dissertation with a |
| 899 |
> |
summary of all results. The chapters are arranged in chronological |
| 900 |
> |
order, and reflect the progression of techniques I employed during my |
| 901 |
> |
research. |
| 902 |
> |
|
| 903 |
> |
The chapter concerning random sequential adsorption simulations is a |
| 904 |
> |
study in applying Statistical Mechanics simulation techniques in order |
| 905 |
> |
to obtain a simple model capable of explaining the results. My |
| 906 |
> |
advisor, Dr. Gezelter, and I were approached by a colleague, |
| 907 |
> |
Dr. Lieberman, about possible explanations for the partial coverage of a |
| 908 |
> |
gold surface by a particular compound of hers. We suggested it might |
| 909 |
> |
be due to the statistical packing fraction of disks on a plane, and |
| 910 |
> |
set about to simulate this system. As the events in our model were |
| 911 |
> |
not dynamic in nature, a Monte Carlo method was employed. Here, if a |
| 912 |
> |
molecule landed on the surface without overlapping another, then its |
| 913 |
> |
landing was accepted. However, if there was overlap, the landing we |
| 914 |
> |
rejected and a new random landing location was chosen. This defined |
| 915 |
> |
our acceptance rules and allowed us to construct a Markov chain whose |
| 916 |
> |
limiting distribution was the surface coverage in which we were |
| 917 |
> |
interested. |
| 918 |
> |
|
| 919 |
> |
The following chapter, about the simulation package {\sc oopse}, |
| 920 |
> |
describes in detail the large body of scientific code that had to be |
| 921 |
> |
written in order to study phospholipid bilayers. Although there are |
| 922 |
> |
pre-existing molecular dynamic simulation packages available, none |
| 923 |
> |
were capable of implementing the models we were developing.{\sc oopse} |
| 924 |
> |
is a unique package capable of not only integrating the equations of |
| 925 |
> |
motion in Cartesian space, but is also able to integrate the |
| 926 |
> |
rotational motion of rigid bodies and dipoles. Add to this the |
| 927 |
> |
ability to perform calculations across parallel processors and a |
| 928 |
> |
flexible script syntax for creating systems, and {\sc oopse} becomes a |
| 929 |
> |
very powerful scientific instrument for the exploration of our model. |
| 930 |
> |
|
| 931 |
> |
Bringing us to Ch.~\ref{chapt:lipid}. Using {\sc oopse}, I have been |
| 932 |
> |
able to parameterize a mesoscale model for phospholipid simulations. |
| 933 |
> |
This model retains information about solvent ordering around the |
| 934 |
> |
bilayer, as well as information regarding the interaction of the |
| 935 |
> |
phospholipid head groups' dipoles with each other and the surrounding |
| 936 |
> |
solvent. These simulations give us insight into the dynamic events |
| 937 |
> |
that lead to the formation of phospholipid bilayers, as well as |
| 938 |
> |
provide the foundation for future exploration of bilayer phase |
| 939 |
> |
behavior with this model. |
| 940 |
> |
|
| 941 |
> |
Which leads into the last chapter, where I discuss future directions |
| 942 |
> |
for both{\sc oopse} and this mesoscale model. Additionally, I will |
| 943 |
> |
give a summary of results for this dissertation. |
| 944 |
> |
|
| 945 |
> |
|
