| 16 |
|
|
| 17 |
|
Integration schemes for the rotational motion of the rigid molecules |
| 18 |
|
in the microcanonical ensemble have been extensively studied over |
| 19 |
< |
the last two decades. Matubayasi\cite{Matubayasi1999} developed a |
| 19 |
> |
the last two decades. Matubayasi developed a |
| 20 |
|
time-reversible integrator for rigid bodies in quaternion |
| 21 |
< |
representation. Although it is not symplectic, this integrator still |
| 21 |
> |
representation\cite{Matubayasi1999}. Although it is not symplectic, this integrator still |
| 22 |
|
demonstrates a better long-time energy conservation than Euler angle |
| 23 |
|
methods because of the time-reversible nature. Extending the |
| 24 |
|
Trotter-Suzuki factorization to general system with a flat phase |
| 46 |
|
{\it symplectic}), |
| 47 |
|
\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
| 48 |
|
Monte Carlo applications, and |
| 49 |
< |
\item the error for a single time step is of order $\mathcal{O}\left(h^4\right)$ |
| 49 |
> |
\item the error for a single time step is of order $\mathcal{O}\left(h^3\right)$ |
| 50 |
|
for timesteps of length $h$. |
| 51 |
|
\end{enumerate} |
| 52 |
|
The integration of the equations of motion is carried out in a |
| 74 |
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
| 75 |
|
\end{equation} |
| 76 |
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
| 77 |
< |
rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed |
| 78 |
< |
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
| 77 |
> |
rotates both the rotation matrix $\mathsf{Q}$ and the body-fixed |
| 78 |
> |
angular momentum ${\bf j}$ by an angle $\theta$ around body-fixed |
| 79 |
|
axis $\alpha$, |
| 80 |
|
\begin{equation} |
| 81 |
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 136 |
|
average 7\% increase in computation time using the DLM method in |
| 137 |
|
place of quaternions. This cost is more than justified when |
| 138 |
|
comparing the energy conservation of the two methods as illustrated |
| 139 |
< |
in Fig.~\ref{methodFig:timestep} where the resulting energy drift at |
| 139 |
> |
in Fig.~\ref{methodFig:timestep} where the resulting energy drifts at |
| 140 |
|
various time steps for both the DLM and quaternion integration |
| 141 |
< |
schemes is compared. All of the 1000 molecule water simulations |
| 141 |
> |
schemes are compared. All of the 1000 molecule water simulations |
| 142 |
|
started with the same configuration, and the only difference was the |
| 143 |
|
method for handling rotational motion. At time steps of 0.1 and 0.5 |
| 144 |
|
fs, both methods for propagating molecule rotation conserve energy |
| 573 |
|
\end{array} |
| 574 |
|
\right. |
| 575 |
|
\end{equation} |
| 576 |
– |
|
| 576 |
|
Note that the iterative schemes for NPAT are identical to those |
| 577 |
|
described for the NPTi integrator. |
| 578 |
|
|
| 581 |
|
|
| 582 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
| 583 |
|
membrane system should be zero since its surface free energy $G$ is |
| 584 |
< |
minimum with respect to surface area $A$, $\gamma = \frac{{\partial |
| 585 |
< |
G}}{{\partial A}}.$ However, a surface tension of zero is not |
| 584 |
> |
minimum with respect to surface area $A$, |
| 585 |
> |
\begin{equation} |
| 586 |
> |
\gamma = \frac{{\partial G}}{{\partial A}}. |
| 587 |
> |
\end{equation}0 |
| 588 |
> |
However, a surface tension of zero is not |
| 589 |
|
appropriate for relatively small patches of membrane. In order to |
| 590 |
|
eliminate the edge effect of membrane simulations, a special |
| 591 |
|
ensemble, NP$\gamma$T, has been proposed to maintain the lateral |
