| 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 developed a time-reversible |
| 20 |
< |
integrator for rigid bodies in quaternion representation. Although |
| 21 |
< |
it is not symplectic, this integrator still demonstrates a better |
| 22 |
< |
long-time energy conservation than Euler angle methods because of |
| 23 |
< |
the time-reversible nature. Extending the Trotter-Suzuki |
| 24 |
< |
factorization to general system with a flat phase space, Miller and |
| 25 |
< |
his colleagues devised a novel symplectic, time-reversible and |
| 26 |
< |
volume-preserving integrator in the quaternion representation, which |
| 27 |
< |
was shown to be superior to the Matubayasi's time-reversible |
| 28 |
< |
integrator. However, all of the integrators in the quaternion |
| 29 |
< |
representation suffer from the computational penalty of constructing |
| 30 |
< |
a rotation matrix from quaternions to evolve coordinates and |
| 31 |
< |
velocities at every time step. An alternative integration scheme |
| 32 |
< |
utilizing the rotation matrix directly proposed by Dullweber, |
| 33 |
< |
Leimkuhler and McLachlan (DLM) also preserved the same structural |
| 34 |
< |
properties of the Hamiltonian flow. In this section, the integration |
| 19 |
> |
the last two decades. Matubayasi\cite{Matubayasi1999} developed a |
| 20 |
> |
time-reversible integrator for rigid bodies in quaternion |
| 21 |
> |
representation. 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 |
| 25 |
> |
space, Miller\cite{Miller2002} and his colleagues devised a novel |
| 26 |
> |
symplectic, time-reversible and volume-preserving integrator in the |
| 27 |
> |
quaternion representation, which was shown to be superior to the |
| 28 |
> |
Matubayasi's time-reversible integrator. However, all of the |
| 29 |
> |
integrators in the quaternion representation suffer from the |
| 30 |
> |
computational penalty of constructing a rotation matrix from |
| 31 |
> |
quaternions to evolve coordinates and velocities at every time step. |
| 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 |
| 36 |
|
scheme of DLM method will be reviewed and extended to other |
| 37 |
|
ensembles. |
| 38 |
|
|
| 63 |
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 64 |
|
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
| 65 |
|
% |
| 66 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 66 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 67 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 68 |
|
\end{align*} |
| 69 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
| 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 ($\mathsf{A}$) and the 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\{ |
| 82 |
|
\begin{array}{lcl} |
| 83 |
< |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 83 |
> |
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 84 |
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
| 85 |
|
j}(0). |
| 86 |
|
\end{array} |
| 112 |
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
| 113 |
|
\times (\frac{\partial V}{\partial {\bf u}})_{u(t+h)}, \\ |
| 114 |
|
% |
| 115 |
< |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
| 115 |
> |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
| 116 |
|
\cdot {\bf \tau}^s(t + h). |
| 117 |
|
\end{align*} |
| 118 |
|
${\bf u}$ is automatically updated when the rotation matrix |
| 119 |
< |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 119 |
> |
$\mathsf{Q}$ is calculated in {\tt moveA}. Once the forces and |
| 120 |
|
torques have been obtained at the new time step, the velocities can |
| 121 |
|
be advanced to the same time value. |
| 122 |
|
|
| 168 |
|
\begin{eqnarray} |
| 169 |
|
\dot{{\bf r}} & = & {\bf v}, \\ |
| 170 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
| 171 |
< |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 171 |
> |
\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
| 172 |
|
\mbox{ skew}\left(\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j}\right), \\ |
| 173 |
|
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 174 |
|
\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 175 |
< |
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial V}{\partial |
| 176 |
< |
\mathsf{A}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
| 175 |
> |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial V}{\partial |
| 176 |
> |
\mathsf{Q}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
| 177 |
|
\end{eqnarray} |
| 178 |
|
$\chi$ is an ``extra'' variable included in the extended system, and |
| 179 |
|
it is propagated using the first order equation of motion |
| 181 |
|
\dot{\chi} = \frac{1}{\tau_{T}^2} \left( |
| 182 |
|
\frac{T}{T_{\mathrm{target}}} - 1 \right). \label{eq:nosehooverext} |
| 183 |
|
\end{equation} |
| 184 |
< |
The instantaneous temperature $T$ is proportional to the total |
| 185 |
< |
kinetic energy (both translational and orientational) and is given |
| 186 |
< |
by |
| 184 |
> |
where $\tau_T$ is the time constant for relaxation of the |
| 185 |
> |
temperature to the target value, and the instantaneous temperature |
| 186 |
> |
$T$ is given by |
| 187 |
|
\begin{equation} |
| 188 |
|
T = \frac{2 K}{f k_B}. |
| 189 |
|
\end{equation} |
| 192 |
|
f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
| 193 |
|
\end{equation} |
| 194 |
|
where $N_{\mathrm{orient}}$ is the number of molecules with |
| 195 |
< |
orientational degrees of freedom, and $K$ is the total kinetic |
| 196 |
< |
energy, |
| 196 |
< |
\begin{equation} |
| 197 |
< |
K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
| 198 |
< |
\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
| 199 |
< |
\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
| 200 |
< |
\end{equation} |
| 201 |
< |
In Eq.~\ref{eq:nosehooverext}, $\tau_T$ is the time constant for |
| 202 |
< |
relaxation of the temperature to the target value. The integration |
| 203 |
< |
of the equations of motion is carried out in a velocity-Verlet style |
| 204 |
< |
2 part algorithm: |
| 195 |
> |
orientational degrees of freedom. The integration of the equations of motion |
| 196 |
> |
is carried out in a velocity-Verlet style 2 part algorithm: |
| 197 |
|
|
| 198 |
|
{\tt moveA:} |
| 199 |
|
\begin{align*} |
| 210 |
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 211 |
|
\chi(t) \right) ,\\ |
| 212 |
|
% |
| 213 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate} |
| 213 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate} |
| 214 |
|
\left(h * {\bf j}(t + h / 2) |
| 215 |
|
\overleftrightarrow{\mathsf{I}}^{-1} \right) ,\\ |
| 216 |
|
% |
| 219 |
|
{T_{\mathrm{target}}} - 1 \right) . |
| 220 |
|
\end{align*} |
| 221 |
|
Here $\mathrm{rotate}(h * {\bf j} |
| 222 |
< |
\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic |
| 223 |
< |
Trotter factorization of the three rotation operations that was |
| 224 |
< |
discussed in the section on the DLM integrator. Note that this |
| 225 |
< |
operation modifies both the rotation matrix $\mathsf{A}$ and the |
| 226 |
< |
angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
| 227 |
< |
half time step, and positional degrees of freedom by a full time |
| 228 |
< |
step. The new positions (and orientations) are then used to |
| 229 |
< |
calculate a new set of forces and torques in exactly the same way |
| 230 |
< |
they are calculated in the {\tt doForces} portion of the DLM |
| 231 |
< |
integrator. Once the forces and torques have been obtained at the |
| 232 |
< |
new time step, the temperature, velocities, and the extended system |
| 233 |
< |
variable can be advanced to the same time value. |
| 222 |
> |
\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic Strang |
| 223 |
> |
factorization of the three rotation operations that was discussed in |
| 224 |
> |
the section on the DLM integrator. Note that this operation |
| 225 |
> |
modifies both the rotation matrix $\mathsf{Q}$ and the angular |
| 226 |
> |
momentum ${\bf j}$. {\tt moveA} propagates velocities by a half |
| 227 |
> |
time step, and positional degrees of freedom by a full time step. |
| 228 |
> |
The new positions (and orientations) are then used to calculate a |
| 229 |
> |
new set of forces and torques in exactly the same way they are |
| 230 |
> |
calculated in the {\tt doForces} portion of the DLM integrator. Once |
| 231 |
> |
the forces and torques have been obtained at the new time step, the |
| 232 |
> |
temperature, velocities, and the extended system variable can be |
| 233 |
> |
advanced to the same time value. |
| 234 |
|
|
| 235 |
|
{\tt moveB:} |
| 236 |
|
\begin{align*} |
| 264 |
|
dt^\prime \right). |
| 265 |
|
\end{equation} |
| 266 |
|
Poor choices of $h$ or $\tau_T$ can result in non-conservation of |
| 267 |
< |
$H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
| 268 |
< |
last column of the {\tt .stat} file to allow checks on the quality |
| 277 |
< |
of the integration. |
| 267 |
> |
$H_{\mathrm{NVT}}$, so the conserved quantity should be checked |
| 268 |
> |
periodically to verify the quality of the integration. |
| 269 |
|
|
| 270 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
| 271 |
|
isotropic box (NPTi)} |
| 276 |
|
\begin{eqnarray} |
| 277 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 278 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v}, \\ |
| 279 |
< |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 279 |
> |
\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
| 280 |
|
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right),\\ |
| 281 |
|
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 282 |
|
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 283 |
< |
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 284 |
< |
V}{\partial \mathsf{A}} \right) - \chi {\bf j}, \\ |
| 283 |
> |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial |
| 284 |
> |
V}{\partial \mathsf{Q}} \right) - \chi {\bf j}, \\ |
| 285 |
|
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 286 |
|
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 287 |
|
\dot{\eta} & = & \frac{1}{\tau_{B}^2 f k_B T_{\mathrm{target}}} V |
| 342 |
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 343 |
|
\chi(t) \right), \\ |
| 344 |
|
% |
| 345 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 345 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 346 |
|
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 347 |
|
\right) ,\\ |
| 348 |
|
% |
| 418 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 419 |
|
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t). |
| 420 |
|
\end{equation} |
| 421 |
< |
Poor choices of $\delta t$, $\tau_T$, or $\tau_B$ can result in |
| 422 |
< |
non-conservation of $H_{\mathrm{NPTi}}$, so the conserved quantity |
| 423 |
< |
is maintained in the last column of the {\tt .stat} file to allow |
| 433 |
< |
checks on the quality of the integration. It is also known that |
| 434 |
< |
this algorithm samples the equilibrium distribution for the enthalpy |
| 435 |
< |
(including contributions for the thermostat and barostat), |
| 421 |
> |
It is also known that this algorithm samples the equilibrium |
| 422 |
> |
distribution for the enthalpy (including contributions for the |
| 423 |
> |
thermostat and barostat), |
| 424 |
|
\begin{equation} |
| 425 |
|
H_{\mathrm{NPTi}} = V + K + \frac{f k_B T_{\mathrm{target}}}{2} |
| 426 |
|
\left( \chi^2 \tau_T^2 + \eta^2 \tau_B^2 \right) + |
| 441 |
|
\dot{{\bf r}} & = & {\bf v} + \overleftrightarrow{\eta} \cdot \left( {\bf r} - {\bf R}_0 \right), \\ |
| 442 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\overleftrightarrow{\eta} + |
| 443 |
|
\chi \cdot \mathsf{1}) {\bf v}, \\ |
| 444 |
< |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 444 |
> |
\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
| 445 |
|
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) ,\\ |
| 446 |
|
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 447 |
|
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 448 |
< |
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
| 449 |
< |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} ,\\ |
| 448 |
> |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial |
| 449 |
> |
V}{\partial \mathsf{Q}} \right) - \chi {\bf j} ,\\ |
| 450 |
|
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 451 |
|
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 452 |
|
\dot{\overleftrightarrow{\eta}} & = & \frac{1}{\tau_{B}^2 f k_B |
| 479 |
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 480 |
|
\chi(t) \right), \\ |
| 481 |
|
% |
| 482 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 482 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 483 |
|
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 484 |
|
\right), \\ |
| 485 |
|
% |
| 559 |
|
functions in biological membrane system ultimately relies on |
| 560 |
|
structure and dynamics of lipid bilayers, which are strongly |
| 561 |
|
affected by the interfacial interaction between lipid molecules and |
| 562 |
< |
surrounding media. One quantity to describe the interfacial |
| 563 |
< |
interaction is so called the average surface area per lipid. |
| 562 |
> |
surrounding media. One quantity used to describe the interfacial |
| 563 |
> |
interaction is the average surface area per lipid. |
| 564 |
|
Constant area and constant lateral pressure simulations can be |
| 565 |
|
achieved by extending the standard NPT ensemble with a different |
| 566 |
|
pressure control strategy |
| 585 |
|
minimum with respect to surface area $A$, $\gamma = \frac{{\partial |
| 586 |
|
G}}{{\partial A}}.$ However, a surface tension of zero is not |
| 587 |
|
appropriate for relatively small patches of membrane. In order to |
| 588 |
< |
eliminate the edge effect of the membrane simulation, a special |
| 588 |
> |
eliminate the edge effect of membrane simulations, a special |
| 589 |
|
ensemble, NP$\gamma$T, has been proposed to maintain the lateral |
| 590 |
|
surface tension and normal pressure. The equation of motion for the |
| 591 |
|
cell size control tensor, $\eta$, in $NP\gamma T$ is |
