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Revision 953 by mmeineke, Sun Jan 18 03:07:59 2004 UTC vs.
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+The Monte Carlo method was developed by Metropolis and Ulam for their
+work in fissionable material.\cite{metropolis:1949} The method is so
-<
+named, because it heavily uses random numbers in the solution of the
-<
+problem.
->
+named, because it heavily uses random numbers in its
->
+solution.\cite{allen87:csl} The Monte Carlo method allows for the
->
+solution of integrals through the stochastic sampling of the values
->
+within the integral. In the simplest case, the evaluation of an
->
+integral would follow a brute force method of
->
+sampling.\cite{Frenkel1996} Consider the following single dimensional
->
+integral:
->
+\begin{equation}
->
+I = f(x)dx
->
+\label{eq:MCex1}
->
+\end{equation}
->
+The equation can be recast as:
->
+\begin{equation}
->
+I = (b-a)<f(x)>
->
+\label{eq:MCex2}
->
+\end{equation}
->
+Where $<f(x)>$ is the unweighted average over the interval
->
+$[a,b]$. The calculation of the integral could then be solved by
->
+randomly choosing points along the interval $[a,b]$ and calculating
->
+the value of $f(x)$ at each point. The accumulated average would then
->
+approach $I$ in the limit where the number of trials is infintely
->
+large.
-+
+However, in Statistical Mechanics, one is typically interested in
-+
+integrals of the form:
-+
+\begin{equation}
-+
+<A> = \frac{A}{exp^{-\beta}}
-+
+\label{eq:mcEnsAvg}
-+
+\end{equation}
-+
+Where $r^N$ stands for the coordinates of all $N$ particles and $A$ is
-+
+some observable that is only dependent on position. $<A>$ is the
-+
+ensemble average of $A$ as presented in
-+
+Sec.~\ref{introSec:statThermo}. Because $A$ is independent of
-+
+momentum, the momenta contribution of the integral can be factored
-+
+out, leaving the configurational integral. Application of the brute
-+
+force method to this system would yield highly inefficient
-+
+results. Due to the Boltzman weighting of this integral, most random
-+
+configurations will have a near zero contribution to the ensemble
-+
+average. This is where a importance sampling comes into
-+
+play.\cite{allen87:csl}
-+
+Importance Sampling is a method where one selects a distribution from
-+
+which the random configurations are chosen in order to more
-+
+efficiently calculate the integral.\cite{Frenkel1996} Consider again
-+
+Eq.~\ref{eq:MCex1} rewritten to be:
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+Where $fix$ is an arbitrary probability distribution in $x$.  If one
-+
+conducts $fix$ trials selecting a random number, $fix$, from the
-+
+distribution $fix$ on the interval $[a,b]$, then Eq.~\ref{fix} becomes
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+Looking at Eq.~ref{fix}, and realizing
-+
+\begin {equation}
-+
+EQ Here
-+
+\end{equation}
-+
+The ensemble average can be rewritten as
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+Appllying Eq.~ref{fix} one obtains
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+By selecting $fix$ to be $fix$ Eq.~ref{fix} becomes
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+The difficulty is selecting points $fix$ such that they are sampled
-+
+from the distribution $fix$.  A solution was proposed by Metropolis et
-+
+al.\cite{fix} which involved the use of a Markov chain whose limiting
-+
+distribution was $fix$.
-+
-+
+\subsection{Markov Chains}
-+
-+
+A Markov chain is a chain of states satisfying the following
-+
+conditions:\cite{fix}
-+
+\begin{itemize}
-+
+\item The outcome of each trial depends only on the outcome of the previous trial.
-+
+\item Each trial belongs to a finite set of outcomes called the state space.
-+
+\end{itemize}
-+
+If given two configuartions, $fix$ and $fix$, $fix$ and $fix$ are the
-+
+probablilities of being in state $fix$ and $fix$ respectively.
-+
+Further, the two states are linked by a transition probability, $fix$,
-+
+which is the probability of going from state $m$ to state $n$.
-+
-+
+The transition probability is given by the following:
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+Where $fix$ is the probability of attempting the move $fix$, and $fix$
-+
+is the probability of accepting the move $fix$.  Defining a
-+
+probability vector, $fix$, such that
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+a transition matrix $fix$ can be defined, whose elements are $fix$,
-+
+for each given transition.  The limiting distribution of the Markov
-+
+chain can then be found by applying the transition matrix an infinite
-+
+number of times to the distribution vector.
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
-+
+The limiting distribution of the chain is independent of the starting
-+
+distribution, and successive applications of the transition matrix
-+
+will only yield the limiting distribution again.
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
-+
+\subsection{fix}
-+
-+
+In the Metropolis method \cite{fix} Eq.~ref{fix} is solved such that
-+
+$fix$ matches the Boltzman distribution of states.  The method
-+
+accomplishes this by imposing the strong condition of microscopic
-+
+reversibility on the equilibrium distribution.  Meaning, that at
-+
+equilibrium the probability of going from $m$ to $n$ is the same as
-+
+going from $n$ to $m$.
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+Further, $fix$ is chosen to be a symetric matrix in the Metropolis
-+
+method.  Using Eq.~\ref{fix}, Eq.~\ref{fix} becomes
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+For a Boltxman limiting distribution
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+This allows for the following set of acceptance rules be defined:
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
-+
+Using the acceptance criteria from Eq.~\ref{fix} the Metropolis method
-+
+proceeds as follows
-+
+\begin{itemize}
-+
+\item Generate an initial configuration $fix$ which has some finite probability in $fix$.
-+
+\item Modify $fix$, to generate configuratioon $fix$.
-+
+\item If configuration $n$ lowers the energy of the system, accept the move with unity ($fix$ becomes $fix$).  Otherwise accept with probability $fix$.
-+
+\item Accumulate the average for the configurational observable of intereest.
-+
+\item Repeat from step 2 until average converges.
-+
+\end{itemize}
-+
+One important note is that the average is accumulated whether the move
-+
+is accepted or not, this ensures proper weighting of the average.
-+
+Using Eq.~\ref{fix} it becomes clear that the accumulated averages are
-+
+the ensemble averages, as this method ensures that the limiting
-+
+distribution is the Boltzman distribution.
-+
+\subsection{\label{introSec:md}Molecular Dynamics Simulations}
-<
+time averages
->
+The main simulation tool used in this research is Molecular Dynamics.
->
+Molecular Dynamics is when the equations of motion for a system are
->
+integrated in order to obtain information about both the positions and
->
+momentum of a system, allowing the calculation of not only
->
+configurational observables, but momenta dependent ones as well:
->
+diffusion constants, velocity auto correlations, folding/unfolding
->
+events, etc.  Due to the principle of ergodicity, Eq.~\ref{fix}, the
->
+average of these observables over the time period of the simulation
->
+are taken to be the ensemble averages for the system.
-<
+time integrating schemes
->
+The choice of when to use molecular dynamics over Monte Carlo
->
+techniques, is normally decided by the observables in which the
->
+researcher is interested.  If the observabvles depend on momenta in
->
+any fashion, then the only choice is molecular dynamics in some form.
->
+However, when the observable is dependent only on the configuration,
->
+then most of the time Monte Carlo techniques will be more efficent.
-<
+time reversible
->
+The focus of research in the second half of this dissertation is
->
+centered around the dynamic properties of phospholipid bilayers,
->
+making molecular dynamics key in the simulation of those properties.
-<
+symplectic methods
->
+\subsection{Molecular dynamics Algorithm}
-<
+Extended ensembles (NVT NPT)
->
+To illustrate how the molecular dynamics technique is applied, the
->
+following sections will describe the sequence involved in a
->
+simulation.  Sec.~\ref{fix} deals with the initialization of a
->
+simulation.  Sec.~\ref{fix} discusses issues involved with the
->
+calculation of the forces.  Sec.~\ref{fix} concludes the algorithm
->
+discussion with the integration of the equations of motion. \cite{fix}
-<
+constrained dynamics
->
+\subsection{initialization}
-+
+When selecting the initial configuration for the simulation it is
-+
+important to consider what dynamics one is hoping to observe.
-+
+Ch.~\ref{fix} deals with the formation and equilibrium dynamics of
-+
+phospholipid membranes.  Therefore in these simulations initial
-+
+positions were selected that in some cases dispersed the lipids in
-+
+water, and in other cases structured the lipids into preformed
-+
+bilayers.  Important considerations at this stage of the simulation are:
-+
+\begin{itemize}
-+
+\item There are no major overlaps of molecular or atomic orbitals
-+
+\item Velocities are chosen in such a way as to not gie the system a non=zero total momentum or angular momentum.
-+
+\item It is also sometimes desireable to select the velocities to correctly sample the target temperature.
-+
+\end{itemize}
-+
-+
+The first point is important due to the amount of potential energy
-+
+generated by having two particles too close together.  If overlap
-+
+occurs, the first evaluation of forces will return numbers so large as
-+
+to render the numerical integration of teh motion meaningless.  The
-+
+second consideration keeps the system from drifting or rotating as a
-+
+whole.  This arises from the fact that most simulations are of systems
-+
+in equilibrium in the absence of outside forces.  Therefore any net
-+
+movement would be unphysical and an artifact of the simulation method
-+
+used.  The final point addresses teh selection of the magnitude of the
-+
+initial velocities.  For many simulations it is convienient to use
-+
+this opportunity to scale the amount of kinetic energy to reflect the
-+
+desired thermal distribution of the system.  However, it must be noted
-+
+that most systems will require further velocity rescaling after the
-+
+first few initial simulation steps due to either loss or gain of
-+
+kinetic energy from energy stored in potential degrees of freedom.
-+
-+
+\subsection{Force Evaluation}
-+
-+
+The evaluation of forces is the most computationally expensive portion
-+
+of a given molecular dynamics simulation.  This is due entirely to the
-+
+evaluation of long range forces in a simulation, typically pair-wise.
-+
+These forces are most commonly the Van der Waals force, and sometimes
-+
+Coulombic forces as well.  For a pair-wise force, there are $fix$
-+
+pairs to be evaluated, where $n$ is the number of particles in the
-+
+system.  This leads to the calculations scaling as $fix$, making large
-+
+simulations prohibitive in the absence of any computation saving
-+
+techniques.
-+
-+
+Another consideration one must resolve, is that in a given simulation
-+
+a disproportionate number of the particles will feel the effects of
-+
+the surface. \cite{fix} For a cubic system of 1000 particles arranged
-+
+in a $10x10x10$ cube, 488 particles will be exposed to the surface.
-+
+Unless one is simulating an isolated particle group in a vacuum, the
-+
+behavior of the system will be far from the desired bulk
-+
+charecteristics.  To offset this, simulations employ the use of
-+
+periodic boundary images. \cite{fix}
-+
-+
+The technique involves the use of an algorithm that replicates the
-+
+simulation box on an infinite lattice in cartesian space.  Any given
-+
+particle leaving the simulation box on one side will have an image of
-+
+itself enter on the opposite side (see Fig.~\ref{fix}).
-+
+\begin{equation}
-+
+EQ Here
-+
+\end{equation}
-+
+In addition, this sets that any given particle pair has an image, real
-+
+or periodic, within $fix$ of each other.  A discussion of the method
-+
+used to calculate the periodic image can be found in Sec.\ref{fix}.
-+
-+
+Returning to the topic of the computational scale of the force
-+
+evaluation, the use of periodic boundary conditions requires that a
-+
+cutoff radius be employed.  Using a cutoff radius improves the
-+
+efficiency of the force evaluation, as particles farther than a
-+
+predetermined distance, $fix$, are not included in the
-+
+calculation. \cite{fix} In a simultation with periodic images, $fix$
-+
+has a maximum value of $fix$.  Fig.~\ref{fix} illustrates how using an
-+
+$fix$ larger than this value, or in the extreme limit of no $fix$ at
-+
+all, the corners of the simulation box are unequally weighted due to
-+
+the lack of particle images in the $x$, $y$, or $z$ directions past a
-+
+disance of $fix$.
-+
-+
+With the use of an $fix$, however, comes a discontinuity in the potential energy curve (Fig.~\ref{fix}).
-+
-+
+\section{\label{introSec:chapterLayout}Chapter Layout}
+\subsection{\label{introSec:RSA}Random Sequential Adsorption}

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Comparing trunk/mattDisertation/Introduction.tex (file contents): Revision 953 by mmeineke, Sun Jan 18 03:07:59 2004 UTC vs. Revision 956 by mmeineke, Sun Jan 18 19:10:32 2004 UTC

Diff Legend

Comparing trunk/mattDisertation/Introduction.tex (file contents):
Revision 953 by mmeineke, Sun Jan 18 03:07:59 2004 UTC vs.
Revision 956 by mmeineke, Sun Jan 18 19:10:32 2004 UTC