foo

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 2eda8a2..c25e858 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -1,13 +1,25 @@
 \chapter{Basics}
 
+\begin{quotation}
+\dq We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.\dq{}
+\cite{laplace}
+\begin{flushright}
+{\em Marquis Pierre Simon de Laplace, 1814.}
+\end{flushright}
+\end{quotation}
+
 \section{Molecular dynamics simulations}
 
+Pierre Simon de Laplace phrased this vision in terms of a controlling, omniscient instance - the {\em Laplace demon} - which would be able to look into the future as well as into the past due to the deterministic nature of processes, governed by the solution of differential equations.
+Although Laplace's vision is nowadays corrected by chaos theory and quantum mechanics, it expresses two main features of classical mechanics, the determinism of processes and time reversibility of the fundamental equations.
+This understanding was one of the first ideas for doing molecular dynamics simulations, considering an isolated system of particles, the behaviour of which is fully determined by the solution of the classical equations of motion.
+
 \subsection{Introduction to molecular dynamics simulations}
 
-Basically, molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, evolving in time.
-The MD method was introduced by Alder and Wainwright in 1957 \cite{alder1,alder2} to study the interactions of hard spheres.
+Basically, molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, with their positions, volocities and forces among each other evolving in time.
+The MD method was first introduced by Alder and Wainwright in 1957 \cite{alder1,alder2} to study the interactions of hard spheres.
 The basis of the approach are Newton's equations of motion to describe classicaly the many-body system.
-MD simulation is the numerical way of solving the $N$-body problem ($N > 3$) which cannot be solved analytically.
+MD simulation is the numerical way of solving the $N$-body problem which cannot be solved analytically ($N>3$).
 Quantum mechanical effects are taken into account by an analytical interaction potential between the nuclei.
 
 By MD simulation techniques a complete description of the system in the sense of classical mechanics on the microscopic level is obtained.
@@ -26,14 +38,23 @@ Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{{\bf r}}_i(t_0)$ the equ
 The solution of these equations provides the complete information of a system evolving in time.
 
 The following chapters cover the tools of the trade necessary for the MD simulation technique.
-First a detailed overview of the available integration algorithms is given, including their advantages and disadvantages.
-After that the interaction potentials and their accuracy for describing certain systems of elements are discussed.
+Three ingredients are required for an MD simulation:
+\begin{enumerate}
+\item A model for the interaction between system constituents is needed.
+      Interaction potentials and their accuracy for describing certain systems of elements will be outlined in chapter \ref{subsection:interact_pot}.
+\item An integrator is needed, which propagtes the particle positions and velocities from time $t$ to $t+\delta t$, realised by a finite difference scheme which moves trajectories discretely in time.
+      In chapter \ref{subsection:integrate_algo} a detailed overview of the available integration algorithms is given, including their advantages and disadvantages.
+\item A statistical ensemble has to be chosen, which allows certain thermodynamic quantities to be controlled or to stay constant.
+      This is discussed in chapter \ref{subsection:statistical_ensembles}
+\end{enumerate}
+
 In addition special techniques will be outlined which reduce the complexity of the MD algorithm, though the force/energy evaluation almost inevitably dictates the overall speed.
 
 \subsection{Integration algorithms}
-
+\label{subsection:integrate_algo}
 
 \subsection{Interaction potentials}
+\label{subsection:interact_pot}
 
 The potential energy of $N$ interacting atoms can be written in the form
 \begin{equation}
@@ -41,10 +62,17 @@ U(\{{\bf r}\}) = \sum_i U_1({\bf r}_i) + \sum_i \sum_{j>i} U_2({\bf r}_i,{\bf r}
 \end{equation}
 where $U$ is the total potential energy.
 $U_1$ is a single particle potential describing external forces.
-This could for instance be the gravitational force or an electric field.
+Examples of single particle potentials are the gravitational force or an electric field.
 $U_2$ is a two body pair potential which only depends on the distance $r_{ij}$ between the two atoms $i$ and $j$.
+If not only pair potentials are considered, three body potentials $U_3$ or multi body potentials $U_n$ can be included.
+Mainly these higher order terms are avoided since they are not easy to model and it is rather time consuming to evaluate potentials and forces originating from these many body terms.
 
-$U_3$ is a three body potential which may have an additional angular dependence describing covalent bonds, plus higher order terms which are expected to be small and thus neglected.
+Ordinary pair potentials have a close-packed structure like face-centered cubic (FCC) or hexagonal close-packed (HCP) as a ground state.
+A pair potential is thus unable to describe properly elements with other structures than FCC or HCP.
+Silicon and carbon for instance, have a diamand/zincblende structure with four covalent bonded neighbours, which is far from a close-packed structure.
+A three body potential has to be included for these types of elements.
+
+In the following, relevant potentials for this work are discussed. 
 
 \subsubsection{The Lennard-Jones potential}
 
@@ -53,7 +81,7 @@ The L-J potential is a realistic two body pair potential and is of the form
 U^{LJ}(r) = 4 \epsilon \Big[ \Big( \frac{\sigma}{r} \Big)^{12} - \Big( \frac{\sigma}{r} \Big)^6 \Big] \, \textrm{,}
 \label{eq:lj-p}
 \end{equation}
-where $r$ denotes the disatnce between the two atoms.
+where $r$ denotes the distance between the two atoms.
 
 The attractive tail for large separations $(\sim r^{-6})$ is essentially due to correlations between electron clouds surrounding the atoms. The attractive part is also known as {\em van der Waals} or {\em London} interaction.
 It can be derived classically by considering how two charged spheres induce dipol-dipol interactions into each other, or by considering the interaction between two oscillators in a quantum mechanical way.
@@ -78,3 +106,10 @@ F_i^j = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
 \label{eq:lj-f}
 \end{equation}
 
+\subsubsection{The Stillinger Weber potential}
+\subsubsection{The Stillinger Weber potential}
+\subsubsection{The Stillinger Weber potential}
+
+\subsection{Statistical ensembles}
+\label{subsection:statistical_ensembles}
+

diff --git a/posic/thesis/literature.tex b/posic/thesis/literature.tex
index d860d2f..e478cad 100644 (file)
--- a/posic/thesis/literature.tex
+++ b/posic/thesis/literature.tex
@@ -1,5 +1,10 @@
 \addcontentsline{toc}{chapter}{References}
 \begin{thebibliography}{99}
+  \bibitem{laplace}
+    P. S. de Laplace.
+    Th\'eorie analytique des probabilit\'es.
+    Oeuvres Compl\`etes de Laplace, volume VII.
+    Paris, 1820, Gauthier-Villars.
   \bibitem{alder1}
     B. J. Alder, T.E. Wainwright.
     J. Chem. Phys. 27 (1957) 1208.