| 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 |
|
|
| 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 |
| 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 |
