| 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}, |
| 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. |
| 465 |
< |
Given vector spaces $V$ and $W$ over same field $F$, $f: V \to W$ is a linear transformation if |
| 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) |
| 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*} |
| 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} |
| 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 $(q,p)$. |
| 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 |
|
|
| 988 |
|
illustration of shifted Coulomb potential.} |
| 989 |
|
\label{introFigure:shiftedCoulomb} |
| 990 |
|
\end{figure} |
| 995 |
– |
|
| 996 |
– |
%multiple time step |
| 991 |
|
|
| 992 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 993 |
|
|
| 994 |
< |
Recently, advanced visualization techniques have been applied to |
| 1001 |
< |
monitor the motions of molecules. Although the dynamics of the |
| 1002 |
< |
system can be described qualitatively from animation, quantitative |
| 1003 |
< |
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 |
| 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 |
|
|
