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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 main classes of |
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molecular simulation: Molecular Dynamics and Monte Carlo. Molecular Dynamic simulations |
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integrate the equations of motion for a given system of particles, allowing the researher |
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to gain insight into the time dependent evolution of a system. Diffusion phenomena are |
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readily studied with this simulation technique, making Molecular Dynamics the main simulation |
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technique used in this research. Other aspects of the research fall under the Monte Carlo |
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class of simulations. In Monte Carlo, the configuration space available to the collection |
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of particles is sampled stochastichally, or randomly. Each configuration is chosen with |
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a given probability 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 only on the state of |
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the system. Structural information about a system is most readily obtained through |
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these types of methods. |
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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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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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Although the two techniques employed seem dissimilar, they are both linked by the overarching |
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principles of Statistical Thermodynamics. Statistical Thermodynamics governs the behavior of |
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both classes of simulations and dictates what each method can and cannot do. When investigating |
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a system, one most first analyze what thermodynamic properties of the system are being probed, |
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then chose which method best suits that objective. |
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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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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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\subsection{\label{introSec:monteCarlo}Monte Carlo Simulations} |
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Stochastic sampling |
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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 = (b-a)<f(x)> |
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\label{eq:MCex2} |
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\end{equation} |
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Where $<f(x)>$ is the unweighted average over the interval |
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$[a,b]$. The calculation of the integral could then be solved by |
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randomly choosing points along the interval $[a,b]$ and calculating |
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the value of $f(x)$ at each point. The accumulated average would then |
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approach $I$ in the limit where the number of trials is infintely |
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large. |
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detatiled balance |
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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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<A> = \frac{A}{exp^{-\beta}} |
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\label{eq:mcEnsAvg} |
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\end{equation} |
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Where $r^N$ stands for the coordinates of all $N$ particles and $A$ is |
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some observable that is only dependent on position. $<A>$ is the |
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ensemble average of $A$ as presented in |
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Sec.~\ref{introSec:statThermo}. Because $A$ is independent of |
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momentum, the momenta contribution of the integral can be factored |
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out, leaving the configurational integral. Application of the brute |
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force method to this system would yield highly inefficient |
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results. Due to the Boltzman 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 a importance sampling comes into |
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play.\cite{allen87:csl} |
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metropilis monte carlo |
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Importance Sampling is a method where one selects a distribution from |
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which the random configurations are chosen in order to more |
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efficiently calculate the integral.\cite{Frenkel1996} Consider again |
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Eq.~\ref{eq:MCex1} rewritten to be: |
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+ |
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+ |
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\subsection{\label{introSec:md}Molecular Dynamics Simulations} |
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time averages |
