| 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)} |
| 986 |
|
|
| 987 |
|
\subsection{\label{introSection:production}Production} |
| 988 |
|
|
| 989 |
< |
Production run is the most important steps of the simulation, in |
| 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 |
| 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} is developed to simulate bulk |
| 1006 |
< |
properties with a relatively small number of particles. In this |
| 1007 |
< |
method, the simulation box is replicated throughout space to form an |
| 1008 |
< |
infinite lattice. During the simulation, when a particle moves in |
| 1009 |
< |
the primary cell, its image in other cells move in exactly the same |
| 1010 |
< |
direction with exactly the same orientation. Thus, as a particle |
| 1011 |
< |
leaves the primary cell, one of its images will enter through the |
| 1012 |
< |
opposite face. |
| 1013 |
< |
%\begin{figure} |
| 1014 |
< |
%\centering |
| 1015 |
< |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
| 1016 |
< |
%\caption[An illustration of periodic boundary conditions]{A 2-D |
| 1017 |
< |
%illustration of periodic boundary conditions. As one particle leaves |
| 1018 |
< |
%the right of the simulation box, an image of it enters the left.} |
| 1019 |
< |
%\label{introFig:pbc} |
| 1020 |
< |
%\end{figure} |
| 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 |
| 1028 |
< |
(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the |
| 1029 |
< |
potential to ensure the potential curve go smoothly to zero at the |
| 1030 |
< |
cutoff radius. Cutoff strategy works pretty well for Lennard-Jones |
| 1031 |
< |
interaction because of its short range nature. However, simply |
| 1032 |
< |
truncating the electrostatic interaction with the use of cutoff has |
| 1033 |
< |
been shown to lead to severe artifacts in simulations. Ewald |
| 1034 |
< |
summation, in which the slowly conditionally convergent Coulomb |
| 1035 |
< |
potential is transformed into direct and reciprocal sums with rapid |
| 1036 |
< |
and absolute convergence, has proved to minimize the periodicity |
| 1037 |
< |
artifacts in liquid simulations. Taking the advantages of the fast |
| 1038 |
< |
Fourier transform (FFT) for calculating discrete Fourier transforms, |
| 1039 |
< |
the particle mesh-based methods are accelerated from $O(N^{3/2})$ to |
| 1040 |
< |
$O(N logN)$. An alternative approach is \emph{fast multipole |
| 1041 |
< |
method}, which treats Coulombic interaction exactly at short range, |
| 1042 |
< |
and approximate the potential at long range through multipolar |
| 1043 |
< |
expansion. In spite of their wide acceptances at the molecular |
| 1044 |
< |
simulation community, these two methods are hard to be implemented |
| 1045 |
< |
correctly and efficiently. Instead, we use a damped and |
| 1046 |
< |
charge-neutralized Coulomb potential method developed by Wolf and |
| 1047 |
< |
his coworkers. The shifted Coulomb potential for particle $i$ and |
| 1048 |
< |
particle $j$ at distance $r_{rj}$ is given by: |
| 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 |
| 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]{pbcFig.eps} |
| 1062 |
< |
%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} |
| 1063 |
< |
%\label{introFigure:shiftedCoulomb} |
| 1064 |
< |
%\end{figure} |
| 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 |
|
|
| 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{Powles73}. The experiment |
| 1112 |
< |
result can serve as a criterion to justify the correctness of the |
| 1113 |
< |
theory. Moreover, various equilibrium thermodynamic and structural |
| 1114 |
< |
properties can also be expressed in terms of radial distribution |
| 1115 |
< |
function \cite{allen87:csl}. |
| 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 |
| 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 |
| 1619 |
– |
\begin{equation} |
| 1620 |
– |
\begin{array}{c} |
| 1621 |
– |
L(x + y) = L(x) + L(y) \\ |
| 1622 |
– |
L(ax) = aL(x) \\ |
| 1623 |
– |
L(\dot x) = pL(x) - px(0) \\ |
| 1624 |
– |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
| 1625 |
– |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
| 1626 |
– |
\end{array} |
| 1627 |
– |
\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 |
< |
\[ |
| 1639 |
< |
\begin{array}{c} |
| 1640 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
| 1641 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1642 |
< |
\end{array} |
| 1635 |
< |
\] |
| 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 |
< |
\[ |
| 1645 |
< |
\begin{array}{c} |
| 1646 |
< |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1647 |
< |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1641 |
< |
\end{array} |
| 1642 |
< |
\] |
| 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 $O$. |
