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 |
|
|
831 |
|
error of splitting method in terms of commutator of the |
832 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
833 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
834 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
834 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
835 |
|
\begin{equation} |
836 |
|
\exp (hX + hY) = \exp (hZ) |
837 |
|
\end{equation} |
847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
848 |
|
can obtain |
849 |
|
\begin{eqnarray*} |
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 ) |
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 |
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 |
1236 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1237 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
1238 |
|
\begin{equation} |
1239 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
1239 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1240 |
|
\end{equation} |
1241 |
|
which is used to ensure rotation matrix's orthogonality. |
1242 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
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 |