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 |
> |
\subsection{\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 |