18 |
|
in the microcanonical ensemble have been extensively studied over |
19 |
|
the last two decades. Matubayasi developed a |
20 |
|
time-reversible integrator for rigid bodies in quaternion |
21 |
< |
representation\cite{Matubayasi1999}. 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 |
32 |
|
An alternative integration scheme utilizing the rotation matrix |
33 |
|
directly proposed by Dullweber, Leimkuhler and McLachlan (DLM) also |
34 |
|
preserved the same structural properties of the Hamiltonian |
35 |
< |
propagator\cite{Dullweber1997}. In this section, the integration |
35 |
> |
propagator.\cite{Dullweber1997} In this section, the integration |
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: |
87 |
|
\right. |
88 |
|
\end{equation} |
89 |
|
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
90 |
< |
rotation matrix. For example, in the small-angle limit, the |
90 |
> |
rotation matrix. For example, in small-angle limit, the |
91 |
|
rotation matrix around the body-fixed x-axis can be approximated as |
92 |
|
\begin{equation} |
93 |
|
\mathsf{R}_x(\theta) \approx \left( |
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) . |
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 $A$, |
584 |
> |
minimum with respect to surface area $A$, |
585 |
|
\begin{equation} |
586 |
< |
\gamma = \frac{{\partial G}}{{\partial A}}. |
587 |
< |
\end{equation}0 |
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} |
618 |
|
|
619 |
|
Based on the fluctuation-dissipation theorem, a force |
620 |
|
auto-correlation method was developed by Roux and Karplus to |
621 |
< |
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
621 |
> |
investigate the dynamics of ions inside ion channels.\cite{Roux1991} |
622 |
|
The time-dependent friction coefficient can be calculated from the |
623 |
< |
deviation of the instantaneous force from its mean force. |
623 |
> |
deviation of the instantaneous force from its mean force: |
624 |
|
\begin{equation} |
625 |
|
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
626 |
|
\end{equation} |