From: hackbard Date: Wed, 14 Jun 2006 15:47:51 +0000 (+0000) Subject: - X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=e86993833978bf71f38e6ca163e3e513e6c27b59;p=lectures%2Flatex.git - --- diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index bc9e8a9..c11a832 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -2,8 +2,30 @@ \section{Molecular dynamics simulations} +\subsection{Theory of melecular dynamics simulations} -\subsection{Potentials} +Basically molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, that evolve 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. +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. +This microscopic information has to 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 + +\subsection{Interaction potentials} \subsubsection{The Lennard-Jones potential} @@ -30,7 +52,7 @@ Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ ( \label{eq:lj-d} \end{equation} one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$. -Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be half the negative well depth. +Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth. The $i$th component of the force $F^j$ on particle $j$ is obtained by \begin{equation} F_i^j = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.} diff --git a/posic/thesis/content.tex b/posic/thesis/content.tex index c886626..90d5122 100644 --- a/posic/thesis/content.tex +++ b/posic/thesis/content.tex @@ -1,2 +1,3 @@ +\selectlanguage{english} \addcontentsline{toc}{chapter}{Contents} \tableofcontents diff --git a/posic/thesis/literature.tex b/posic/thesis/literature.tex index e2f4f38..d860d2f 100644 --- a/posic/thesis/literature.tex +++ b/posic/thesis/literature.tex @@ -1,5 +1,11 @@ \addcontentsline{toc}{chapter}{References} \begin{thebibliography}{99} + \bibitem{alder1} + B. J. Alder, T.E. Wainwright. + J. Chem. Phys. 27 (1957) 1208. + \bibitem{alder2} + B. J. Alder, T.E. Wainwright. + J. Chem. Phys. 31 (1959) 459. \bibitem{example} \selectlanguage{german} F. Zirkelbach, M. H"aberlen, J. K. N. Lindner, B. Stritzker. diff --git a/posic/thesis/title.tex b/posic/thesis/title.tex index 341e50c..e4625a7 100644 --- a/posic/thesis/title.tex +++ b/posic/thesis/title.tex @@ -53,7 +53,6 @@ \raisebox{600pt}{ } -\selectlanguage{german} \begin{tabular}{ll} Erstkorrektor: & Prof. Dr. Bernd Stritzker \\ Zweitkorrektor: & Prof. Dr. Kai Nordlund \\