+\label{section:md}
+
+\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{}
+\begin{flushright}
+{\em Marquis Pierre Simon de Laplace, 1814.} \cite{laplace}
+\end{flushright}
+\end{quotation}
+
+\noindent
+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}
+
+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{alder57,alder59} 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 is the numerical way of solving the $N$-body problem which cannot be solved analytically for $N>3$.
+A potential is necessary describing the interaction of the particles.
+By MD a complete description of the system in the sense of classical mechanics on the microscopic level is obtained.
+The microscopic information can then be translated to macroscopic observables by means of statistical mechanics.
+
+The basic idea is to integrate Newton's equations numerically.
+A system of $N$ particles of masses $m_i$ ($i=1,\ldots,N$) at positions ${\bf r}_i$ and velocities $\dot{{\bf r}}_i$ is given by
+\begin{equation}
+m_i \frac{d^2}{dt^2} {\bf r}_i = {\bf F}_i \, \textrm{.}
+\end{equation}
+The forces ${\bf F}_i$ are obtained from the potential energy $U(\{{\bf r}\})$:
+\begin{equation}
+{\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 evolving in time.
+
+The following sections cover the tools of the trade necessary for the MD simulation technique.
+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 section \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.
+ This is explained in section \ref{subsection:integrate_algo}.
+\item A statistical ensemble has to be chosen, which allows certain thermodynamic quantities to be controlled or to stay constant.
+ This is discussed in section \ref{subsection:statistical_ensembles}.
+\end{enumerate}
+Furthermore special techniques will be outlined which reduce the complexity of the MD algorithm, though the evaluation of the energy and force 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}
+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.
+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.
+
+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 and 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.