831 |
|
error of splitting method in terms of commutator of the |
832 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
833 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
834 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
834 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
835 |
|
\begin{equation} |
836 |
|
\exp (hX + hY) = \exp (hZ) |
837 |
|
\end{equation} |
847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
848 |
|
can obtain |
849 |
|
\begin{eqnarray*} |
850 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
851 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
852 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
853 |
< |
\ldots ) |
850 |
> |
\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 \\ |
851 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
852 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
853 |
|
\end{eqnarray*} |
854 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
855 |
|
error of Spring splitting is proportional to $h^3$. The same |
858 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
859 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
860 |
|
\end{equation} |
861 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
861 |
> |
Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher |
862 |
|
order method. Yoshida proposed an elegant way to compose higher |
863 |
|
order methods based on symmetric splitting. Given a symmetric second |
864 |
|
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
891 |
|
simulations. For instance, instantaneous temperature of an |
892 |
|
Hamiltonian system of $N$ particle can be measured by |
893 |
|
\[ |
894 |
< |
T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
894 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
895 |
|
\] |
896 |
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
897 |
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
912 |
|
These three individual steps will be covered in the following |
913 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
914 |
|
initialization of a simulation. Sec.~\ref{introSec:production} will |
915 |
< |
discusses issues in production run, including the force evaluation |
916 |
< |
and the numerical integration schemes of the equations of motion . |
918 |
< |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
919 |
< |
trajectory analysis. |
915 |
> |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
916 |
> |
provides the theoretical tools for trajectory analysis. |
917 |
|
|
918 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
919 |
|
|
983 |
|
|
984 |
|
\subsection{\label{introSection:production}Production} |
985 |
|
|
986 |
< |
\subsubsection{\label{introSec:forceCalculation}The Force Calculation} |
986 |
> |
Production run is the most important steps of the simulation, in |
987 |
> |
which the equilibrated structure is used as a starting point and the |
988 |
> |
motions of the molecules are collected for later analysis. In order |
989 |
> |
to capture the macroscopic properties of the system, the molecular |
990 |
> |
dynamics simulation must be performed in correct and efficient way. |
991 |
|
|
992 |
< |
\subsubsection{\label{introSection:integrationSchemes} Integration |
993 |
< |
Schemes} |
992 |
> |
The most expensive part of a molecular dynamics simulation is the |
993 |
> |
calculation of non-bonded forces, such as van der Waals force and |
994 |
> |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
995 |
> |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
996 |
> |
which making large simulations prohibitive in the absence of any |
997 |
> |
computation saving techniques. |
998 |
> |
|
999 |
> |
A natural approach to avoid system size issue is to represent the |
1000 |
> |
bulk behavior by a finite number of the particles. However, this |
1001 |
> |
approach will suffer from the surface effect. To offset this, |
1002 |
> |
\textit{Periodic boundary condition} is developed to simulate bulk |
1003 |
> |
properties with a relatively small number of particles. In this |
1004 |
> |
method, the simulation box is replicated throughout space to form an |
1005 |
> |
infinite lattice. During the simulation, when a particle moves in |
1006 |
> |
the primary cell, its image in other cells move in exactly the same |
1007 |
> |
direction with exactly the same orientation. Thus, as a particle |
1008 |
> |
leaves the primary cell, one of its images will enter through the |
1009 |
> |
opposite face. |
1010 |
> |
%\begin{figure} |
1011 |
> |
%\centering |
1012 |
> |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1013 |
> |
%\caption[An illustration of periodic boundary conditions]{A 2-D |
1014 |
> |
%illustration of periodic boundary conditions. As one particle leaves |
1015 |
> |
%the right of the simulation box, an image of it enters the left.} |
1016 |
> |
%\label{introFig:pbc} |
1017 |
> |
%\end{figure} |
1018 |
|
|
1019 |
+ |
%cutoff and minimum image convention |
1020 |
+ |
Another important technique to improve the efficiency of force |
1021 |
+ |
evaluation is to apply cutoff where particles farther than a |
1022 |
+ |
predetermined distance, are not included in the calculation |
1023 |
+ |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1024 |
+ |
discontinuity in the potential energy curve. Fortunately, one can |
1025 |
+ |
shift the potential to ensure the potential curve go smoothly to |
1026 |
+ |
zero at the cutoff radius. Cutoff strategy works pretty well for |
1027 |
+ |
Lennard-Jones interaction because of its short range nature. |
1028 |
+ |
However, simply truncating the electrostatic interaction with the |
1029 |
+ |
use of cutoff has been shown to lead to severe artifacts in |
1030 |
+ |
simulations. Ewald summation, in which the slowly conditionally |
1031 |
+ |
convergent Coulomb potential is transformed into direct and |
1032 |
+ |
reciprocal sums with rapid and absolute convergence, has proved to |
1033 |
+ |
minimize the periodicity artifacts in liquid simulations. Taking the |
1034 |
+ |
advantages of the fast Fourier transform (FFT) for calculating |
1035 |
+ |
discrete Fourier transforms, the particle mesh-based methods are |
1036 |
+ |
accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative |
1037 |
+ |
approach is \emph{fast multipole method}, which treats Coulombic |
1038 |
+ |
interaction exactly at short range, and approximate the potential at |
1039 |
+ |
long range through multipolar expansion. In spite of their wide |
1040 |
+ |
acceptances at the molecular simulation community, these two methods |
1041 |
+ |
are hard to be implemented correctly and efficiently. Instead, we |
1042 |
+ |
use a damped and charge-neutralized Coulomb potential method |
1043 |
+ |
developed by Wolf and his coworkers. The shifted Coulomb potential |
1044 |
+ |
for particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
1045 |
+ |
\begin{equation} |
1046 |
+ |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1047 |
+ |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1048 |
+ |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
1049 |
+ |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
1050 |
+ |
\end{equation} |
1051 |
+ |
where $\alpha$ is the convergence parameter. Due to the lack of |
1052 |
+ |
inherent periodicity and rapid convergence,this method is extremely |
1053 |
+ |
efficient and easy to implement. |
1054 |
+ |
%\begin{figure} |
1055 |
+ |
%\centering |
1056 |
+ |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1057 |
+ |
%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} |
1058 |
+ |
%\label{introFigure:shiftedCoulomb} |
1059 |
+ |
%\end{figure} |
1060 |
+ |
|
1061 |
+ |
%multiple time step |
1062 |
+ |
|
1063 |
|
\subsection{\label{introSection:Analysis} Analysis} |
1064 |
|
|
1065 |
|
Recently, advanced visualization technique are widely applied to |
1074 |
|
|
1075 |
|
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
1076 |
|
|
1077 |
+ |
Thermodynamics properties, which can be expressed in terms of some |
1078 |
+ |
function of the coordinates and momenta of all particles in the |
1079 |
+ |
system, can be directly computed from molecular dynamics. The usual |
1080 |
+ |
way to measure the pressure is based on virial theorem of Clausius |
1081 |
+ |
which states that the virial is equal to $-3Nk_BT$. For a system |
1082 |
+ |
with forces between particles, the total virial, $W$, contains the |
1083 |
+ |
contribution from external pressure and interaction between the |
1084 |
+ |
particles: |
1085 |
+ |
\[ |
1086 |
+ |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
1087 |
+ |
f_{ij} } } \right\rangle |
1088 |
+ |
\] |
1089 |
+ |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
1090 |
+ |
distance $r_{ij}$. Thus, the expression for the pressure is given |
1091 |
+ |
by: |
1092 |
+ |
\begin{equation} |
1093 |
+ |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
1094 |
+ |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1095 |
+ |
\end{equation} |
1096 |
+ |
|
1097 |
|
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
1098 |
|
|
1099 |
|
Structural Properties of a simple fluid can be described by a set of |
1100 |
|
distribution functions. Among these functions,\emph{pair |
1101 |
|
distribution function}, also known as \emph{radial distribution |
1102 |
< |
function}, are of most fundamental importance to liquid-state |
1103 |
< |
theory. Pair distribution function can be gathered by Fourier |
1104 |
< |
transforming raw data from a series of neutron diffraction |
1105 |
< |
experiments and integrating over the surface factor \cite{Powles73}. |
1106 |
< |
The experiment result can serve as a criterion to justify the |
1107 |
< |
correctness of the theory. Moreover, various equilibrium |
1108 |
< |
thermodynamic and structural properties can also be expressed in |
1109 |
< |
terms of radial distribution function \cite{allen87:csl}. |
1102 |
> |
function}, is of most fundamental importance to liquid-state theory. |
1103 |
> |
Pair distribution function can be gathered by Fourier transforming |
1104 |
> |
raw data from a series of neutron diffraction experiments and |
1105 |
> |
integrating over the surface factor \cite{Powles73}. The experiment |
1106 |
> |
result can serve as a criterion to justify the correctness of the |
1107 |
> |
theory. Moreover, various equilibrium thermodynamic and structural |
1108 |
> |
properties can also be expressed in terms of radial distribution |
1109 |
> |
function \cite{allen87:csl}. |
1110 |
|
|
1111 |
|
A pair distribution functions $g(r)$ gives the probability that a |
1112 |
|
particle $i$ will be located at a distance $r$ from a another |
1148 |
|
function is called \emph{auto correlation function}. One example of |
1149 |
|
auto correlation function is velocity auto-correlation function |
1150 |
|
which is directly related to transport properties of molecular |
1151 |
< |
liquids. Another example is the calculation of the IR spectrum |
1152 |
< |
through a Fourier transform of the dipole autocorrelation function. |
1151 |
> |
liquids: |
1152 |
> |
\[ |
1153 |
> |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1154 |
> |
\right\rangle } dt |
1155 |
> |
\] |
1156 |
> |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1157 |
> |
function which is averaging over time origins and over all the |
1158 |
> |
atoms, dipole autocorrelation are calculated for the entire system. |
1159 |
> |
The dipole autocorrelation function is given by: |
1160 |
> |
\[ |
1161 |
> |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1162 |
> |
\right\rangle |
1163 |
> |
\] |
1164 |
> |
Here $u_{tot}$ is the net dipole of the entire system and is given |
1165 |
> |
by |
1166 |
> |
\[ |
1167 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1168 |
> |
\] |
1169 |
> |
In principle, many time correlation functions can be related with |
1170 |
> |
Fourier transforms of the infrared, Raman, and inelastic neutron |
1171 |
> |
scattering spectra of molecular liquids. In practice, one can |
1172 |
> |
extract the IR spectrum from the intensity of dipole fluctuation at |
1173 |
> |
each frequency using the following relationship: |
1174 |
> |
\[ |
1175 |
> |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1176 |
> |
i2\pi vt} dt} |
1177 |
> |
\] |
1178 |
|
|
1179 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1180 |
|
|
1236 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1237 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
1238 |
|
\begin{equation} |
1239 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
1239 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1240 |
|
\end{equation} |
1241 |
|
which is used to ensure rotation matrix's orthogonality. |
1242 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1261 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1262 |
|
We can use constraint force provided by lagrange multiplier on the |
1263 |
|
normal manifold to keep the motion on constraint space. Or we can |
1264 |
< |
simply evolve the system in constraint manifold. The two method are |
1265 |
< |
proved to be equivalent. The holonomic constraint and equations of |
1266 |
< |
motions define a constraint manifold for rigid body |
1264 |
> |
simply evolve the system in constraint manifold. These two methods |
1265 |
> |
are proved to be equivalent. The holonomic constraint and equations |
1266 |
> |
of motions define a constraint manifold for rigid body |
1267 |
|
\[ |
1268 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1269 |
|
\right\}. |
1476 |
|
\] |
1477 |
|
The equations of motion corresponding to potential energy and |
1478 |
|
kinetic energy are listed in the below table, |
1479 |
+ |
\begin{table} |
1480 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
1481 |
|
\begin{center} |
1482 |
|
\begin{tabular}{|l|l|} |
1483 |
|
\hline |
1490 |
|
\hline |
1491 |
|
\end{tabular} |
1492 |
|
\end{center} |
1493 |
< |
A second-order symplectic method is now obtained by the composition |
1494 |
< |
of the flow maps, |
1493 |
> |
\end{table} |
1494 |
> |
A second-order symplectic method is now obtained by the |
1495 |
> |
composition of the flow maps, |
1496 |
|
\[ |
1497 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1498 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1788 |
|
when the system become more and more complicate. Instead, various |
1789 |
|
approaches based on hydrodynamics have been developed to calculate |
1790 |
|
the friction coefficients. The friction effect is isotropic in |
1791 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
1792 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
1791 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
1792 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
1793 |
|
\[ |
1794 |
|
\Xi = \left( {\begin{array}{*{20}c} |
1795 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |