93 |
|
The actual trajectory, along which a dynamical system may move from |
94 |
|
one point to another within a specified time, is derived by finding |
95 |
|
the path which minimizes the time integral of the difference between |
96 |
< |
the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
96 |
> |
the kinetic, $K$, and potential energies, $U$ \cite{Tolman1979}. |
97 |
|
\begin{equation} |
98 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
99 |
|
\label{introEquation:halmitonianPrinciple1} |
189 |
|
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
190 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
191 |
|
equation of motion. Due to their symmetrical formula, they are also |
192 |
< |
known as the canonical equations of motions \cite{Goldstein01}. |
192 |
> |
known as the canonical equations of motions \cite{Goldstein2001}. |
193 |
|
|
194 |
|
An important difference between Lagrangian approach and the |
195 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
200 |
|
appropriate for application to statistical mechanics and quantum |
201 |
|
mechanics, since it treats the coordinate and its time derivative as |
202 |
|
independent variables and it only works with 1st-order differential |
203 |
< |
equations\cite{Marion90}. |
203 |
> |
equations\cite{Marion1990}. |
204 |
|
|
205 |
|
In Newtonian Mechanics, a system described by conservative forces |
206 |
|
conserves the total energy \ref{introEquation:energyConservation}. |
470 |
|
many-body system in Statistical Mechanics. Fortunately, Ergodic |
471 |
|
Hypothesis is proposed to make a connection between time average and |
472 |
|
ensemble average. It states that time average and average over the |
473 |
< |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
473 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
474 |
|
\begin{equation} |
475 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
476 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
484 |
|
a properly weighted statistical average. This allows the researcher |
485 |
|
freedom of choice when deciding how best to measure a given |
486 |
|
observable. In case an ensemble averaged approach sounds most |
487 |
< |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
487 |
> |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
488 |
|
utilized. Or if the system lends itself to a time averaging |
489 |
|
approach, the Molecular Dynamics techniques in |
490 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
498 |
|
within the equations. Since 1990, geometric integrators, which |
499 |
|
preserve various phase-flow invariants such as symplectic structure, |
500 |
|
volume and time reversal symmetry, are developed to address this |
501 |
< |
issue. The velocity verlet method, which happens to be a simple |
502 |
< |
example of symplectic integrator, continues to gain its popularity |
503 |
< |
in molecular dynamics community. This fact can be partly explained |
504 |
< |
by its geometric nature. |
501 |
> |
issue\cite{Dullweber1997, McLachlan1998, Leimkuhler1999}. The |
502 |
> |
velocity verlet method, which happens to be a simple example of |
503 |
> |
symplectic integrator, continues to gain its popularity in molecular |
504 |
> |
dynamics community. This fact can be partly explained by its |
505 |
> |
geometric nature. |
506 |
|
|
507 |
|
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
508 |
|
A \emph{manifold} is an abstract mathematical space. It locally |
566 |
|
\end{equation}In this case, $f$ is |
567 |
|
called a \emph{Hamiltonian vector field}. |
568 |
|
|
569 |
< |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
569 |
> |
Another generalization of Hamiltonian dynamics is Poisson |
570 |
> |
Dynamics\cite{Olver1986}, |
571 |
|
\begin{equation} |
572 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
573 |
|
\end{equation} |
614 |
|
|
615 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
616 |
|
|
617 |
< |
The hidden geometric properties of ODE and its flow play important |
618 |
< |
roles in numerical studies. Many of them can be found in systems |
619 |
< |
which occur naturally in applications. |
617 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE |
618 |
> |
and its flow play important roles in numerical studies. Many of them |
619 |
> |
can be found in systems which occur naturally in applications. |
620 |
|
|
621 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
622 |
|
a \emph{symplectic} flow if it satisfies, |
660 |
|
which is the condition for conserving \emph{first integral}. For a |
661 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
662 |
|
smooth function $G$ is given by, |
663 |
< |
\begin{equation} |
664 |
< |
\begin{array}{c} |
665 |
< |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
666 |
< |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
665 |
< |
\end{array} |
663 |
> |
|
664 |
> |
\begin{eqnarray} |
665 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
666 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
667 |
|
\label{introEquation:firstIntegral1} |
668 |
< |
\end{equation} |
668 |
> |
\end{eqnarray} |
669 |
> |
|
670 |
> |
|
671 |
|
Using poisson bracket notion, Equation |
672 |
|
\ref{introEquation:firstIntegral1} can be rewritten as |
673 |
|
\[ |
682 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
683 |
|
0$. |
684 |
|
|
685 |
< |
|
683 |
< |
When designing any numerical methods, one should always try to |
685 |
> |
When designing any numerical methods, one should always try to |
686 |
|
preserve the structural properties of the original ODE and its flow. |
687 |
|
|
688 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
699 |
|
\item Splitting methods |
700 |
|
\end{enumerate} |
701 |
|
|
702 |
< |
Generating function tends to lead to methods which are cumbersome |
703 |
< |
and difficult to use. In dissipative systems, variational methods |
704 |
< |
can capture the decay of energy accurately. Since their |
705 |
< |
geometrically unstable nature against non-Hamiltonian perturbations, |
706 |
< |
ordinary implicit Runge-Kutta methods are not suitable for |
707 |
< |
Hamiltonian system. Recently, various high-order explicit |
708 |
< |
Runge--Kutta methods have been developed to overcome this |
702 |
> |
Generating function\cite{Channell1990} tends to lead to methods |
703 |
> |
which are cumbersome and difficult to use. In dissipative systems, |
704 |
> |
variational methods can capture the decay of energy |
705 |
> |
accurately\cite{Kane2000}. Since their geometrically unstable nature |
706 |
> |
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
707 |
> |
methods are not suitable for Hamiltonian system. Recently, various |
708 |
> |
high-order explicit Runge-Kutta methods |
709 |
> |
\cite{Owren1992,Chen2003}have been developed to overcome this |
710 |
|
instability. However, due to computational penalty involved in |
711 |
|
implementing the Runge-Kutta methods, they do not attract too much |
712 |
|
attention from Molecular Dynamics community. Instead, splitting have |
713 |
|
been widely accepted since they exploit natural decompositions of |
714 |
< |
the system\cite{Tuckerman92}. |
714 |
> |
the system\cite{Tuckerman1992, McLachlan1998}. |
715 |
|
|
716 |
|
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
717 |
|
|
834 |
|
error of splitting method in terms of commutator of the |
835 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
836 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
837 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
837 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
838 |
|
\begin{equation} |
839 |
|
\exp (hX + hY) = \exp (hZ) |
840 |
|
\end{equation} |
847 |
|
\[ |
848 |
|
[X,Y] = XY - YX . |
849 |
|
\] |
850 |
< |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
851 |
< |
can obtain |
850 |
> |
Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to |
851 |
> |
Sprang splitting, we can obtain |
852 |
|
\begin{eqnarray*} |
853 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
854 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
855 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
853 |
< |
\ldots ) |
853 |
> |
\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 \\ |
854 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
855 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
856 |
|
\end{eqnarray*} |
857 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
858 |
|
error of Spring splitting is proportional to $h^3$. The same |
861 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
862 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
863 |
|
\end{equation} |
864 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
864 |
> |
Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher |
865 |
|
order method. Yoshida proposed an elegant way to compose higher |
866 |
< |
order methods based on symmetric splitting. Given a symmetric second |
867 |
< |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
868 |
< |
method can be constructed by composing, |
866 |
> |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
867 |
> |
a symmetric second order base method $ \varphi _h^{(2)} $, a |
868 |
> |
fourth-order symmetric method can be constructed by composing, |
869 |
|
\[ |
870 |
|
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
871 |
|
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
894 |
|
simulations. For instance, instantaneous temperature of an |
895 |
|
Hamiltonian system of $N$ particle can be measured by |
896 |
|
\[ |
897 |
< |
T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
897 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
898 |
|
\] |
899 |
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
900 |
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
915 |
|
These three individual steps will be covered in the following |
916 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
917 |
|
initialization of a simulation. Sec.~\ref{introSec:production} will |
918 |
< |
discusses issues in production run, including the force evaluation |
919 |
< |
and the numerical integration schemes of the equations of motion . |
918 |
< |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
919 |
< |
trajectory analysis. |
918 |
> |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
919 |
> |
provides the theoretical tools for trajectory analysis. |
920 |
|
|
921 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
922 |
|
|
986 |
|
|
987 |
|
\subsection{\label{introSection:production}Production} |
988 |
|
|
989 |
< |
\subsubsection{\label{introSec:forceCalculation}The Force Calculation} |
989 |
> |
Production run is the most important step of the simulation, in |
990 |
> |
which the equilibrated structure is used as a starting point and the |
991 |
> |
motions of the molecules are collected for later analysis. In order |
992 |
> |
to capture the macroscopic properties of the system, the molecular |
993 |
> |
dynamics simulation must be performed in correct and efficient way. |
994 |
|
|
995 |
< |
\subsubsection{\label{introSection:integrationSchemes} Integration |
996 |
< |
Schemes} |
995 |
> |
The most expensive part of a molecular dynamics simulation is the |
996 |
> |
calculation of non-bonded forces, such as van der Waals force and |
997 |
> |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
998 |
> |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
999 |
> |
which making large simulations prohibitive in the absence of any |
1000 |
> |
computation saving techniques. |
1001 |
|
|
1002 |
+ |
A natural approach to avoid system size issue is to represent the |
1003 |
+ |
bulk behavior by a finite number of the particles. However, this |
1004 |
+ |
approach will suffer from the surface effect. To offset this, |
1005 |
+ |
\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc}) |
1006 |
+ |
is developed to simulate bulk properties with a relatively small |
1007 |
+ |
number of particles. In this method, the simulation box is |
1008 |
+ |
replicated throughout space to form an infinite lattice. During the |
1009 |
+ |
simulation, when a particle moves in the primary cell, its image in |
1010 |
+ |
other cells move in exactly the same direction with exactly the same |
1011 |
+ |
orientation. Thus, as a particle leaves the primary cell, one of its |
1012 |
+ |
images will enter through the opposite face. |
1013 |
+ |
\begin{figure} |
1014 |
+ |
\centering |
1015 |
+ |
\includegraphics[width=\linewidth]{pbc.eps} |
1016 |
+ |
\caption[An illustration of periodic boundary conditions]{A 2-D |
1017 |
+ |
illustration of periodic boundary conditions. As one particle leaves |
1018 |
+ |
the left of the simulation box, an image of it enters the right.} |
1019 |
+ |
\label{introFig:pbc} |
1020 |
+ |
\end{figure} |
1021 |
+ |
|
1022 |
+ |
%cutoff and minimum image convention |
1023 |
+ |
Another important technique to improve the efficiency of force |
1024 |
+ |
evaluation is to apply cutoff where particles farther than a |
1025 |
+ |
predetermined distance, are not included in the calculation |
1026 |
+ |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1027 |
+ |
discontinuity in the potential energy curve. Fortunately, one can |
1028 |
+ |
shift the potential to ensure the potential curve go smoothly to |
1029 |
+ |
zero at the cutoff radius. Cutoff strategy works pretty well for |
1030 |
+ |
Lennard-Jones interaction because of its short range nature. |
1031 |
+ |
However, simply truncating the electrostatic interaction with the |
1032 |
+ |
use of cutoff has been shown to lead to severe artifacts in |
1033 |
+ |
simulations. Ewald summation, in which the slowly conditionally |
1034 |
+ |
convergent Coulomb potential is transformed into direct and |
1035 |
+ |
reciprocal sums with rapid and absolute convergence, has proved to |
1036 |
+ |
minimize the periodicity artifacts in liquid simulations. Taking the |
1037 |
+ |
advantages of the fast Fourier transform (FFT) for calculating |
1038 |
+ |
discrete Fourier transforms, the particle mesh-based |
1039 |
+ |
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
1040 |
+ |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast |
1041 |
+ |
multipole method}\cite{Greengard1987, Greengard1994}, which treats |
1042 |
+ |
Coulombic interaction exactly at short range, and approximate the |
1043 |
+ |
potential at long range through multipolar expansion. In spite of |
1044 |
+ |
their wide acceptances at the molecular simulation community, these |
1045 |
+ |
two methods are hard to be implemented correctly and efficiently. |
1046 |
+ |
Instead, we use a damped and charge-neutralized Coulomb potential |
1047 |
+ |
method developed by Wolf and his coworkers\cite{Wolf1999}. The |
1048 |
+ |
shifted Coulomb potential for particle $i$ and particle $j$ at |
1049 |
+ |
distance $r_{rj}$ is given by: |
1050 |
+ |
\begin{equation} |
1051 |
+ |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1052 |
+ |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1053 |
+ |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
1054 |
+ |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
1055 |
+ |
\end{equation} |
1056 |
+ |
where $\alpha$ is the convergence parameter. Due to the lack of |
1057 |
+ |
inherent periodicity and rapid convergence,this method is extremely |
1058 |
+ |
efficient and easy to implement. |
1059 |
+ |
\begin{figure} |
1060 |
+ |
\centering |
1061 |
+ |
\includegraphics[width=\linewidth]{shifted_coulomb.eps} |
1062 |
+ |
\caption[An illustration of shifted Coulomb potential]{An |
1063 |
+ |
illustration of shifted Coulomb potential.} |
1064 |
+ |
\label{introFigure:shiftedCoulomb} |
1065 |
+ |
\end{figure} |
1066 |
+ |
|
1067 |
+ |
%multiple time step |
1068 |
+ |
|
1069 |
|
\subsection{\label{introSection:Analysis} Analysis} |
1070 |
|
|
1071 |
+ |
Recently, advanced visualization technique are widely applied to |
1072 |
+ |
monitor the motions of molecules. Although the dynamics of the |
1073 |
+ |
system can be described qualitatively from animation, quantitative |
1074 |
+ |
trajectory analysis are more appreciable. According to the |
1075 |
+ |
principles of Statistical Mechanics, |
1076 |
+ |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1077 |
+ |
thermodynamics properties, analyze fluctuations of structural |
1078 |
+ |
parameters, and investigate time-dependent processes of the molecule |
1079 |
+ |
from the trajectories. |
1080 |
+ |
|
1081 |
+ |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
1082 |
+ |
|
1083 |
+ |
Thermodynamics properties, which can be expressed in terms of some |
1084 |
+ |
function of the coordinates and momenta of all particles in the |
1085 |
+ |
system, can be directly computed from molecular dynamics. The usual |
1086 |
+ |
way to measure the pressure is based on virial theorem of Clausius |
1087 |
+ |
which states that the virial is equal to $-3Nk_BT$. For a system |
1088 |
+ |
with forces between particles, the total virial, $W$, contains the |
1089 |
+ |
contribution from external pressure and interaction between the |
1090 |
+ |
particles: |
1091 |
+ |
\[ |
1092 |
+ |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
1093 |
+ |
f_{ij} } } \right\rangle |
1094 |
+ |
\] |
1095 |
+ |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
1096 |
+ |
distance $r_{ij}$. Thus, the expression for the pressure is given |
1097 |
+ |
by: |
1098 |
+ |
\begin{equation} |
1099 |
+ |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
1100 |
+ |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1101 |
+ |
\end{equation} |
1102 |
+ |
|
1103 |
+ |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
1104 |
+ |
|
1105 |
+ |
Structural Properties of a simple fluid can be described by a set of |
1106 |
+ |
distribution functions. Among these functions,\emph{pair |
1107 |
+ |
distribution function}, also known as \emph{radial distribution |
1108 |
+ |
function}, is of most fundamental importance to liquid-state theory. |
1109 |
+ |
Pair distribution function can be gathered by Fourier transforming |
1110 |
+ |
raw data from a series of neutron diffraction experiments and |
1111 |
+ |
integrating over the surface factor \cite{Powles1973}. The |
1112 |
+ |
experiment result can serve as a criterion to justify the |
1113 |
+ |
correctness of the theory. Moreover, various equilibrium |
1114 |
+ |
thermodynamic and structural properties can also be expressed in |
1115 |
+ |
terms of radial distribution function \cite{Allen1987}. |
1116 |
+ |
|
1117 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
1118 |
+ |
particle $i$ will be located at a distance $r$ from a another |
1119 |
+ |
particle $j$ in the system |
1120 |
+ |
\[ |
1121 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1122 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1123 |
+ |
\] |
1124 |
+ |
Note that the delta function can be replaced by a histogram in |
1125 |
+ |
computer simulation. Figure |
1126 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1127 |
+ |
distribution function for the liquid argon system. The occurrence of |
1128 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1129 |
+ |
to find particles at certain radial values than at others. This is a |
1130 |
+ |
result of the attractive interaction at such distances. Because of |
1131 |
+ |
the strong repulsive forces at short distance, the probability of |
1132 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
1133 |
+ |
other is essentially zero. |
1134 |
+ |
|
1135 |
+ |
%\begin{figure} |
1136 |
+ |
%\centering |
1137 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
1138 |
+ |
%\caption[Pair distribution function for the liquid argon |
1139 |
+ |
%]{Pair distribution function for the liquid argon} |
1140 |
+ |
%\label{introFigure:pairDistributionFunction} |
1141 |
+ |
%\end{figure} |
1142 |
+ |
|
1143 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
1144 |
+ |
Properties} |
1145 |
+ |
|
1146 |
+ |
Time-dependent properties are usually calculated using \emph{time |
1147 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
1148 |
+ |
at two different time |
1149 |
+ |
\begin{equation} |
1150 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1151 |
+ |
\label{introEquation:timeCorrelationFunction} |
1152 |
+ |
\end{equation} |
1153 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
1154 |
+ |
function is called \emph{auto correlation function}. One example of |
1155 |
+ |
auto correlation function is velocity auto-correlation function |
1156 |
+ |
which is directly related to transport properties of molecular |
1157 |
+ |
liquids: |
1158 |
+ |
\[ |
1159 |
+ |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1160 |
+ |
\right\rangle } dt |
1161 |
+ |
\] |
1162 |
+ |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1163 |
+ |
function which is averaging over time origins and over all the |
1164 |
+ |
atoms, dipole autocorrelation are calculated for the entire system. |
1165 |
+ |
The dipole autocorrelation function is given by: |
1166 |
+ |
\[ |
1167 |
+ |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1168 |
+ |
\right\rangle |
1169 |
+ |
\] |
1170 |
+ |
Here $u_{tot}$ is the net dipole of the entire system and is given |
1171 |
+ |
by |
1172 |
+ |
\[ |
1173 |
+ |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1174 |
+ |
\] |
1175 |
+ |
In principle, many time correlation functions can be related with |
1176 |
+ |
Fourier transforms of the infrared, Raman, and inelastic neutron |
1177 |
+ |
scattering spectra of molecular liquids. In practice, one can |
1178 |
+ |
extract the IR spectrum from the intensity of dipole fluctuation at |
1179 |
+ |
each frequency using the following relationship: |
1180 |
+ |
\[ |
1181 |
+ |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1182 |
+ |
i2\pi vt} dt} |
1183 |
+ |
\] |
1184 |
+ |
|
1185 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1186 |
|
|
1187 |
|
Rigid bodies are frequently involved in the modeling of different |
1190 |
|
movement of the objects in 3D gaming engine or other physics |
1191 |
|
simulator is governed by the rigid body dynamics. In molecular |
1192 |
|
simulation, rigid body is used to simplify the model in |
1193 |
< |
protein-protein docking study{\cite{Gray03}}. |
1193 |
> |
protein-protein docking study\cite{Gray2003}. |
1194 |
|
|
1195 |
|
It is very important to develop stable and efficient methods to |
1196 |
|
integrate the equations of motion of orientational degrees of |
1198 |
|
rotational degrees of freedom. However, due to its singularity, the |
1199 |
|
numerical integration of corresponding equations of motion is very |
1200 |
|
inefficient and inaccurate. Although an alternative integrator using |
1201 |
< |
different sets of Euler angles can overcome this difficulty\cite{}, |
1202 |
< |
the computational penalty and the lost of angular momentum |
1203 |
< |
conservation still remain. A singularity free representation |
1204 |
< |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
1205 |
< |
this approach suffer from the nonseparable Hamiltonian resulted from |
1206 |
< |
quaternion representation, which prevents the symplectic algorithm |
1207 |
< |
to be utilized. Another different approach is to apply holonomic |
1208 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
1209 |
< |
moves independently under the normal forces deriving from potential |
1210 |
< |
energy and constraint forces which are used to guarantee the |
1211 |
< |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
1212 |
< |
algorithm converge very slowly when the number of constraint |
1213 |
< |
increases. |
1201 |
> |
different sets of Euler angles can overcome this |
1202 |
> |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
1203 |
> |
of angular momentum conservation still remain. A singularity free |
1204 |
> |
representation utilizing quaternions was developed by Evans in |
1205 |
> |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
1206 |
> |
nonseparable Hamiltonian resulted from quaternion representation, |
1207 |
> |
which prevents the symplectic algorithm to be utilized. Another |
1208 |
> |
different approach is to apply holonomic constraints to the atoms |
1209 |
> |
belonging to the rigid body. Each atom moves independently under the |
1210 |
> |
normal forces deriving from potential energy and constraint forces |
1211 |
> |
which are used to guarantee the rigidness. However, due to their |
1212 |
> |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
1213 |
> |
when the number of constraint increases\cite{Ryckaert1977, |
1214 |
> |
Andersen1983}. |
1215 |
|
|
1216 |
|
The break through in geometric literature suggests that, in order to |
1217 |
|
develop a long-term integration scheme, one should preserve the |
1218 |
|
symplectic structure of the flow. Introducing conjugate momentum to |
1219 |
|
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
1220 |
< |
symplectic integrator, RSHAKE, was proposed to evolve the |
1221 |
< |
Hamiltonian system in a constraint manifold by iteratively |
1220 |
> |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
1221 |
> |
the Hamiltonian system in a constraint manifold by iteratively |
1222 |
|
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
1223 |
< |
method using quaternion representation was developed by Omelyan. |
1224 |
< |
However, both of these methods are iterative and inefficient. In |
1225 |
< |
this section, we will present a symplectic Lie-Poisson integrator |
1226 |
< |
for rigid body developed by Dullweber and his |
1227 |
< |
coworkers\cite{Dullweber1997} in depth. |
1223 |
> |
method using quaternion representation was developed by |
1224 |
> |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
1225 |
> |
iterative and inefficient. In this section, we will present a |
1226 |
> |
symplectic Lie-Poisson integrator for rigid body developed by |
1227 |
> |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1228 |
|
|
1229 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
1230 |
|
The motion of the rigid body is Hamiltonian with the Hamiltonian |
1243 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1244 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
1245 |
|
\begin{equation} |
1246 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
1246 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1247 |
|
\end{equation} |
1248 |
|
which is used to ensure rotation matrix's orthogonality. |
1249 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1268 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1269 |
|
We can use constraint force provided by lagrange multiplier on the |
1270 |
|
normal manifold to keep the motion on constraint space. Or we can |
1271 |
< |
simply evolve the system in constraint manifold. The two method are |
1272 |
< |
proved to be equivalent. The holonomic constraint and equations of |
1273 |
< |
motions define a constraint manifold for rigid body |
1271 |
> |
simply evolve the system in constraint manifold. These two methods |
1272 |
> |
are proved to be equivalent. The holonomic constraint and equations |
1273 |
> |
of motions define a constraint manifold for rigid body |
1274 |
|
\[ |
1275 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1276 |
|
\right\}. |
1357 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1358 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
1359 |
|
unique property eliminate the requirement of iterations which can |
1360 |
< |
not be avoided in other methods\cite{}. |
1360 |
> |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
1361 |
|
|
1362 |
|
Applying hat-map isomorphism, we obtain the equation of motion for |
1363 |
|
angular momentum on body frame |
1483 |
|
\] |
1484 |
|
The equations of motion corresponding to potential energy and |
1485 |
|
kinetic energy are listed in the below table, |
1486 |
+ |
\begin{table} |
1487 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
1488 |
|
\begin{center} |
1489 |
|
\begin{tabular}{|l|l|} |
1490 |
|
\hline |
1497 |
|
\hline |
1498 |
|
\end{tabular} |
1499 |
|
\end{center} |
1500 |
< |
A second-order symplectic method is now obtained by the composition |
1501 |
< |
of the flow maps, |
1500 |
> |
\end{table} |
1501 |
> |
A second-order symplectic method is now obtained by the |
1502 |
> |
composition of the flow maps, |
1503 |
|
\[ |
1504 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1505 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1624 |
|
\] |
1625 |
|
where $p$ is real and $L$ is called the Laplace Transform |
1626 |
|
Operator. Below are some important properties of Laplace transform |
1434 |
– |
\begin{equation} |
1435 |
– |
\begin{array}{c} |
1436 |
– |
L(x + y) = L(x) + L(y) \\ |
1437 |
– |
L(ax) = aL(x) \\ |
1438 |
– |
L(\dot x) = pL(x) - px(0) \\ |
1439 |
– |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
1440 |
– |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
1441 |
– |
\end{array} |
1442 |
– |
\end{equation} |
1627 |
|
|
1628 |
+ |
\begin{eqnarray*} |
1629 |
+ |
L(x + y) & = & L(x) + L(y) \\ |
1630 |
+ |
L(ax) & = & aL(x) \\ |
1631 |
+ |
L(\dot x) & = & pL(x) - px(0) \\ |
1632 |
+ |
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
1633 |
+ |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
1634 |
+ |
\end{eqnarray*} |
1635 |
+ |
|
1636 |
+ |
|
1637 |
|
Applying Laplace transform to the bath coordinates, we obtain |
1638 |
< |
\[ |
1639 |
< |
\begin{array}{c} |
1640 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1641 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1642 |
< |
\end{array} |
1450 |
< |
\] |
1638 |
> |
\begin{eqnarray*} |
1639 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1640 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1641 |
> |
\end{eqnarray*} |
1642 |
> |
|
1643 |
|
By the same way, the system coordinates become |
1644 |
< |
\[ |
1645 |
< |
\begin{array}{c} |
1646 |
< |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1647 |
< |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1456 |
< |
\end{array} |
1457 |
< |
\] |
1644 |
> |
\begin{eqnarray*} |
1645 |
> |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1646 |
> |
& & \mbox{} - \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1647 |
> |
\end{eqnarray*} |
1648 |
|
|
1649 |
|
With the help of some relatively important inverse Laplace |
1650 |
|
transformations: |
1656 |
|
\end{array} |
1657 |
|
\] |
1658 |
|
, we obtain |
1659 |
< |
\begin{align} |
1660 |
< |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
1659 |
> |
\[ |
1660 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
1661 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1662 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
1663 |
|
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
1664 |
|
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
1665 |
|
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
1666 |
|
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
1667 |
< |
% |
1668 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1667 |
> |
\] |
1668 |
> |
\[ |
1669 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1670 |
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1671 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1672 |
|
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
1674 |
|
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
1675 |
|
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
1676 |
|
(\omega _\alpha t)} \right\}} |
1677 |
< |
\end{align} |
1677 |
> |
\] |
1678 |
|
|
1679 |
|
Introducing a \emph{dynamic friction kernel} |
1680 |
|
\begin{equation} |
1769 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1770 |
|
\] |
1771 |
|
And since the $q$ coordinates are harmonic oscillators, |
1772 |
< |
\[ |
1773 |
< |
\begin{array}{c} |
1774 |
< |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1775 |
< |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1776 |
< |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1777 |
< |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1778 |
< |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1779 |
< |
= kT\xi (t) \\ |
1780 |
< |
\end{array} |
1781 |
< |
\] |
1772 |
> |
|
1773 |
> |
\begin{eqnarray*} |
1774 |
> |
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1775 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1776 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1777 |
> |
\left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1778 |
> |
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1779 |
> |
& = &kT\xi (t) \\ |
1780 |
> |
\end{eqnarray*} |
1781 |
> |
|
1782 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1783 |
|
\begin{equation} |
1784 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1793 |
|
when the system become more and more complicate. Instead, various |
1794 |
|
approaches based on hydrodynamics have been developed to calculate |
1795 |
|
the friction coefficients. The friction effect is isotropic in |
1796 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
1797 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
1796 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
1797 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
1798 |
|
\[ |
1799 |
|
\Xi = \left( {\begin{array}{*{20}c} |
1800 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
1850 |
|
hydrodynamics theory, because their properties can be calculated |
1851 |
|
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
1852 |
|
also called a triaxial ellipsoid, which is given in Cartesian |
1853 |
< |
coordinates by |
1853 |
> |
coordinates by\cite{Perrin1934, Perrin1936} |
1854 |
|
\[ |
1855 |
|
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
1856 |
|
}} = 1 |
1894 |
|
hydrodynamic properties of rigid bodies. However, since the mapping |
1895 |
|
from all possible ellipsoidal space, $r$-space, to all possible |
1896 |
|
combination of rotational diffusion coefficients, $D$-space is not |
1897 |
< |
unique\cite{Wegener79} as well as the intrinsic coupling between |
1898 |
< |
translational and rotational motion of rigid body\cite{}, general |
1899 |
< |
ellipsoid is not always suitable for modeling arbitrarily shaped |
1900 |
< |
rigid molecule. A number of studies have been devoted to determine |
1901 |
< |
the friction tensor for irregularly shaped rigid bodies using more |
1902 |
< |
advanced method\cite{} where the molecule of interest was modeled by |
1903 |
< |
combinations of spheres(beads)\cite{} and the hydrodynamics |
1904 |
< |
properties of the molecule can be calculated using the hydrodynamic |
1905 |
< |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
1906 |
< |
immersed in a continuous medium. Due to hydrodynamics interaction, |
1907 |
< |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
1908 |
< |
unperturbed velocity $v_i$, |
1897 |
> |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
1898 |
> |
translational and rotational motion of rigid body, general ellipsoid |
1899 |
> |
is not always suitable for modeling arbitrarily shaped rigid |
1900 |
> |
molecule. A number of studies have been devoted to determine the |
1901 |
> |
friction tensor for irregularly shaped rigid bodies using more |
1902 |
> |
advanced method where the molecule of interest was modeled by |
1903 |
> |
combinations of spheres(beads)\cite{Carrasco1999} and the |
1904 |
> |
hydrodynamics properties of the molecule can be calculated using the |
1905 |
> |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
1906 |
> |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
1907 |
> |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
1908 |
> |
than its unperturbed velocity $v_i$, |
1909 |
|
\[ |
1910 |
|
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
1911 |
|
\] |
1926 |
|
\end{equation} |
1927 |
|
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
1928 |
|
A second order expression for element of different size was |
1929 |
< |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
1930 |
< |
la Torre and Bloomfield, |
1929 |
> |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
1930 |
> |
Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
1931 |
|
\begin{equation} |
1932 |
|
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
1933 |
|
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
2028 |
|
Using Equations \ref{introEquation:definitionCR} and |
2029 |
|
\ref{introEquation:resistanceTensorTransformation}, one can locate |
2030 |
|
the position of center of resistance, |
2031 |
< |
\[ |
2032 |
< |
\left( \begin{array}{l} |
2031 |
> |
\begin{eqnarray*} |
2032 |
> |
\left( \begin{array}{l} |
2033 |
|
x_{OR} \\ |
2034 |
|
y_{OR} \\ |
2035 |
|
z_{OR} \\ |
2036 |
< |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
2036 |
> |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
2037 |
|
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
2038 |
|
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
2039 |
|
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
2040 |
< |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
2040 |
> |
\end{array}} \right)^{ - 1} \\ |
2041 |
> |
& & \left( \begin{array}{l} |
2042 |
|
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
2043 |
|
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
2044 |
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
2045 |
< |
\end{array} \right). |
2046 |
< |
\] |
2045 |
> |
\end{array} \right) \\ |
2046 |
> |
\end{eqnarray*} |
2047 |
> |
|
2048 |
> |
|
2049 |
> |
|
2050 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
2051 |
|
joining center of resistance $R$ and origin $O$. |
1857 |
– |
|
1858 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |