X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=2eda8a226a87bd7efb49ddb7c7303d7ceaa955ce;hb=cbacb767e889a7a8d7196b37286b9eeec2eb802c;hp=c11a8321e266878857f9a29709c08b29a8917d99;hpb=e86993833978bf71f38e6ca163e3e513e6c27b59;p=lectures%2Flatex.git

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index c11a832..2eda8a2 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -2,9 +2,9 @@
 
 \section{Molecular dynamics simulations}
 
-\subsection{Theory of melecular dynamics simulations}
+\subsection{Introduction to molecular dynamics simulations}
 
-Basically molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, that evolve in time.
+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.
 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.
@@ -23,10 +23,29 @@ The forces ${\bf F}_i$ are obtained from the potential energy $U(\{{\bf r}\})$:
 {\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.}
 \end{equation}
 Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{{\bf r}}_i(t_0)$ the equations can be integrated by a certain integration algorithm.
-The solution of these equations provides the complete information of a system 
+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.
+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}
+
 
 \subsection{Interaction potentials}
 
+The potential energy of $N$ interacting atoms can be written in the form
+\begin{equation}
+U(\{{\bf r}\}) = \sum_i U_1({\bf r}_i) + \sum_i \sum_{j>i} U_2({\bf r}_i,{\bf r}_j) + \sum_i \sum_{j>i} \sum_{k>j>i} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) \ldots
+\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.
+$U_2$ is a two body pair potential which only depends on the distance $r_{ij}$ between the two atoms $i$ and $j$.
+
+$U_3$ 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.
+
 \subsubsection{The Lennard-Jones potential}
 
 The L-J potential is a realistic two body pair potential and is of the form