| 281 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
| 282 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 283 |
|
\label{introEquation:ensembelAverage} |
| 284 |
– |
\end{equation} |
| 285 |
– |
|
| 286 |
– |
There are several different types of ensembles with different |
| 287 |
– |
statistical characteristics. As a function of macroscopic |
| 288 |
– |
parameters, such as temperature \textit{etc}, the partition function |
| 289 |
– |
can be used to describe the statistical properties of a system in |
| 290 |
– |
thermodynamic equilibrium. As an ensemble of systems, each of which |
| 291 |
– |
is known to be thermally isolated and conserve energy, the |
| 292 |
– |
Microcanonical ensemble (NVE) has a partition function like, |
| 293 |
– |
\begin{equation} |
| 294 |
– |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
| 295 |
– |
\end{equation} |
| 296 |
– |
A canonical ensemble (NVT) is an ensemble of systems, each of which |
| 297 |
– |
can share its energy with a large heat reservoir. The distribution |
| 298 |
– |
of the total energy amongst the possible dynamical states is given |
| 299 |
– |
by the partition function, |
| 300 |
– |
\begin{equation} |
| 301 |
– |
\Omega (N,V,T) = e^{ - \beta A}. |
| 302 |
– |
\label{introEquation:NVTPartition} |
| 284 |
|
\end{equation} |
| 304 |
– |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 305 |
– |
TS$. Since most experiments are carried out under constant pressure |
| 306 |
– |
condition, the isothermal-isobaric ensemble (NPT) plays a very |
| 307 |
– |
important role in molecular simulations. The isothermal-isobaric |
| 308 |
– |
ensemble allow the system to exchange energy with a heat bath of |
| 309 |
– |
temperature $T$ and to change the volume as well. Its partition |
| 310 |
– |
function is given as |
| 311 |
– |
\begin{equation} |
| 312 |
– |
\Delta (N,P,T) = - e^{\beta G}. |
| 313 |
– |
\label{introEquation:NPTPartition} |
| 314 |
– |
\end{equation} |
| 315 |
– |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
| 285 |
|
|
| 286 |
|
\subsection{\label{introSection:liouville}Liouville's theorem} |
| 287 |
|
|
| 1139 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1140 |
|
\label{introEquation:RBHamiltonian} |
| 1141 |
|
\end{equation} |
| 1142 |
< |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 1143 |
< |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 1144 |
< |
$J$, a diagonal matrix, is defined by |
| 1142 |
> |
Here, $q$ and $Q$ are the position vector and rotation matrix for |
| 1143 |
> |
the rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , |
| 1144 |
> |
and $J$, a diagonal matrix, is defined by |
| 1145 |
|
\[ |
| 1146 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1147 |
|
\] |
| 1151 |
|
\begin{equation} |
| 1152 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1153 |
|
\end{equation} |
| 1154 |
< |
which is used to ensure rotation matrix's unitarity. Differentiating |
| 1155 |
< |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
| 1187 |
< |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1188 |
< |
\begin{equation} |
| 1189 |
< |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1190 |
< |
\label{introEquation:RBFirstOrderConstraint} |
| 1191 |
< |
\end{equation} |
| 1192 |
< |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1154 |
> |
which is used to ensure the rotation matrix's unitarity. Using |
| 1155 |
> |
Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1156 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1157 |
|
the equations of motion, |
| 1158 |
|
\begin{eqnarray} |
| 1161 |
|
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
| 1162 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1163 |
|
\end{eqnarray} |
| 1164 |
+ |
Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and |
| 1165 |
+ |
using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1166 |
+ |
\begin{equation} |
| 1167 |
+ |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1168 |
+ |
\label{introEquation:RBFirstOrderConstraint} |
| 1169 |
+ |
\end{equation} |
| 1170 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1171 |
|
We can use a constraint force provided by a Lagrange multiplier on |
| 1172 |
< |
the normal manifold to keep the motion on constraint space. Or we |
| 1173 |
< |
can simply evolve the system on the constraint manifold. These two |
| 1174 |
< |
methods have been proved to be equivalent. The holonomic constraint |
| 1175 |
< |
and equations of motions define a constraint manifold for rigid |
| 1176 |
< |
bodies |
| 1172 |
> |
the normal manifold to keep the motion on the constraint space. Or |
| 1173 |
> |
we can simply evolve the system on the constraint manifold. These |
| 1174 |
> |
two methods have been proved to be equivalent. The holonomic |
| 1175 |
> |
constraint and equations of motions define a constraint manifold for |
| 1176 |
> |
rigid bodies |
| 1177 |
|
\[ |
| 1178 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1179 |
|
\right\}. |
| 1180 |
|
\] |
| 1181 |
< |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1182 |
< |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
| 1183 |
< |
rotation group $SO(3)$. However, it turns out that under symplectic |
| 1184 |
< |
transformation, the cotangent space and the phase space are |
| 1216 |
< |
diffeomorphic. By introducing |
| 1181 |
> |
Unfortunately, this constraint manifold is not $T^* SO(3)$ which is |
| 1182 |
> |
a symplectic manifold on Lie rotation group $SO(3)$. However, it |
| 1183 |
> |
turns out that under symplectic transformation, the cotangent space |
| 1184 |
> |
and the phase space are diffeomorphic. By introducing |
| 1185 |
|
\[ |
| 1186 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1187 |
|
\] |
| 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 on body frame |
| 1252 |
> |
equation of motion for angular momentum in the body frame |
| 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 }. |
| 1262 |
|
\] |
| 1263 |
|
|
| 1264 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1265 |
< |
Lie-Poisson Integrator for Free Rigid Body} |
| 1265 |
> |
Lie-Poisson Integrator for Free Rigid Bodies} |
| 1266 |
|
|
| 1267 |
|
If there are no external forces exerted on the rigid body, the only |
| 1268 |
|
contribution to the rotational motion is from the kinetic energy |
| 1314 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1315 |
|
\] |
| 1316 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1317 |
< |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
| 1318 |
< |
propagator, |
| 1319 |
< |
\[ |
| 1320 |
< |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1321 |
< |
). |
| 1322 |
< |
\] |
| 1323 |
< |
The propagator maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1317 |
> |
tR_1 }$, we can use the Cayley transformation to obtain a |
| 1318 |
> |
single-aixs propagator, |
| 1319 |
> |
\begin{eqnarray*} |
| 1320 |
> |
e^{\Delta tR_1 } & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta |
| 1321 |
> |
tR_1 ) \\ |
| 1322 |
> |
% |
| 1323 |
> |
& \approx & \left( \begin{array}{ccc} |
| 1324 |
> |
1 & 0 & 0 \\ |
| 1325 |
> |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
| 1326 |
> |
\theta^2 / 4} \\ |
| 1327 |
> |
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
| 1328 |
> |
\theta^2 / 4} |
| 1329 |
> |
\end{array} |
| 1330 |
> |
\right). |
| 1331 |
> |
\end{eqnarray*} |
| 1332 |
> |
The propagators for $T_2^r$ and $T_3^r$ can be found in the same |
| 1333 |
|
manner. In order to construct a second-order symplectic method, we |
| 1334 |
|
split the angular kinetic Hamiltonian function into five terms |
| 1335 |
|
\[ |
| 1355 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1356 |
|
conserved quantity in Poisson system. We can easily verify that the |
| 1357 |
|
norm of the angular momentum, $\parallel \pi |
| 1358 |
< |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1358 |
> |
\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1359 |
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1360 |
|
then by the chain rule |
| 1361 |
|
\[ |
| 1425 |
|
\begin{eqnarray} |
| 1426 |
|
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
| 1427 |
|
& & \circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \notag\\ |
| 1428 |
< |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1428 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} . |
| 1429 |
|
\label{introEquation:overallRBFlowMaps} |
| 1430 |
|
\end{eqnarray} |
| 1431 |
|
|
| 1502 |
|
differential equations,the Laplace transform is the appropriate tool |
| 1503 |
|
to solve this problem. The basic idea is to transform the difficult |
| 1504 |
|
differential equations into simple algebra problems which can be |
| 1505 |
< |
solved easily. Then, by applying the inverse Laplace transform, also |
| 1506 |
< |
known as the Bromwich integral, we can retrieve the solutions of the |
| 1507 |
< |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
| 1508 |
< |
$, the Laplace transform of $f(t)$ is a new function defined as |
| 1505 |
> |
solved easily. Then, by applying the inverse Laplace transform, we |
| 1506 |
> |
can retrieve the solutions of the original problems. Let $f(t)$ be a |
| 1507 |
> |
function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$ |
| 1508 |
> |
is a new function defined as |
| 1509 |
|
\[ |
| 1510 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1511 |
|
\] |
| 1523 |
|
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x), \\ |
| 1524 |
|
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
| 1525 |
|
\end{eqnarray*} |
| 1526 |
< |
By the same way, the system coordinates become |
| 1526 |
> |
In the same way, the system coordinates become |
| 1527 |
|
\begin{eqnarray*} |
| 1528 |
|
mL(\ddot x) & = & |
| 1529 |
|
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1547 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1548 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1549 |
|
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1550 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1551 |
< |
\end{eqnarray*} |
| 1552 |
< |
\begin{eqnarray*} |
| 1553 |
< |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1554 |
< |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1555 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 1550 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}}\\ |
| 1551 |
> |
% |
| 1552 |
> |
& = & - |
| 1553 |
> |
\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha |
| 1554 |
> |
= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha |
| 1555 |
> |
^2 }}} \right)\cos (\omega _\alpha |
| 1556 |
|
t)\dot x(t - \tau )d} \tau } \\ |
| 1557 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1558 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1630 |
|
which is known as the Langevin equation. The static friction |
| 1631 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1632 |
|
or be determined by Stokes' law for regular shaped particles. A |
| 1633 |
< |
briefly review on calculating friction tensor for arbitrary shaped |
| 1633 |
> |
brief review on calculating friction tensors for arbitrary shaped |
| 1634 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1635 |
|
|
| 1636 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1651 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1652 |
|
\left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
| 1653 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1654 |
< |
& = &kT\xi (t) \\ |
| 1654 |
> |
& = &kT\xi (t) |
| 1655 |
|
\end{eqnarray*} |
| 1656 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1657 |
|
\begin{equation} |
