| 67 |
|
\begin{equation}E = T + V. \label{introEquation:energyConservation} |
| 68 |
|
\end{equation} |
| 69 |
|
All of these conserved quantities are important factors to determine |
| 70 |
< |
the quality of numerical integration schemes for rigid bodies |
| 71 |
< |
\cite{Dullweber1997}. |
| 70 |
> |
the quality of numerical integration schemes for rigid |
| 71 |
> |
bodies.\cite{Dullweber1997} |
| 72 |
|
|
| 73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 74 |
|
|
| 178 |
|
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 179 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
| 180 |
|
equation of motion. Due to their symmetrical formula, they are also |
| 181 |
< |
known as the canonical equations of motions \cite{Goldstein2001}. |
| 181 |
> |
known as the canonical equations of motions.\cite{Goldstein2001} |
| 182 |
|
|
| 183 |
|
An important difference between Lagrangian approach and the |
| 184 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
| 188 |
|
Hamiltonian Mechanics is more appropriate for application to |
| 189 |
|
statistical mechanics and quantum mechanics, since it treats the |
| 190 |
|
coordinate and its time derivative as independent variables and it |
| 191 |
< |
only works with 1st-order differential equations\cite{Marion1990}. |
| 191 |
> |
only works with 1st-order differential equations.\cite{Marion1990} |
| 192 |
|
In Newtonian Mechanics, a system described by conservative forces |
| 193 |
|
conserves the total energy |
| 194 |
|
(Eq.~\ref{introEquation:energyConservation}). It follows that |
| 416 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 417 |
|
Hypothesis makes a connection between time average and the ensemble |
| 418 |
|
average. It states that the time average and average over the |
| 419 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
| 419 |
> |
statistical ensemble are identical:\cite{Frenkel1996, Leach2001} |
| 420 |
|
\begin{equation} |
| 421 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 422 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 434 |
|
utilized. Or if the system lends itself to a time averaging |
| 435 |
|
approach, the Molecular Dynamics techniques in |
| 436 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 437 |
< |
choice\cite{Frenkel1996}. |
| 437 |
> |
choice.\cite{Frenkel1996} |
| 438 |
|
|
| 439 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 440 |
|
A variety of numerical integrators have been proposed to simulate |
| 441 |
|
the motions of atoms in MD simulation. They usually begin with |
| 442 |
< |
initial conditions and move the objects in the direction governed |
| 443 |
< |
by the differential equations. However, most of them ignore the |
| 444 |
< |
hidden physical laws contained within the equations. Since 1990, |
| 445 |
< |
geometric integrators, which preserve various phase-flow invariants |
| 446 |
< |
such as symplectic structure, volume and time reversal symmetry, |
| 447 |
< |
were developed to address this issue\cite{Dullweber1997, |
| 448 |
< |
McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which |
| 449 |
< |
happens to be a simple example of symplectic integrator, continues |
| 450 |
< |
to gain popularity in the molecular dynamics community. This fact |
| 451 |
< |
can be partly explained by its geometric nature. |
| 442 |
> |
initial conditions and move the objects in the direction governed by |
| 443 |
> |
the differential equations. However, most of them ignore the hidden |
| 444 |
> |
physical laws contained within the equations. Since 1990, geometric |
| 445 |
> |
integrators, which preserve various phase-flow invariants such as |
| 446 |
> |
symplectic structure, volume and time reversal symmetry, were |
| 447 |
> |
developed to address this issue.\cite{Dullweber1997, McLachlan1998, |
| 448 |
> |
Leimkuhler1999} The velocity Verlet method, which happens to be a |
| 449 |
> |
simple example of symplectic integrator, continues to gain |
| 450 |
> |
popularity in the molecular dynamics community. This fact can be |
| 451 |
> |
partly explained by its geometric nature. |
| 452 |
|
|
| 453 |
|
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 454 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
| 457 |
|
surface of Earth. It seems to be flat locally, but it is round if |
| 458 |
|
viewed as a whole. A \emph{differentiable manifold} (also known as |
| 459 |
|
\emph{smooth manifold}) is a manifold on which it is possible to |
| 460 |
< |
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
| 460 |
> |
apply calculus.\cite{Hirsch1997} A \emph{symplectic manifold} is |
| 461 |
|
defined as a pair $(M, \omega)$ which consists of a |
| 462 |
|
\emph{differentiable manifold} $M$ and a close, non-degenerate, |
| 463 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 464 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
| 465 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 466 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 467 |
< |
$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in |
| 467 |
> |
$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in |
| 468 |
|
vector field is an example of symplectic form. One of the |
| 469 |
|
motivations to study \emph{symplectic manifolds} in Hamiltonian |
| 470 |
|
Mechanics is that a symplectic manifold can represent all possible |
| 471 |
|
configurations of the system and the phase space of the system can |
| 472 |
< |
be described by it's cotangent bundle\cite{Jost2002}. Every |
| 472 |
> |
be described by it's cotangent bundle.\cite{Jost2002} Every |
| 473 |
|
symplectic manifold is even dimensional. For instance, in Hamilton |
| 474 |
|
equations, coordinate and momentum always appear in pairs. |
| 475 |
|
|
| 496 |
|
\label{introEquation:compactHamiltonian} |
| 497 |
|
\end{equation}In this case, $f$ is |
| 498 |
|
called a \emph{Hamiltonian vector field}. Another generalization of |
| 499 |
< |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
| 499 |
> |
Hamiltonian dynamics is Poisson Dynamics,\cite{Olver1986} |
| 500 |
|
\begin{equation} |
| 501 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 502 |
|
\end{equation} |
| 503 |
< |
The most obvious change being that matrix $J$ now depends on $x$. |
| 503 |
> |
where the most obvious change being that matrix $J$ now depends on |
| 504 |
> |
$x$. |
| 505 |
|
|
| 506 |
|
\subsection{\label{introSection:exactFlow}Exact Propagator} |
| 507 |
|
|
| 621 |
|
Generating functions\cite{Channell1990} tend to lead to methods |
| 622 |
|
which are cumbersome and difficult to use. In dissipative systems, |
| 623 |
|
variational methods can capture the decay of energy |
| 624 |
< |
accurately\cite{Kane2000}. Since they are geometrically unstable |
| 624 |
> |
accurately.\cite{Kane2000} Since they are geometrically unstable |
| 625 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 626 |
< |
methods are not suitable for Hamiltonian system. Recently, various |
| 627 |
< |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
| 628 |
< |
have been developed to overcome this instability. However, due to |
| 629 |
< |
computational penalty involved in implementing the Runge-Kutta |
| 630 |
< |
methods, they have not attracted much attention from the Molecular |
| 631 |
< |
Dynamics community. Instead, splitting methods have been widely |
| 632 |
< |
accepted since they exploit natural decompositions of the |
| 633 |
< |
system\cite{Tuckerman1992, McLachlan1998}. |
| 626 |
> |
methods are not suitable for Hamiltonian |
| 627 |
> |
system.\cite{Cartwright1992} Recently, various high-order explicit |
| 628 |
> |
Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to |
| 629 |
> |
overcome this instability. However, due to computational penalty |
| 630 |
> |
involved in implementing the Runge-Kutta methods, they have not |
| 631 |
> |
attracted much attention from the Molecular Dynamics community. |
| 632 |
> |
Instead, splitting methods have been widely accepted since they |
| 633 |
> |
exploit natural decompositions of the system.\cite{McLachlan1998, |
| 634 |
> |
Tuckerman1992} |
| 635 |
|
|
| 636 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 637 |
|
|
| 654 |
|
problem. If $H_1$ and $H_2$ can be integrated using exact |
| 655 |
|
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
| 656 |
|
simple first order expression is then given by the Lie-Trotter |
| 657 |
< |
formula |
| 657 |
> |
formula\cite{Trotter1959} |
| 658 |
|
\begin{equation} |
| 659 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 660 |
|
\label{introEquation:firstOrderSplitting} |
| 675 |
|
The Lie-Trotter |
| 676 |
|
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
| 677 |
|
local errors proportional to $h^2$, while the Strang splitting gives |
| 678 |
< |
a second-order decomposition, |
| 678 |
> |
a second-order decomposition,\cite{Strang1968} |
| 679 |
|
\begin{equation} |
| 680 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 681 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 750 |
|
|
| 751 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 752 |
|
|
| 753 |
< |
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 754 |
< |
local error of a splitting method in terms of the commutator of the |
| 755 |
< |
operators(Eq.~\ref{introEquation:exponentialOperator}) associated with |
| 756 |
< |
the sub-propagator. For operators $hX$ and $hY$ which are associated |
| 757 |
< |
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 753 |
> |
The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used |
| 754 |
> |
to determine the local error of a splitting method in terms of the |
| 755 |
> |
commutator of the |
| 756 |
> |
operators(Eq.~\ref{introEquation:exponentialOperator}) associated |
| 757 |
> |
with the sub-propagator. For operators $hX$ and $hY$ which are |
| 758 |
> |
associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we |
| 759 |
> |
have |
| 760 |
|
\begin{equation} |
| 761 |
|
\exp (hX + hY) = \exp (hZ) |
| 762 |
|
\end{equation} |
| 786 |
|
\end{equation} |
| 787 |
|
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 788 |
|
order methods. Yoshida proposed an elegant way to compose higher |
| 789 |
< |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 789 |
> |
order methods based on symmetric splitting.\cite{Yoshida1990} Given |
| 790 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 791 |
|
fourth-order symmetric method can be constructed by composing, |
| 792 |
|
\[ |
| 947 |
|
%cutoff and minimum image convention |
| 948 |
|
Another important technique to improve the efficiency of force |
| 949 |
|
evaluation is to apply spherical cutoffs where particles farther |
| 950 |
< |
than a predetermined distance are not included in the calculation |
| 951 |
< |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 952 |
< |
discontinuity in the potential energy curve. Fortunately, one can |
| 950 |
> |
than a predetermined distance are not included in the |
| 951 |
> |
calculation.\cite{Frenkel1996} The use of a cutoff radius will cause |
| 952 |
> |
a discontinuity in the potential energy curve. Fortunately, one can |
| 953 |
|
shift a simple radial potential to ensure the potential curve go |
| 954 |
|
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 955 |
|
well for Lennard-Jones interaction because of its short range |
| 958 |
|
in simulations. The Ewald summation, in which the slowly decaying |
| 959 |
|
Coulomb potential is transformed into direct and reciprocal sums |
| 960 |
|
with rapid and absolute convergence, has proved to minimize the |
| 961 |
< |
periodicity artifacts in liquid simulations. Taking advantage |
| 962 |
< |
of fast Fourier transform (FFT) techniques for calculating discrete Fourier |
| 963 |
< |
transforms, the particle mesh-based |
| 961 |
> |
periodicity artifacts in liquid simulations. Taking advantage of |
| 962 |
> |
fast Fourier transform (FFT) techniques for calculating discrete |
| 963 |
> |
Fourier transforms, the particle mesh-based |
| 964 |
|
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 965 |
|
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
| 966 |
|
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
| 970 |
|
simulation community, these two methods are difficult to implement |
| 971 |
|
correctly and efficiently. Instead, we use a damped and |
| 972 |
|
charge-neutralized Coulomb potential method developed by Wolf and |
| 973 |
< |
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
| 973 |
> |
his coworkers.\cite{Wolf1999} The shifted Coulomb potential for |
| 974 |
|
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 975 |
|
\begin{equation} |
| 976 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 1033 |
|
function}, is of most fundamental importance to liquid theory. |
| 1034 |
|
Experimentally, pair distribution functions can be gathered by |
| 1035 |
|
Fourier transforming raw data from a series of neutron diffraction |
| 1036 |
< |
experiments and integrating over the surface factor |
| 1037 |
< |
\cite{Powles1973}. The experimental results can serve as a criterion |
| 1038 |
< |
to justify the correctness of a liquid model. Moreover, various |
| 1039 |
< |
equilibrium thermodynamic and structural properties can also be |
| 1040 |
< |
expressed in terms of the radial distribution function |
| 1041 |
< |
\cite{Allen1987}. The pair distribution functions $g(r)$ gives the |
| 1042 |
< |
probability that a particle $i$ will be located at a distance $r$ |
| 1043 |
< |
from a another particle $j$ in the system |
| 1036 |
> |
experiments and integrating over the surface |
| 1037 |
> |
factor.\cite{Powles1973} The experimental results can serve as a |
| 1038 |
> |
criterion to justify the correctness of a liquid model. Moreover, |
| 1039 |
> |
various equilibrium thermodynamic and structural properties can also |
| 1040 |
> |
be expressed in terms of the radial distribution |
| 1041 |
> |
function.\cite{Allen1987} The pair distribution functions $g(r)$ |
| 1042 |
> |
gives the probability that a particle $i$ will be located at a |
| 1043 |
> |
distance $r$ from a another particle $j$ in the system |
| 1044 |
|
\begin{equation} |
| 1045 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1046 |
|
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
| 1101 |
|
movement of the objects in 3D gaming engines or other physics |
| 1102 |
|
simulators is governed by rigid body dynamics. In molecular |
| 1103 |
|
simulations, rigid bodies are used to simplify protein-protein |
| 1104 |
< |
docking studies\cite{Gray2003}. |
| 1104 |
> |
docking studies.\cite{Gray2003} |
| 1105 |
|
|
| 1106 |
|
It is very important to develop stable and efficient methods to |
| 1107 |
|
integrate the equations of motion for orientational degrees of |
| 1113 |
|
angles can overcome this difficulty\cite{Barojas1973}, the |
| 1114 |
|
computational penalty and the loss of angular momentum conservation |
| 1115 |
|
still remain. A singularity-free representation utilizing |
| 1116 |
< |
quaternions was developed by Evans in 1977\cite{Evans1977}. |
| 1116 |
> |
quaternions was developed by Evans in 1977.\cite{Evans1977} |
| 1117 |
|
Unfortunately, this approach used a nonseparable Hamiltonian |
| 1118 |
|
resulting from the quaternion representation, which prevented the |
| 1119 |
|
symplectic algorithm from being utilized. Another different approach |
| 1122 |
|
deriving from potential energy and constraint forces which are used |
| 1123 |
|
to guarantee the rigidness. However, due to their iterative nature, |
| 1124 |
|
the SHAKE and Rattle algorithms also converge very slowly when the |
| 1125 |
< |
number of constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1125 |
> |
number of constraints increases.\cite{Ryckaert1977, Andersen1983} |
| 1126 |
|
|
| 1127 |
|
A break-through in geometric literature suggests that, in order to |
| 1128 |
|
develop a long-term integration scheme, one should preserve the |
| 1132 |
|
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1133 |
|
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 1134 |
|
An alternative method using the quaternion representation was |
| 1135 |
< |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
| 1135 |
> |
developed by Omelyan.\cite{Omelyan1998} However, both of these |
| 1136 |
|
methods are iterative and inefficient. In this section, we descibe a |
| 1137 |
|
symplectic Lie-Poisson integrator for rigid bodies developed by |
| 1138 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1139 |
|
|
| 1140 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1141 |
< |
The Hamiltonian of a rigid body is given by |
| 1141 |
> |
The Hamiltonian of a rigid body is given by |
| 1142 |
|
\begin{equation} |
| 1143 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1144 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1252 |
|
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
| 1253 |
|
Lagrange multiplier $\Lambda$ is absent from the equations of |
| 1254 |
|
motion. This unique property eliminates the requirement of |
| 1255 |
< |
iterations which can not be avoided in other methods\cite{Kol1997, |
| 1256 |
< |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
| 1255 |
> |
iterations which can not be avoided in other methods.\cite{Kol1997, |
| 1256 |
> |
Omelyan1998} Applying the hat-map isomorphism, we obtain the |
| 1257 |
|
equation of motion for angular momentum in the body frame |
| 1258 |
|
\begin{equation} |
| 1259 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1359 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1360 |
|
conserved quantity in Poisson system. We can easily verify that the |
| 1361 |
|
norm of the angular momentum, $\parallel \pi |
| 1362 |
< |
\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let $F(\pi ) = S(\frac{{\parallel |
| 1362 |
> |
\parallel$, is a \emph{Casimir}.\cite{McLachlan1993} Let $F(\pi ) = S(\frac{{\parallel |
| 1363 |
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1364 |
|
then by the chain rule |
| 1365 |
|
\[ |
| 1380 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1381 |
|
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations |
| 1382 |
|
of motion corresponding to potential energy and kinetic energy are |
| 1383 |
< |
listed in Table~\ref{introTable:rbEquations}. |
| 1383 |
> |
listed in Table~\ref{introTable:rbEquations}. |
| 1384 |
|
\begin{table} |
| 1385 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1386 |
|
\label{introTable:rbEquations} |
