| 117 |
|
\cdot {\bf \tau}^s(t + h). |
| 118 |
|
\end{align*} |
| 119 |
|
|
| 120 |
< |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
| 120 |
> |
${\bf u}$ will be automatically updated when the rotation matrix |
| 121 |
|
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 122 |
|
torques have been obtained at the new time step, the velocities can |
| 123 |
|
be advanced to the same time value. |
| 278 |
|
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
| 279 |
|
depend on their own values at time $t + h$. {\tt moveB} is |
| 280 |
|
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
| 281 |
< |
self-consistent. The relative tolerance for the self-consistency |
| 282 |
< |
check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will |
| 283 |
< |
terminate the iteration after 4 loops even if the consistency check |
| 284 |
< |
has not been satisfied. |
| 281 |
> |
self-consistent. |
| 282 |
|
|
| 283 |
|
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
| 284 |
|
the extended system that is, to within a constant, identical to the |
| 296 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
| 297 |
|
isotropic box deformations (NPTi)} |
| 298 |
|
|
| 299 |
< |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
| 300 |
< |
implements the Melchionna modifications to the |
| 301 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
| 299 |
> |
Isobaric-isothermal ensemble integrator is implemented using the |
| 300 |
> |
Melchionna modifications to the Nos\'e-Hoover-Andersen equations of |
| 301 |
> |
motion,\cite{Melchionna1993} |
| 302 |
|
|
| 303 |
|
\begin{eqnarray} |
| 304 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 394 |
|
|
| 395 |
|
Most of these equations are identical to their counterparts in the |
| 396 |
|
NVT integrator, but the propagation of positions to time $t + h$ |
| 397 |
< |
depends on the positions at the same time. {\sc oopse} carries out |
| 398 |
< |
this step iteratively (with a limit of 5 passes through the |
| 399 |
< |
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
| 400 |
< |
uniformly for one full time step by an exponential factor that |
| 401 |
< |
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
| 405 |
< |
box uniformly also scales the volume of the box by |
| 397 |
> |
depends on the positions at the same time. The simulation box |
| 398 |
> |
$\mathsf{H}$ is scaled uniformly for one full time step by an |
| 399 |
> |
exponential factor that depends on the value of $\eta$ at time $t + |
| 400 |
> |
h / 2$. Reshaping the box uniformly also scales the volume of the |
| 401 |
> |
box by |
| 402 |
|
\begin{equation} |
| 403 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
| 404 |
|
\mathcal{V}(t) |
| 440 |
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
| 441 |
|
h)$, they indirectly depend on their own values at time $t + h$. |
| 442 |
|
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
| 443 |
< |
+ h)$ and $\eta(t + h)$ become self-consistent. The relative |
| 448 |
< |
tolerance for the self-consistency check defaults to a value of |
| 449 |
< |
$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after |
| 450 |
< |
4 loops even if the consistency check has not been satisfied. |
| 443 |
> |
+ h)$ and $\eta(t + h)$ become self-consistent. |
| 444 |
|
|
| 445 |
|
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
| 446 |
|
is known to conserve a Hamiltonian for the extended system that is, |
| 535 |
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
| 536 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
| 537 |
|
\end{align*} |
| 538 |
< |
{\sc oopse} uses a power series expansion truncated at second order |
| 539 |
< |
for the exponential operation which scales the simulation box. |
| 538 |
> |
Here, a power series expansion truncated at second order for the |
| 539 |
> |
exponential operation is used to scale the simulation box. |
| 540 |
|
|
| 541 |
|
The {\tt moveB} portion of the algorithm is largely unchanged from |
| 542 |
|
the NPTi integrator: |
| 575 |
|
identical to those described for the NPTi integrator. |
| 576 |
|
|
| 577 |
|
The NPTf integrator is known to conserve the following Hamiltonian: |
| 578 |
< |
\begin{equation} |
| 579 |
< |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 578 |
> |
\begin{eqnarray*} |
| 579 |
> |
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
| 580 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 581 |
< |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
| 582 |
< |
T_{\mathrm{target}}}{2} |
| 581 |
> |
dt^\prime \right) \\ |
| 582 |
> |
+ P_{\mathrm{target}} \mathcal{V}(t) + \frac{f |
| 583 |
> |
k_B T_{\mathrm{target}}}{2} |
| 584 |
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
| 585 |
< |
\end{equation} |
| 585 |
> |
\end{eqnarray*} |
| 586 |
|
|
| 587 |
|
This integrator must be used with care, particularly in liquid |
| 588 |
|
simulations. Liquids have very small restoring forces in the |
| 592 |
|
finds most use in simulating crystals or liquid crystals which |
| 593 |
|
assume non-orthorhombic geometries. |
| 594 |
|
|
| 595 |
< |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
| 595 |
> |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 596 |
|
|
| 603 |
– |
\subsubsection{\label{methodSection:NPAT}\textbf{NPAT Ensemble}} |
| 604 |
– |
|
| 597 |
|
A comprehensive understanding of structure¨Cfunction relations of |
| 598 |
|
biological membrane system ultimately relies on structure and |
| 599 |
|
dynamics of lipid bilayer, which are strongly affected by the |
| 615 |
|
Note that the iterative schemes for NPAT are identical to those |
| 616 |
|
described for the NPTi integrator. |
| 617 |
|
|
| 618 |
< |
\subsubsection{\label{methodSection:NPrT}\textbf{NP$\gamma$T Ensemble}} |
| 618 |
> |
\subsection{\label{methodSection:NPrT}NP$\gamma$T |
| 619 |
> |
Ensemble} |
| 620 |
|
|
| 621 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
| 622 |
|
membrane system should be zero since its surface free energy $G$ is |
