| 36 |
|
scheme of DLM method will be reviewed and extended to other |
| 37 |
|
ensembles. |
| 38 |
|
|
| 39 |
< |
\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
| 40 |
< |
DLM method} |
| 39 |
> |
\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: The |
| 40 |
> |
DLM Method} |
| 41 |
|
|
| 42 |
|
The DLM method uses a Trotter factorization of the orientational |
| 43 |
|
propagator. This has three effects: |
| 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 |
| 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 |
| 211 |
|
\chi(t) \right) ,\\ |
| 212 |
|
% |
| 213 |
|
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate} |
| 214 |
< |
\left(h * {\bf j}(t + h / 2) |
| 214 |
> |
\left(h {\bf j}(t + h / 2) |
| 215 |
|
\overleftrightarrow{\mathsf{I}}^{-1} \right) ,\\ |
| 216 |
|
% |
| 217 |
|
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) |
| 318 |
|
r}_{ij}(t) \otimes {\bf f}_{ij}(t). |
| 319 |
|
\end{equation} |
| 320 |
|
The instantaneous pressure is then simply obtained from the trace of |
| 321 |
< |
the Pressure tensor, |
| 321 |
> |
the pressure tensor, |
| 322 |
|
\begin{equation} |
| 323 |
|
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
| 324 |
|
\overleftrightarrow{\mathsf{P}}(t) \right) . |
| 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}}=0. |
| 587 |
> |
\end{equation} |
| 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 |
| 591 |
> |
ensemble NP$\gamma$T has been proposed to maintain the lateral |
| 592 |
|
surface tension and normal pressure. The equation of motion for the |
| 593 |
|
cell size control tensor, $\eta$, in $NP\gamma T$ is |
| 594 |
|
\begin{equation} |
