1022 |
|
evaluation is to apply cutoff where particles farther than a |
1023 |
|
predetermined distance, are not included in the calculation |
1024 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1025 |
< |
discontinuity in the potential energy curve |
1026 |
< |
(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the |
1027 |
< |
potential to ensure the potential curve go smoothly to zero at the |
1028 |
< |
cutoff radius. Cutoff strategy works pretty well for Lennard-Jones |
1029 |
< |
interaction because of its short range nature. However, simply |
1030 |
< |
truncating the electrostatic interaction with the use of cutoff has |
1031 |
< |
been shown to lead to severe artifacts in simulations. Ewald |
1032 |
< |
summation, in which the slowly conditionally convergent Coulomb |
1033 |
< |
potential is transformed into direct and reciprocal sums with rapid |
1034 |
< |
and absolute convergence, has proved to minimize the periodicity |
1035 |
< |
artifacts in liquid simulations. Taking the advantages of the fast |
1036 |
< |
Fourier transform (FFT) for calculating discrete Fourier transforms, |
1037 |
< |
the particle mesh-based methods are accelerated from $O(N^{3/2})$ to |
1038 |
< |
$O(N logN)$. An alternative approach is \emph{fast multipole |
1039 |
< |
method}, which treats Coulombic interaction exactly at short range, |
1040 |
< |
and approximate the potential at long range through multipolar |
1041 |
< |
expansion. In spite of their wide acceptances at the molecular |
1042 |
< |
simulation community, these two methods are hard to be implemented |
1043 |
< |
correctly and efficiently. Instead, we use a damped and |
1044 |
< |
charge-neutralized Coulomb potential method developed by Wolf and |
1045 |
< |
his coworkers. The shifted Coulomb potential for particle $i$ and |
1046 |
< |
particle $j$ at distance $r_{rj}$ is given by: |
1025 |
> |
discontinuity in the potential energy curve. Fortunately, one can |
1026 |
> |
shift the potential to ensure the potential curve go smoothly to |
1027 |
> |
zero at the cutoff radius. Cutoff strategy works pretty well for |
1028 |
> |
Lennard-Jones interaction because of its short range nature. |
1029 |
> |
However, simply truncating the electrostatic interaction with the |
1030 |
> |
use of cutoff has been shown to lead to severe artifacts in |
1031 |
> |
simulations. Ewald summation, in which the slowly conditionally |
1032 |
> |
convergent Coulomb potential is transformed into direct and |
1033 |
> |
reciprocal sums with rapid and absolute convergence, has proved to |
1034 |
> |
minimize the periodicity artifacts in liquid simulations. Taking the |
1035 |
> |
advantages of the fast Fourier transform (FFT) for calculating |
1036 |
> |
discrete Fourier transforms, the particle mesh-based methods are |
1037 |
> |
accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative |
1038 |
> |
approach is \emph{fast multipole method}, which treats Coulombic |
1039 |
> |
interaction exactly at short range, and approximate the potential at |
1040 |
> |
long range through multipolar expansion. In spite of their wide |
1041 |
> |
acceptances at the molecular simulation community, these two methods |
1042 |
> |
are hard to be implemented correctly and efficiently. Instead, we |
1043 |
> |
use a damped and charge-neutralized Coulomb potential method |
1044 |
> |
developed by Wolf and his coworkers. The shifted Coulomb potential |
1045 |
> |
for particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
1046 |
|
\begin{equation} |
1047 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1048 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1262 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1263 |
|
We can use constraint force provided by lagrange multiplier on the |
1264 |
|
normal manifold to keep the motion on constraint space. Or we can |
1265 |
< |
simply evolve the system in constraint manifold. The two method are |
1266 |
< |
proved to be equivalent. The holonomic constraint and equations of |
1267 |
< |
motions define a constraint manifold for rigid body |
1265 |
> |
simply evolve the system in constraint manifold. These two methods |
1266 |
> |
are proved to be equivalent. The holonomic constraint and equations |
1267 |
> |
of motions define a constraint manifold for rigid body |
1268 |
|
\[ |
1269 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1270 |
|
\right\}. |
1477 |
|
\] |
1478 |
|
The equations of motion corresponding to potential energy and |
1479 |
|
kinetic energy are listed in the below table, |
1480 |
+ |
\begin{table} |
1481 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
1482 |
|
\begin{center} |
1483 |
|
\begin{tabular}{|l|l|} |
1484 |
|
\hline |
1491 |
|
\hline |
1492 |
|
\end{tabular} |
1493 |
|
\end{center} |
1494 |
< |
A second-order symplectic method is now obtained by the composition |
1495 |
< |
of the flow maps, |
1494 |
> |
\end{table} |
1495 |
> |
A second-order symplectic method is now obtained by the |
1496 |
> |
composition of the flow maps, |
1497 |
|
\[ |
1498 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1499 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1789 |
|
when the system become more and more complicate. Instead, various |
1790 |
|
approaches based on hydrodynamics have been developed to calculate |
1791 |
|
the friction coefficients. The friction effect is isotropic in |
1792 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
1793 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
1792 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
1793 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
1794 |
|
\[ |
1795 |
|
\Xi = \left( {\begin{array}{*{20}c} |
1796 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |