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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2720 by tim, Wed Apr 19 19:46:53 2006 UTC vs.
Revision 2726 by tim, Fri Apr 21 15:37:53 2006 UTC

+error of splitting method in terms of commutator of the
+operators(\ref{introEquation:exponentialOperator}) associated with
+the sub-flow. For operators $hX$ and $hY$ which are associate to
-<
+$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have
->
+$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have
+\begin{equation}
+\exp (hX + hY) = \exp (hZ)
+\end{equation}
+\]
+Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we
+can obtain
-<
+\begin{eqnarray*}
->
+\begin{equation}
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2
+[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
+& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} +
+\ldots )
-<
+\end{eqnarray*}
->
+\end{equation}
+Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local
+error of Spring splitting is proportional to $h^3$. The same
+procedure can be applied to general splitting,  of the form
+simulations. For instance, instantaneous temperature of an
+Hamiltonian system of $N$ particle can be measured by
+\[
-<
+T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
->
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
+\]
+where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
+respectively, $f$ is the number of degrees of freedom, and $k_B$ is
+These three individual steps will be covered in the following
+sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
+initialization of a simulation. Sec.~\ref{introSec:production} will
-<
+discusses issues in production run, including the force evaluation
-<
+and the numerical integration schemes of the equations of motion .
-<
+Sec.~\ref{introSection:Analysis} provides the theoretical tools for
-<
+trajectory analysis.
->
+discusses issues in production run. Sec.~\ref{introSection:Analysis}
->
+provides the theoretical tools for trajectory analysis.
+\subsection{\label{introSec:initialSystemSettings}Initialization}
+\subsection{\label{introSection:production}Production}
-<
+\subsubsection{\label{introSec:forceCalculation}The Force Calculation}
->
+Production run is the most important steps of the simulation, in
->
+which the equilibrated structure is used as a starting point and the
->
+motions of the molecules are collected for later analysis. In order
->
+to capture the macroscopic properties of the system, the molecular
->
+dynamics simulation must be performed in correct and efficient way.
-<
+\subsubsection{\label{introSection:integrationSchemes} Integration
-<
+Schemes}
->
+The most expensive part of a molecular dynamics simulation is the
->
+calculation of non-bonded forces, such as van der Waals force and
->
+Coulombic forces \textit{etc}. For a system of $N$ particles, the
->
+complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
->
+which making large simulations prohibitive in the absence of any
->
+computation saving techniques.
-+
+A natural approach to avoid system size issue is to represent the
-+
+bulk behavior by a finite number of the particles. However, this
-+
+approach will suffer from the surface effect. To offset this,
-+
+\textit{Periodic boundary condition} is developed to simulate bulk
-+
+properties with a relatively small number of particles. In this
-+
+method, the simulation box is replicated throughout space to form an
-+
+infinite lattice. During the simulation, when a particle moves in
-+
+the primary cell, its image in other cells move in exactly the same
-+
+direction with exactly the same orientation. Thus, as a particle
-+
+leaves the primary cell, one of its images will enter through the
-+
+opposite face.
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pbcFig.eps}
-+
+%\caption[An illustration of periodic boundary conditions]{A 2-D
-+
+%illustration of periodic boundary conditions. As one particle leaves
-+
+%the right of the simulation box, an image of it enters the left.}
-+
+%\label{introFig:pbc}
-+
+%\end{figure}
-+
-+
+%cutoff and minimum image convention
-+
+Another important technique to improve the efficiency of force
-+
+evaluation is to apply cutoff where particles farther than a
-+
+predetermined distance, are not included in the calculation
-+
+\cite{Frenkel1996}. The use of a cutoff radius will cause a
-+
+discontinuity in the potential energy curve
-+
+(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the
-+
+potential to ensure the potential curve go smoothly to zero at the
-+
+cutoff radius. Cutoff strategy works pretty well for Lennard-Jones
-+
+interaction because of its short range nature. However, simply
-+
+truncating the electrostatic interaction with the use of cutoff has
-+
+been shown to lead to severe artifacts in simulations. Ewald
-+
+summation, in which the slowly conditionally convergent Coulomb
-+
+potential is transformed into direct and reciprocal sums with rapid
-+
+and absolute convergence, has proved to minimize the periodicity
-+
+artifacts in liquid simulations. Taking the advantages of the fast
-+
+Fourier transform (FFT) for calculating discrete Fourier transforms,
-+
+the particle mesh-based methods are accelerated from $O(N^{3/2})$ to
-+
+$O(N logN)$. An alternative approach is \emph{fast multipole
-+
+method}, which treats Coulombic interaction exactly at short range,
-+
+and approximate the potential at long range through multipolar
-+
+expansion. In spite of their wide acceptances at the molecular
-+
+simulation community, these two methods are hard to be implemented
-+
+correctly and efficiently. Instead, we use a damped and
-+
+charge-neutralized Coulomb potential method developed by Wolf and
-+
+his coworkers. The shifted Coulomb potential for particle $i$ and
-+
+particle $j$ at distance $r_{rj}$ is given by:
-+
+\begin{equation}
-+
+V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
-+
+r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
-+
+R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
-+
+r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb}
-+
+\end{equation}
-+
+where $\alpha$ is the convergence parameter. Due to the lack of
-+
+inherent periodicity and rapid convergence,this method is extremely
-+
+efficient and easy to implement.
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pbcFig.eps}
-+
+%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.}
-+
+%\label{introFigure:shiftedCoulomb}
-+
+%\end{figure}
-+
-+
+%multiple time step
-+
+\subsection{\label{introSection:Analysis} Analysis}
-+
+Recently, advanced visualization technique are widely applied to
-+
+monitor the motions of molecules. Although the dynamics of the
-+
+system can be described qualitatively from animation, quantitative
-+
+trajectory analysis are more appreciable. According to the
-+
+principles of Statistical Mechanics,
-+
+Sec.~\ref{introSection:statisticalMechanics}, one can compute
-+
+thermodynamics properties, analyze fluctuations of structural
-+
+parameters, and investigate time-dependent processes of the molecule
-+
+from the trajectories.
-+
-+
+\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties}
-+
-+
+Thermodynamics properties, which can be expressed in terms of some
-+
+function of the coordinates and momenta of all particles in the
-+
+system, can be directly computed from molecular dynamics. The usual
-+
+way to measure the pressure is based on virial theorem of Clausius
-+
+which states that the virial is equal to $-3Nk_BT$. For a system
-+
+with forces between particles, the total virial, $W$, contains the
-+
+contribution from external pressure and interaction between the
-+
+particles:
-+
+\[
-+
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
-+
+f_{ij} } } \right\rangle
-+
+\]
-+
+where $f_{ij}$ is the force between particle $i$ and $j$ at a
-+
+distance $r_{ij}$. Thus, the expression for the pressure is given
-+
+by:
-+
+\begin{equation}
-+
+P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
-+
+< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
-+
+\end{equation}
-+
-+
+\subsubsection{\label{introSection:structuralProperties}Structural Properties}
-+
-+
+Structural Properties of a simple fluid can be described by a set of
-+
+distribution functions. Among these functions,\emph{pair
-+
+distribution function}, also known as \emph{radial distribution
-+
+function}, is of most fundamental importance to liquid-state theory.
-+
+Pair distribution function can be gathered by Fourier transforming
-+
+raw data from a series of neutron diffraction experiments and
-+
+integrating over the surface factor \cite{Powles73}. The experiment
-+
+result can serve as a criterion to justify the correctness of the
-+
+theory. Moreover, various equilibrium thermodynamic and structural
-+
+properties can also be expressed in terms of radial distribution
-+
+function \cite{allen87:csl}.
-+
-+
+A pair distribution functions $g(r)$ gives the probability that a
-+
+particle $i$ will be located at a distance $r$ from a another
-+
+particle $j$ in the system
-+
+\[
-+
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
-+
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle.
-+
+\]
-+
+Note that the delta function can be replaced by a histogram in
-+
+computer simulation. Figure
-+
+\ref{introFigure:pairDistributionFunction} shows a typical pair
-+
+distribution function for the liquid argon system. The occurrence of
-+
+several peaks in the plot of $g(r)$ suggests that it is more likely
-+
+to find particles at certain radial values than at others. This is a
-+
+result of the attractive interaction at such distances. Because of
-+
+the strong repulsive forces at short distance, the probability of
-+
+locating particles at distances less than about 2.5{\AA} from each
-+
+other is essentially zero.
-+
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pdf.eps}
-+
+%\caption[Pair distribution function for the liquid argon
-+
+%]{Pair distribution function for the liquid argon}
-+
+%\label{introFigure:pairDistributionFunction}
-+
+%\end{figure}
-+
-+
+\subsubsection{\label{introSection:timeDependentProperties}Time-dependent
-+
+Properties}
-+
-+
+Time-dependent properties are usually calculated using \emph{time
-+
+correlation function}, which correlates random variables $A$ and $B$
-+
+at two different time
-+
+\begin{equation}
-+
+C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
-+
+\label{introEquation:timeCorrelationFunction}
-+
+\end{equation}
-+
+If $A$ and $B$ refer to same variable, this kind of correlation
-+
+function is called \emph{auto correlation function}. One example of
-+
+auto correlation function is velocity auto-correlation function
-+
+which is directly related to transport properties of molecular
-+
+liquids:
-+
+\[
-+
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
-+
+\right\rangle } dt
-+
+\]
-+
+where $D$ is diffusion constant. Unlike velocity autocorrelation
-+
+function which is averaging over time origins and over all the
-+
+atoms, dipole autocorrelation are calculated for the entire system.
-+
+The dipole autocorrelation function is given by:
-+
+\[
-+
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
-+
+\right\rangle
-+
+\]
-+
+Here $u_{tot}$ is the net dipole of the entire system and is given
-+
+by
-+
+\[
-+
+u_{tot} (t) = \sum\limits_i {u_i (t)}
-+
+\]
-+
+In principle, many time correlation functions can be related with
-+
+Fourier transforms of the infrared, Raman, and inelastic neutron
-+
+scattering spectra of molecular liquids. In practice, one can
-+
+extract the IR spectrum from the intensity of dipole fluctuation at
-+
+each frequency using the following relationship:
-+
+\[
-+
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
-+
+i2\pi vt} dt}
-+
+\]
-+
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+Rigid bodies are frequently involved in the modeling of different
+where $I_{ii}$ is the diagonal element of the inertia tensor. This
+constrained Hamiltonian equation subjects to a holonomic constraint,
+\begin{equation}
-<
+Q^T Q = 1$, \label{introEquation:orthogonalConstraint}
->
+Q^T Q = 1, \label{introEquation:orthogonalConstraint}
+\end{equation}
+which is used to ensure rotation matrix's orthogonality.
+Differentiating \ref{introEquation:orthogonalConstraint} and using
+\]
+where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
+joining center of resistance $R$ and origin $O$.
-–
-–
+%\section{\label{introSection:correlationFunctions}Correlation Functions}

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-–
+Removed lines
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-<
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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2720 by tim, Wed Apr 19 19:46:53 2006 UTC vs. Revision 2726 by tim, Fri Apr 21 15:37:53 2006 UTC

Diff Legend

Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2720 by tim, Wed Apr 19 19:46:53 2006 UTC vs.
Revision 2726 by tim, Fri Apr 21 15:37:53 2006 UTC