| 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} |
| 846 |
|
\] |
| 847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
|
can obtain |
| 849 |
< |
\begin{eqnarray*} |
| 849 |
> |
\begin{equation} |
| 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 ) |
| 854 |
< |
\end{eqnarray*} |
| 854 |
> |
\end{equation} |
| 855 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 856 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 857 |
|
procedure can be applied to general splitting, of the form |
| 883 |
|
|
| 884 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 885 |
|
|
| 886 |
< |
As a special discipline of molecular modeling, Molecular dynamics |
| 887 |
< |
has proven to be a powerful tool for studying the functions of |
| 888 |
< |
biological systems, providing structural, thermodynamic and |
| 889 |
< |
dynamical information. |
| 890 |
< |
|
| 891 |
< |
One of the principal tools for modeling proteins, nucleic acids and |
| 892 |
< |
their complexes. Stability of proteins Folding of proteins. |
| 893 |
< |
Molecular recognition by:proteins, DNA, RNA, lipids, hormones STP, |
| 894 |
< |
etc. Enzyme reactions Rational design of biologically active |
| 895 |
< |
molecules (drug design) Small and large-scale conformational |
| 896 |
< |
changes. determination and construction of 3D structures (homology, |
| 897 |
< |
Xray diffraction, NMR) Dynamic processes such as ion transport in |
| 898 |
< |
biological systems. |
| 899 |
< |
|
| 900 |
< |
Macroscopic properties are related to microscopic behavior. |
| 886 |
> |
As one of the principal tools of molecular modeling, Molecular |
| 887 |
> |
dynamics has proven to be a powerful tool for studying the functions |
| 888 |
> |
of biological systems, providing structural, thermodynamic and |
| 889 |
> |
dynamical information. The basic idea of molecular dynamics is that |
| 890 |
> |
macroscopic properties are related to microscopic behavior and |
| 891 |
> |
microscopic behavior can be calculated from the trajectories in |
| 892 |
> |
simulations. For instance, instantaneous temperature of an |
| 893 |
> |
Hamiltonian system of $N$ particle can be measured by |
| 894 |
> |
\[ |
| 895 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
| 896 |
> |
\] |
| 897 |
> |
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
| 898 |
> |
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
| 899 |
> |
the boltzman constant. |
| 900 |
|
|
| 901 |
< |
Time dependent (and independent) microscopic behavior of a molecule |
| 902 |
< |
can be calculated by molecular dynamics simulations. |
| 903 |
< |
|
| 904 |
< |
\subsection{\label{introSec:mdInit}Initialization} |
| 905 |
< |
|
| 906 |
< |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 901 |
> |
A typical molecular dynamics run consists of three essential steps: |
| 902 |
> |
\begin{enumerate} |
| 903 |
> |
\item Initialization |
| 904 |
> |
\begin{enumerate} |
| 905 |
> |
\item Preliminary preparation |
| 906 |
> |
\item Minimization |
| 907 |
> |
\item Heating |
| 908 |
> |
\item Equilibration |
| 909 |
> |
\end{enumerate} |
| 910 |
> |
\item Production |
| 911 |
> |
\item Analysis |
| 912 |
> |
\end{enumerate} |
| 913 |
> |
These three individual steps will be covered in the following |
| 914 |
> |
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 915 |
> |
initialization of a simulation. Sec.~\ref{introSec:production} will |
| 916 |
> |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
| 917 |
> |
provides the theoretical tools for trajectory analysis. |
| 918 |
|
|
| 919 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 919 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 920 |
> |
|
| 921 |
> |
\subsubsection{Preliminary preparation} |
| 922 |
> |
|
| 923 |
> |
When selecting the starting structure of a molecule for molecular |
| 924 |
> |
simulation, one may retrieve its Cartesian coordinates from public |
| 925 |
> |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
| 926 |
> |
thousands of crystal structures of molecules are discovered every |
| 927 |
> |
year, many more remain unknown due to the difficulties of |
| 928 |
> |
purification and crystallization. Even for the molecule with known |
| 929 |
> |
structure, some important information is missing. For example, the |
| 930 |
> |
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 931 |
> |
be added. Moreover, in order to include electrostatic interaction, |
| 932 |
> |
one may need to specify the partial charges for individual atoms. |
| 933 |
> |
Under some circumstances, we may even need to prepare the system in |
| 934 |
> |
a special setup. For instance, when studying transport phenomenon in |
| 935 |
> |
membrane system, we may prepare the lipids in bilayer structure |
| 936 |
> |
instead of placing lipids randomly in solvent, since we are not |
| 937 |
> |
interested in self-aggregation and it takes a long time to happen. |
| 938 |
> |
|
| 939 |
> |
\subsubsection{Minimization} |
| 940 |
> |
|
| 941 |
> |
It is quite possible that some of molecules in the system from |
| 942 |
> |
preliminary preparation may be overlapped with each other. This |
| 943 |
> |
close proximity leads to high potential energy which consequently |
| 944 |
> |
jeopardizes any molecular dynamics simulations. To remove these |
| 945 |
> |
steric overlaps, one typically performs energy minimization to find |
| 946 |
> |
a more reasonable conformation. Several energy minimization methods |
| 947 |
> |
have been developed to exploit the energy surface and to locate the |
| 948 |
> |
local minimum. While converging slowly near the minimum, steepest |
| 949 |
> |
descent method is extremely robust when systems are far from |
| 950 |
> |
harmonic. Thus, it is often used to refine structure from |
| 951 |
> |
crystallographic data. Relied on the gradient or hessian, advanced |
| 952 |
> |
methods like conjugate gradient and Newton-Raphson converge rapidly |
| 953 |
> |
to a local minimum, while become unstable if the energy surface is |
| 954 |
> |
far from quadratic. Another factor must be taken into account, when |
| 955 |
> |
choosing energy minimization method, is the size of the system. |
| 956 |
> |
Steepest descent and conjugate gradient can deal with models of any |
| 957 |
> |
size. Because of the limit of computation power to calculate hessian |
| 958 |
> |
matrix and insufficient storage capacity to store them, most |
| 959 |
> |
Newton-Raphson methods can not be used with very large models. |
| 960 |
|
|
| 961 |
+ |
\subsubsection{Heating} |
| 962 |
+ |
|
| 963 |
+ |
Typically, Heating is performed by assigning random velocities |
| 964 |
+ |
according to a Gaussian distribution for a temperature. Beginning at |
| 965 |
+ |
a lower temperature and gradually increasing the temperature by |
| 966 |
+ |
assigning greater random velocities, we end up with setting the |
| 967 |
+ |
temperature of the system to a final temperature at which the |
| 968 |
+ |
simulation will be conducted. In heating phase, we should also keep |
| 969 |
+ |
the system from drifting or rotating as a whole. Equivalently, the |
| 970 |
+ |
net linear momentum and angular momentum of the system should be |
| 971 |
+ |
shifted to zero. |
| 972 |
+ |
|
| 973 |
+ |
\subsubsection{Equilibration} |
| 974 |
+ |
|
| 975 |
+ |
The purpose of equilibration is to allow the system to evolve |
| 976 |
+ |
spontaneously for a period of time and reach equilibrium. The |
| 977 |
+ |
procedure is continued until various statistical properties, such as |
| 978 |
+ |
temperature, pressure, energy, volume and other structural |
| 979 |
+ |
properties \textit{etc}, become independent of time. Strictly |
| 980 |
+ |
speaking, minimization and heating are not necessary, provided the |
| 981 |
+ |
equilibration process is long enough. However, these steps can serve |
| 982 |
+ |
as a means to arrive at an equilibrated structure in an effective |
| 983 |
+ |
way. |
| 984 |
+ |
|
| 985 |
+ |
\subsection{\label{introSection:production}Production} |
| 986 |
+ |
|
| 987 |
+ |
Production run is the most important steps of the simulation, in |
| 988 |
+ |
which the equilibrated structure is used as a starting point and the |
| 989 |
+ |
motions of the molecules are collected for later analysis. In order |
| 990 |
+ |
to capture the macroscopic properties of the system, the molecular |
| 991 |
+ |
dynamics simulation must be performed in correct and efficient way. |
| 992 |
+ |
|
| 993 |
+ |
The most expensive part of a molecular dynamics simulation is the |
| 994 |
+ |
calculation of non-bonded forces, such as van der Waals force and |
| 995 |
+ |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
| 996 |
+ |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
| 997 |
+ |
which making large simulations prohibitive in the absence of any |
| 998 |
+ |
computation saving techniques. |
| 999 |
+ |
|
| 1000 |
+ |
A natural approach to avoid system size issue is to represent the |
| 1001 |
+ |
bulk behavior by a finite number of the particles. However, this |
| 1002 |
+ |
approach will suffer from the surface effect. To offset this, |
| 1003 |
+ |
\textit{Periodic boundary condition} is developed to simulate bulk |
| 1004 |
+ |
properties with a relatively small number of particles. In this |
| 1005 |
+ |
method, the simulation box is replicated throughout space to form an |
| 1006 |
+ |
infinite lattice. During the simulation, when a particle moves in |
| 1007 |
+ |
the primary cell, its image in other cells move in exactly the same |
| 1008 |
+ |
direction with exactly the same orientation. Thus, as a particle |
| 1009 |
+ |
leaves the primary cell, one of its images will enter through the |
| 1010 |
+ |
opposite face. |
| 1011 |
+ |
%\begin{figure} |
| 1012 |
+ |
%\centering |
| 1013 |
+ |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
| 1014 |
+ |
%\caption[An illustration of periodic boundary conditions]{A 2-D |
| 1015 |
+ |
%illustration of periodic boundary conditions. As one particle leaves |
| 1016 |
+ |
%the right of the simulation box, an image of it enters the left.} |
| 1017 |
+ |
%\label{introFig:pbc} |
| 1018 |
+ |
%\end{figure} |
| 1019 |
+ |
|
| 1020 |
+ |
%cutoff and minimum image convention |
| 1021 |
+ |
Another important technique to improve the efficiency of force |
| 1022 |
+ |
evaluation is to apply cutoff where particles farther than a |
| 1023 |
+ |
predetermined distance, are not included in the calculation |
| 1024 |
+ |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 1025 |
+ |
discontinuity in the potential energy curve. Fortunately, one can |
| 1026 |
+ |
shift the potential to ensure the potential curve go smoothly to |
| 1027 |
+ |
zero at the cutoff radius. Cutoff strategy works pretty well for |
| 1028 |
+ |
Lennard-Jones interaction because of its short range nature. |
| 1029 |
+ |
However, simply truncating the electrostatic interaction with the |
| 1030 |
+ |
use of cutoff has been shown to lead to severe artifacts in |
| 1031 |
+ |
simulations. Ewald summation, in which the slowly conditionally |
| 1032 |
+ |
convergent Coulomb potential is transformed into direct and |
| 1033 |
+ |
reciprocal sums with rapid and absolute convergence, has proved to |
| 1034 |
+ |
minimize the periodicity artifacts in liquid simulations. Taking the |
| 1035 |
+ |
advantages of the fast Fourier transform (FFT) for calculating |
| 1036 |
+ |
discrete Fourier transforms, the particle mesh-based methods are |
| 1037 |
+ |
accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative |
| 1038 |
+ |
approach is \emph{fast multipole method}, which treats Coulombic |
| 1039 |
+ |
interaction exactly at short range, and approximate the potential at |
| 1040 |
+ |
long range through multipolar expansion. In spite of their wide |
| 1041 |
+ |
acceptances at the molecular simulation community, these two methods |
| 1042 |
+ |
are hard to be implemented correctly and efficiently. Instead, we |
| 1043 |
+ |
use a damped and charge-neutralized Coulomb potential method |
| 1044 |
+ |
developed by Wolf and his coworkers. The shifted Coulomb potential |
| 1045 |
+ |
for particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 1046 |
+ |
\begin{equation} |
| 1047 |
+ |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 1048 |
+ |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 1049 |
+ |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
| 1050 |
+ |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
| 1051 |
+ |
\end{equation} |
| 1052 |
+ |
where $\alpha$ is the convergence parameter. Due to the lack of |
| 1053 |
+ |
inherent periodicity and rapid convergence,this method is extremely |
| 1054 |
+ |
efficient and easy to implement. |
| 1055 |
+ |
%\begin{figure} |
| 1056 |
+ |
%\centering |
| 1057 |
+ |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
| 1058 |
+ |
%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} |
| 1059 |
+ |
%\label{introFigure:shiftedCoulomb} |
| 1060 |
+ |
%\end{figure} |
| 1061 |
+ |
|
| 1062 |
+ |
%multiple time step |
| 1063 |
+ |
|
| 1064 |
+ |
\subsection{\label{introSection:Analysis} Analysis} |
| 1065 |
+ |
|
| 1066 |
+ |
Recently, advanced visualization technique are widely applied to |
| 1067 |
+ |
monitor the motions of molecules. Although the dynamics of the |
| 1068 |
+ |
system can be described qualitatively from animation, quantitative |
| 1069 |
+ |
trajectory analysis are more appreciable. According to the |
| 1070 |
+ |
principles of Statistical Mechanics, |
| 1071 |
+ |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 1072 |
+ |
thermodynamics properties, analyze fluctuations of structural |
| 1073 |
+ |
parameters, and investigate time-dependent processes of the molecule |
| 1074 |
+ |
from the trajectories. |
| 1075 |
+ |
|
| 1076 |
+ |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
| 1077 |
+ |
|
| 1078 |
+ |
Thermodynamics properties, which can be expressed in terms of some |
| 1079 |
+ |
function of the coordinates and momenta of all particles in the |
| 1080 |
+ |
system, can be directly computed from molecular dynamics. The usual |
| 1081 |
+ |
way to measure the pressure is based on virial theorem of Clausius |
| 1082 |
+ |
which states that the virial is equal to $-3Nk_BT$. For a system |
| 1083 |
+ |
with forces between particles, the total virial, $W$, contains the |
| 1084 |
+ |
contribution from external pressure and interaction between the |
| 1085 |
+ |
particles: |
| 1086 |
+ |
\[ |
| 1087 |
+ |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
| 1088 |
+ |
f_{ij} } } \right\rangle |
| 1089 |
+ |
\] |
| 1090 |
+ |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
| 1091 |
+ |
distance $r_{ij}$. Thus, the expression for the pressure is given |
| 1092 |
+ |
by: |
| 1093 |
+ |
\begin{equation} |
| 1094 |
+ |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
| 1095 |
+ |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
| 1096 |
+ |
\end{equation} |
| 1097 |
+ |
|
| 1098 |
+ |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
| 1099 |
+ |
|
| 1100 |
+ |
Structural Properties of a simple fluid can be described by a set of |
| 1101 |
+ |
distribution functions. Among these functions,\emph{pair |
| 1102 |
+ |
distribution function}, also known as \emph{radial distribution |
| 1103 |
+ |
function}, is of most fundamental importance to liquid-state theory. |
| 1104 |
+ |
Pair distribution function can be gathered by Fourier transforming |
| 1105 |
+ |
raw data from a series of neutron diffraction experiments and |
| 1106 |
+ |
integrating over the surface factor \cite{Powles73}. The experiment |
| 1107 |
+ |
result can serve as a criterion to justify the correctness of the |
| 1108 |
+ |
theory. Moreover, various equilibrium thermodynamic and structural |
| 1109 |
+ |
properties can also be expressed in terms of radial distribution |
| 1110 |
+ |
function \cite{allen87:csl}. |
| 1111 |
+ |
|
| 1112 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
| 1113 |
+ |
particle $i$ will be located at a distance $r$ from a another |
| 1114 |
+ |
particle $j$ in the system |
| 1115 |
+ |
\[ |
| 1116 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1117 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
| 1118 |
+ |
\] |
| 1119 |
+ |
Note that the delta function can be replaced by a histogram in |
| 1120 |
+ |
computer simulation. Figure |
| 1121 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
| 1122 |
+ |
distribution function for the liquid argon system. The occurrence of |
| 1123 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
| 1124 |
+ |
to find particles at certain radial values than at others. This is a |
| 1125 |
+ |
result of the attractive interaction at such distances. Because of |
| 1126 |
+ |
the strong repulsive forces at short distance, the probability of |
| 1127 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
| 1128 |
+ |
other is essentially zero. |
| 1129 |
+ |
|
| 1130 |
+ |
%\begin{figure} |
| 1131 |
+ |
%\centering |
| 1132 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
| 1133 |
+ |
%\caption[Pair distribution function for the liquid argon |
| 1134 |
+ |
%]{Pair distribution function for the liquid argon} |
| 1135 |
+ |
%\label{introFigure:pairDistributionFunction} |
| 1136 |
+ |
%\end{figure} |
| 1137 |
+ |
|
| 1138 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
| 1139 |
+ |
Properties} |
| 1140 |
+ |
|
| 1141 |
+ |
Time-dependent properties are usually calculated using \emph{time |
| 1142 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
| 1143 |
+ |
at two different time |
| 1144 |
+ |
\begin{equation} |
| 1145 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
| 1146 |
+ |
\label{introEquation:timeCorrelationFunction} |
| 1147 |
+ |
\end{equation} |
| 1148 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1149 |
+ |
function is called \emph{auto correlation function}. One example of |
| 1150 |
+ |
auto correlation function is velocity auto-correlation function |
| 1151 |
+ |
which is directly related to transport properties of molecular |
| 1152 |
+ |
liquids: |
| 1153 |
+ |
\[ |
| 1154 |
+ |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
| 1155 |
+ |
\right\rangle } dt |
| 1156 |
+ |
\] |
| 1157 |
+ |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
| 1158 |
+ |
function which is averaging over time origins and over all the |
| 1159 |
+ |
atoms, dipole autocorrelation are calculated for the entire system. |
| 1160 |
+ |
The dipole autocorrelation function is given by: |
| 1161 |
+ |
\[ |
| 1162 |
+ |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1163 |
+ |
\right\rangle |
| 1164 |
+ |
\] |
| 1165 |
+ |
Here $u_{tot}$ is the net dipole of the entire system and is given |
| 1166 |
+ |
by |
| 1167 |
+ |
\[ |
| 1168 |
+ |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
| 1169 |
+ |
\] |
| 1170 |
+ |
In principle, many time correlation functions can be related with |
| 1171 |
+ |
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1172 |
+ |
scattering spectra of molecular liquids. In practice, one can |
| 1173 |
+ |
extract the IR spectrum from the intensity of dipole fluctuation at |
| 1174 |
+ |
each frequency using the following relationship: |
| 1175 |
+ |
\[ |
| 1176 |
+ |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1177 |
+ |
i2\pi vt} dt} |
| 1178 |
+ |
\] |
| 1179 |
+ |
|
| 1180 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1181 |
|
|
| 1182 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1237 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1238 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 1239 |
|
\begin{equation} |
| 1240 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 1240 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1241 |
|
\end{equation} |
| 1242 |
|
which is used to ensure rotation matrix's orthogonality. |
| 1243 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 1262 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1263 |
|
We can use constraint force provided by lagrange multiplier on the |
| 1264 |
|
normal manifold to keep the motion on constraint space. Or we can |
| 1265 |
< |
simply evolve the system in constraint manifold. The two method are |
| 1266 |
< |
proved to be equivalent. The holonomic constraint and equations of |
| 1267 |
< |
motions define a constraint manifold for rigid body |
| 1265 |
> |
simply evolve the system in constraint manifold. These two methods |
| 1266 |
> |
are proved to be equivalent. The holonomic constraint and equations |
| 1267 |
> |
of motions define a constraint manifold for rigid body |
| 1268 |
|
\[ |
| 1269 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1270 |
|
\right\}. |
| 1425 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1426 |
|
) |
| 1427 |
|
\] |
| 1428 |
< |
|
| 1160 |
< |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
| 1428 |
> |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1429 |
|
manner. |
| 1430 |
|
|
| 1431 |
|
In order to construct a second-order symplectic method, we split the |
| 1477 |
|
\] |
| 1478 |
|
The equations of motion corresponding to potential energy and |
| 1479 |
|
kinetic energy are listed in the below table, |
| 1480 |
+ |
\begin{table} |
| 1481 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
| 1482 |
|
\begin{center} |
| 1483 |
|
\begin{tabular}{|l|l|} |
| 1484 |
|
\hline |
| 1491 |
|
\hline |
| 1492 |
|
\end{tabular} |
| 1493 |
|
\end{center} |
| 1494 |
< |
A second-order symplectic method is now obtained by the composition |
| 1495 |
< |
of the flow maps, |
| 1494 |
> |
\end{table} |
| 1495 |
> |
A second-order symplectic method is now obtained by the |
| 1496 |
> |
composition of the flow maps, |
| 1497 |
|
\[ |
| 1498 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1499 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1751 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1752 |
|
or be determined by Stokes' law for regular shaped particles.A |
| 1753 |
|
briefly review on calculating friction tensor for arbitrary shaped |
| 1754 |
< |
particles is given in section \ref{introSection:frictionTensor}. |
| 1754 |
> |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1755 |
|
|
| 1756 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 1757 |
|
|
| 1789 |
|
when the system become more and more complicate. Instead, various |
| 1790 |
|
approaches based on hydrodynamics have been developed to calculate |
| 1791 |
|
the friction coefficients. The friction effect is isotropic in |
| 1792 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1793 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
| 1792 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 1793 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 1794 |
|
\[ |
| 1795 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 1796 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 2041 |
|
\] |
| 2042 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 2043 |
|
joining center of resistance $R$ and origin $O$. |
| 1773 |
– |
|
| 1774 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
