| 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{}. The velocity verlet method, which happens to be a |
| 502 |
> |
simple example of symplectic integrator, continues to gain its |
| 503 |
> |
popularity in molecular dynamics community. This fact can be partly |
| 504 |
> |
explained by its geometric nature. |
| 505 |
|
|
| 506 |
|
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 507 |
|
A \emph{manifold} is an abstract mathematical space. It locally |
| 708 |
|
implementing the Runge-Kutta methods, they do not attract too much |
| 709 |
|
attention from Molecular Dynamics community. Instead, splitting have |
| 710 |
|
been widely accepted since they exploit natural decompositions of |
| 711 |
< |
the system\cite{Tuckerman92}. |
| 711 |
> |
the system\cite{Tuckerman1992}. |
| 712 |
|
|
| 713 |
|
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
| 714 |
|
|
| 846 |
|
\] |
| 847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
|
can obtain |
| 849 |
< |
\begin{equation} |
| 850 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 851 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 852 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 853 |
< |
\ldots ) |
| 854 |
< |
\end{equation} |
| 849 |
> |
\begin{eqnarray*} |
| 850 |
> |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
| 851 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 852 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
| 853 |
> |
\end{eqnarray*} |
| 854 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 855 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 856 |
|
procedure can be applied to general splitting, of the form |
| 858 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 859 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 860 |
|
\end{equation} |
| 861 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
| 861 |
> |
Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher |
| 862 |
|
order method. Yoshida proposed an elegant way to compose higher |
| 863 |
|
order methods based on symmetric splitting. Given a symmetric second |
| 864 |
|
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
| 1021 |
|
evaluation is to apply cutoff where particles farther than a |
| 1022 |
|
predetermined distance, are not included in the calculation |
| 1023 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 1024 |
< |
discontinuity in the potential energy curve |
| 1025 |
< |
(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the |
| 1026 |
< |
potential to ensure the potential curve go smoothly to zero at the |
| 1027 |
< |
cutoff radius. Cutoff strategy works pretty well for Lennard-Jones |
| 1028 |
< |
interaction because of its short range nature. However, simply |
| 1029 |
< |
truncating the electrostatic interaction with the use of cutoff has |
| 1030 |
< |
been shown to lead to severe artifacts in simulations. Ewald |
| 1031 |
< |
summation, in which the slowly conditionally convergent Coulomb |
| 1032 |
< |
potential is transformed into direct and reciprocal sums with rapid |
| 1033 |
< |
and absolute convergence, has proved to minimize the periodicity |
| 1034 |
< |
artifacts in liquid simulations. Taking the advantages of the fast |
| 1035 |
< |
Fourier transform (FFT) for calculating discrete Fourier transforms, |
| 1036 |
< |
the particle mesh-based methods are accelerated from $O(N^{3/2})$ to |
| 1037 |
< |
$O(N logN)$. An alternative approach is \emph{fast multipole |
| 1038 |
< |
method}, which treats Coulombic interaction exactly at short range, |
| 1039 |
< |
and approximate the potential at long range through multipolar |
| 1040 |
< |
expansion. In spite of their wide acceptances at the molecular |
| 1041 |
< |
simulation community, these two methods are hard to be implemented |
| 1042 |
< |
correctly and efficiently. Instead, we use a damped and |
| 1043 |
< |
charge-neutralized Coulomb potential method developed by Wolf and |
| 1044 |
< |
his coworkers. The shifted Coulomb potential for particle $i$ and |
| 1046 |
< |
particle $j$ at distance $r_{rj}$ is given by: |
| 1024 |
> |
discontinuity in the potential energy curve. Fortunately, one can |
| 1025 |
> |
shift the potential to ensure the potential curve go smoothly to |
| 1026 |
> |
zero at the cutoff radius. Cutoff strategy works pretty well for |
| 1027 |
> |
Lennard-Jones interaction because of its short range nature. |
| 1028 |
> |
However, simply truncating the electrostatic interaction with the |
| 1029 |
> |
use of cutoff has been shown to lead to severe artifacts in |
| 1030 |
> |
simulations. Ewald summation, in which the slowly conditionally |
| 1031 |
> |
convergent Coulomb potential is transformed into direct and |
| 1032 |
> |
reciprocal sums with rapid and absolute convergence, has proved to |
| 1033 |
> |
minimize the periodicity artifacts in liquid simulations. Taking the |
| 1034 |
> |
advantages of the fast Fourier transform (FFT) for calculating |
| 1035 |
> |
discrete Fourier transforms, the particle mesh-based methods are |
| 1036 |
> |
accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative |
| 1037 |
> |
approach is \emph{fast multipole method}, which treats Coulombic |
| 1038 |
> |
interaction exactly at short range, and approximate the potential at |
| 1039 |
> |
long range through multipolar expansion. In spite of their wide |
| 1040 |
> |
acceptances at the molecular simulation community, these two methods |
| 1041 |
> |
are hard to be implemented correctly and efficiently. Instead, we |
| 1042 |
> |
use a damped and charge-neutralized Coulomb potential method |
| 1043 |
> |
developed by Wolf and his coworkers. The shifted Coulomb potential |
| 1044 |
> |
for particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 1045 |
|
\begin{equation} |
| 1046 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 1047 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 1102 |
|
function}, is of most fundamental importance to liquid-state theory. |
| 1103 |
|
Pair distribution function can be gathered by Fourier transforming |
| 1104 |
|
raw data from a series of neutron diffraction experiments and |
| 1105 |
< |
integrating over the surface factor \cite{Powles73}. The experiment |
| 1106 |
< |
result can serve as a criterion to justify the correctness of the |
| 1107 |
< |
theory. Moreover, various equilibrium thermodynamic and structural |
| 1108 |
< |
properties can also be expressed in terms of radial distribution |
| 1109 |
< |
function \cite{allen87:csl}. |
| 1105 |
> |
integrating over the surface factor \cite{Powles1973}. The |
| 1106 |
> |
experiment result can serve as a criterion to justify the |
| 1107 |
> |
correctness of the theory. Moreover, various equilibrium |
| 1108 |
> |
thermodynamic and structural properties can also be expressed in |
| 1109 |
> |
terms of radial distribution function \cite{Allen1987}. |
| 1110 |
|
|
| 1111 |
|
A pair distribution functions $g(r)$ gives the probability that a |
| 1112 |
|
particle $i$ will be located at a distance $r$ from a another |
| 1184 |
|
movement of the objects in 3D gaming engine or other physics |
| 1185 |
|
simulator is governed by the rigid body dynamics. In molecular |
| 1186 |
|
simulation, rigid body is used to simplify the model in |
| 1187 |
< |
protein-protein docking study{\cite{Gray03}}. |
| 1187 |
> |
protein-protein docking study{\cite{Gray2003}}. |
| 1188 |
|
|
| 1189 |
|
It is very important to develop stable and efficient methods to |
| 1190 |
|
integrate the equations of motion of orientational degrees of |
| 1261 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1262 |
|
We can use constraint force provided by lagrange multiplier on the |
| 1263 |
|
normal manifold to keep the motion on constraint space. Or we can |
| 1264 |
< |
simply evolve the system in constraint manifold. The two method are |
| 1265 |
< |
proved to be equivalent. The holonomic constraint and equations of |
| 1266 |
< |
motions define a constraint manifold for rigid body |
| 1264 |
> |
simply evolve the system in constraint manifold. These two methods |
| 1265 |
> |
are proved to be equivalent. The holonomic constraint and equations |
| 1266 |
> |
of motions define a constraint manifold for rigid body |
| 1267 |
|
\[ |
| 1268 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1269 |
|
\right\}. |
| 1476 |
|
\] |
| 1477 |
|
The equations of motion corresponding to potential energy and |
| 1478 |
|
kinetic energy are listed in the below table, |
| 1479 |
+ |
\begin{table} |
| 1480 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
| 1481 |
|
\begin{center} |
| 1482 |
|
\begin{tabular}{|l|l|} |
| 1483 |
|
\hline |
| 1490 |
|
\hline |
| 1491 |
|
\end{tabular} |
| 1492 |
|
\end{center} |
| 1493 |
< |
A second-order symplectic method is now obtained by the composition |
| 1494 |
< |
of the flow maps, |
| 1493 |
> |
\end{table} |
| 1494 |
> |
A second-order symplectic method is now obtained by the |
| 1495 |
> |
composition of the flow maps, |
| 1496 |
|
\[ |
| 1497 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1498 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1788 |
|
when the system become more and more complicate. Instead, various |
| 1789 |
|
approaches based on hydrodynamics have been developed to calculate |
| 1790 |
|
the friction coefficients. The friction effect is isotropic in |
| 1791 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1792 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
| 1791 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 1792 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 1793 |
|
\[ |
| 1794 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 1795 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1889 |
|
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1890 |
|
from all possible ellipsoidal space, $r$-space, to all possible |
| 1891 |
|
combination of rotational diffusion coefficients, $D$-space is not |
| 1892 |
< |
unique\cite{Wegener79} as well as the intrinsic coupling between |
| 1892 |
> |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 1893 |
|
translational and rotational motion of rigid body\cite{}, general |
| 1894 |
|
ellipsoid is not always suitable for modeling arbitrarily shaped |
| 1895 |
|
rigid molecule. A number of studies have been devoted to determine |
