| 337 |
|
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
| 338 |
|
distribution, |
| 339 |
|
\begin{equation} |
| 340 |
< |
\rho \propto e^{ - \beta H} |
| 340 |
> |
\rho \propto e^{ - \beta H}. |
| 341 |
|
\label{introEquation:densityAndHamiltonian} |
| 342 |
|
\end{equation} |
| 343 |
|
|
| 349 |
|
If this region is small enough, the density $\rho$ can be regarded |
| 350 |
|
as uniform over the whole integral. Thus, the number of phase points |
| 351 |
|
inside this region is given by, |
| 352 |
< |
\begin{equation} |
| 353 |
< |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
| 354 |
< |
dp_1 } ..dp_f. |
| 355 |
< |
\end{equation} |
| 356 |
< |
|
| 357 |
< |
\begin{equation} |
| 358 |
< |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
| 352 |
> |
\begin{eqnarray} |
| 353 |
> |
\delta N &=& \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f,\\ |
| 354 |
> |
\frac{{d(\delta N)}}{{dt}} &=& \frac{{d\rho }}{{dt}}\delta v + \rho |
| 355 |
|
\frac{d}{{dt}}(\delta v) = 0. |
| 356 |
< |
\end{equation} |
| 356 |
> |
\end{eqnarray} |
| 357 |
|
With the help of the stationary assumption |
| 358 |
|
(Eq.~\ref{introEquation:stationary}), we obtain the principle of |
| 359 |
|
\emph{conservation of volume in phase space}, |
| 446 |
|
popularity in the molecular dynamics community. This fact can be |
| 447 |
|
partly explained by its geometric nature. |
| 448 |
|
|
| 449 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 449 |
> |
\subsection{\label{introSection:symplecticManifold}Manifolds and Bundles} |
| 450 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
| 451 |
|
locally like Euclidean space, but when viewed globally, it may have |
| 452 |
|
more complicated structure. A good example of manifold is the |
| 461 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 462 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 463 |
|
$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in |
| 464 |
< |
vector field is an example of symplectic form. One of the |
| 465 |
< |
motivations to study \emph{symplectic manifolds} in Hamiltonian |
| 466 |
< |
Mechanics is that a symplectic manifold can represent all possible |
| 467 |
< |
configurations of the system and the phase space of the system can |
| 468 |
< |
be described by it's cotangent bundle.\cite{Jost2002} Every |
| 469 |
< |
symplectic manifold is even dimensional. For instance, in Hamilton |
| 470 |
< |
equations, coordinate and momentum always appear in pairs. |
| 464 |
> |
vector field is an example of symplectic form. |
| 465 |
> |
Given vector spaces $V$ and $W$ over same field $F$, $f: V \to W$ is a linear transformation if |
| 466 |
> |
\begin{eqnarray*} |
| 467 |
> |
f(x+y) & = & f(x) + f(y) \\ |
| 468 |
> |
f(ax) & = & af(x) |
| 469 |
> |
\end{eqnarray*} |
| 470 |
> |
are always satisfied for any two vectors $x$ and $y$ in $V$ and any scalar $a$ in $F$. One can define the dual vector space $V^*$ of $V$ if any two built-in linear transformations $\phi$ and $\psi$ in $V^*$ satisfy the following definition of addition and scalar multiplication: |
| 471 |
> |
\begin{eqnarray*} |
| 472 |
> |
(\phi+\psi)(x) & = & \phi(x)+\psi(x) \\ |
| 473 |
> |
(a\phi)(x) & = & a \phi(x) |
| 474 |
> |
\end{eqnarray*} |
| 475 |
> |
for all $a$ in $F$ and $x$ in $V$. For a manifold $M$, one can define a tangent vector of a tangent space $TM_q$ at every point $q$ |
| 476 |
> |
\begin{equation} |
| 477 |
> |
\dot q = \mathop {\lim }\limits_{t \to 0} \frac{{\phi (t) - \phi (0)}}{t} |
| 478 |
> |
\end{equation} |
| 479 |
> |
where $\phi(0)=q$ and $\phi(t) \in M$. One may also define a cotangent space $T^*M_q$ as the dual space of the tangent space $TM_q$. The tangent space and the cotangent space are isomorphic to each other, since they are both real vector spaces with same dimension. |
| 480 |
> |
The union of tangent spaces at every point of $M$ is called the tangent bundle of $M$ and is denoted by $TM$, while cotangent bundle $T^*M$ is defined as the union of the cotangent spaces to $M$.\cite{Jost2002} For a Hamiltonian system with configuration manifold $V$, the $(q,\dot q)$ phase space is the tangent bundle of the configuration manifold $V$, while the cotangent bundle is represented by $(q,p)$. |
| 481 |
|
|
| 482 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 483 |
|
|
| 534 |
|
\begin{equation} |
| 535 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 536 |
|
\end{equation} |
| 531 |
– |
The exact propagator can also be written as an operator, |
| 532 |
– |
\begin{equation} |
| 533 |
– |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 534 |
– |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 535 |
– |
\label{introEquation:exponentialOperator} |
| 536 |
– |
\end{equation} |
| 537 |
|
In most cases, it is not easy to find the exact propagator |
| 538 |
|
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
| 539 |
|
which is usually called an integrator. The order of an integrator |
| 623 |
|
variational methods can capture the decay of energy |
| 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{McLachlan1998, Tuckerman1992} |
| 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} |
| 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 |
| 756 |
> |
operators 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 |
| 989 |
|
\label{introFigure:shiftedCoulomb} |
| 990 |
|
\end{figure} |
| 991 |
|
|
| 991 |
– |
%multiple time step |
| 992 |
– |
|
| 992 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 993 |
|
|
| 994 |
< |
Recently, advanced visualization techniques have been applied to |
| 996 |
< |
monitor the motions of molecules. Although the dynamics of the |
| 997 |
< |
system can be described qualitatively from animation, quantitative |
| 998 |
< |
trajectory analysis is more useful. According to the principles of |
| 994 |
> |
According to the principles of |
| 995 |
|
Statistical Mechanics in |
| 996 |
|
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 997 |
|
thermodynamic properties, analyze fluctuations of structural |
| 1192 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1193 |
|
\] |
| 1194 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1195 |
< |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1195 |
> |
the rigid body, its corresponding lab fixed vector $X_i^{lab}$ is |
| 1196 |
|
given as |
| 1197 |
|
\begin{equation} |
| 1198 |
|
X_i^{lab} = Q X_i + q. |
| 1249 |
|
motion. This unique property eliminates the requirement of |
| 1250 |
|
iterations which can not be avoided in other methods.\cite{Kol1997, |
| 1251 |
|
Omelyan1998} Applying the hat-map isomorphism, we obtain the |
| 1252 |
< |
equation of motion for angular momentum in the body frame |
| 1252 |
> |
equation of motion for angular momentum |
| 1253 |
|
\begin{equation} |
| 1254 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1255 |
|
F_i (r,Q)} \right) \times X_i }. |
| 1620 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1621 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1622 |
|
\] |
| 1623 |
< |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1623 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes the |
| 1624 |
> |
Langevin equation |
| 1625 |
|
\begin{equation} |
| 1626 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1627 |
< |
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1627 |
> |
x(t) + R(t) \label{introEquation:LangevinEquation}. |
| 1628 |
|
\end{equation} |
| 1629 |
< |
which is known as the Langevin equation. The static friction |
| 1630 |
< |
coefficient $\xi _0$ can either be calculated from spectral density |
| 1631 |
< |
or be determined by Stokes' law for regular shaped particles. A |
| 1632 |
< |
brief review on calculating friction tensors for arbitrary shaped |
| 1633 |
< |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1629 |
> |
The static friction coefficient $\xi _0$ can either be calculated |
| 1630 |
> |
from spectral density or be determined by Stokes' law for regular |
| 1631 |
> |
shaped particles. A brief review on calculating friction tensors for |
| 1632 |
> |
arbitrary shaped particles is given in |
| 1633 |
> |
Sec.~\ref{introSection:frictionTensor}. |
| 1634 |
|
|
| 1635 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1636 |
|
|
