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 developed a |
20 |
> |
time-reversible integrator for rigid bodies in quaternion |
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 |
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 |
|
|
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} |
51 |
– |
|
52 |
|
The integration of the equations of motion is carried out in a |
53 |
|
velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
54 |
|
|
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 |
– |
|
69 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
70 |
|
product of the three body-fixed rotations, |
71 |
|
\begin{equation} |
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 |
78 |
< |
angular momentum (${\bf j}$) by an angle $\theta$ around 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} |
100 |
|
\end{array} |
101 |
|
\right). |
102 |
|
\end{equation} |
103 |
< |
All other rotations follow in a straightforward manner. |
103 |
> |
All other rotations follow in a straightforward manner. After the |
104 |
> |
first part of the propagation, the forces and body-fixed torques are |
105 |
> |
calculated at the new positions and orientations |
106 |
|
|
106 |
– |
After the first part of the propagation, the forces and body-fixed |
107 |
– |
torques are calculated at the new positions and orientations |
108 |
– |
|
107 |
|
{\tt doForces:} |
108 |
|
\begin{align*} |
109 |
|
{\bf f}(t + h) &\leftarrow |
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*} |
120 |
– |
|
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 |
|
|
130 |
|
\right) |
131 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
132 |
|
\end{align*} |
136 |
– |
|
133 |
|
The matrix rotations used in the DLM method end up being more costly |
134 |
|
computationally than the simpler arithmetic quaternion propagation. |
135 |
|
With the same time step, a 1000-molecule water simulation shows an |
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}. |
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 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 |
145 |
> |
fairly well, with the quaternion method showing a slight energy |
146 |
> |
drift over time in the 0.5 fs time step simulation. At time steps of |
147 |
> |
1 and 2 fs, the energy conservation benefits of the DLM method are |
148 |
> |
clearly demonstrated. Thus, while maintaining the same degree of |
149 |
> |
energy conservation, one can take considerably longer time steps, |
150 |
> |
leading to an overall reduction in computation time. |
151 |
|
|
152 |
|
\begin{figure} |
153 |
|
\centering |
162 |
|
\label{methodFig:timestep} |
163 |
|
\end{figure} |
164 |
|
|
158 |
– |
In Fig.~\ref{methodFig:timestep}, the resulting energy drift at |
159 |
– |
various time steps for both the DLM and quaternion integration |
160 |
– |
schemes is compared. All of the 1000 molecule water simulations |
161 |
– |
started with the same configuration, and the only difference was the |
162 |
– |
method for handling rotational motion. At time steps of 0.1 and 0.5 |
163 |
– |
fs, both methods for propagating molecule rotation conserve energy |
164 |
– |
fairly well, with the quaternion method showing a slight energy |
165 |
– |
drift over time in the 0.5 fs time step simulation. At time steps of |
166 |
– |
1 and 2 fs, the energy conservation benefits of the DLM method are |
167 |
– |
clearly demonstrated. Thus, while maintaining the same degree of |
168 |
– |
energy conservation, one can take considerably longer time steps, |
169 |
– |
leading to an overall reduction in computation time. |
170 |
– |
|
165 |
|
\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
166 |
|
|
167 |
|
The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
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} |
184 |
– |
|
178 |
|
$\chi$ is an ``extra'' variable included in the extended system, and |
179 |
|
it is propagated using the first order equation of motion |
180 |
|
\begin{equation} |
181 |
|
\dot{\chi} = \frac{1}{\tau_{T}^2} \left( |
182 |
|
\frac{T}{T_{\mathrm{target}}} - 1 \right). \label{eq:nosehooverext} |
183 |
|
\end{equation} |
184 |
< |
|
185 |
< |
The instantaneous temperature $T$ is proportional to the total |
186 |
< |
kinetic energy (both translational and orientational) and is given |
194 |
< |
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} |
188 |
> |
T = \frac{2 K}{f k_B}. |
189 |
|
\end{equation} |
190 |
|
Here, $f$ is the total number of degrees of freedom in the system, |
191 |
|
\begin{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, |
205 |
< |
\begin{equation} |
206 |
< |
K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
207 |
< |
\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
208 |
< |
\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
209 |
< |
\end{equation} |
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 |
|
|
211 |
– |
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
212 |
– |
relaxation of the temperature to the target value. The integration |
213 |
– |
of the equations of motion is carried out in a velocity-Verlet style |
214 |
– |
2 part algorithm: |
215 |
– |
|
198 |
|
{\tt moveA:} |
199 |
|
\begin{align*} |
200 |
|
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
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} |
214 |
< |
\left(h * {\bf j}(t + h / 2) |
213 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate} |
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) |
218 |
|
+ \frac{h}{2 \tau_T^2} \left( \frac{T(t)} |
219 |
|
{T_{\mathrm{target}}} - 1 \right) . |
220 |
|
\end{align*} |
239 |
– |
|
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. |
232 |
< |
|
252 |
< |
Once the forces and torques have been obtained at the new time step, |
253 |
< |
the temperature, velocities, and the extended system variable can be |
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:} |
251 |
|
\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
252 |
|
\chi(t + h) \right) . |
253 |
|
\end{align*} |
275 |
– |
|
254 |
|
Since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required to |
255 |
|
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
256 |
|
depend on their own values at time $t + h$. {\tt moveB} is |
257 |
|
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
258 |
< |
self-consistent. |
259 |
< |
|
260 |
< |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
283 |
< |
the extended system that is, to within a constant, identical to the |
284 |
< |
Helmholtz free energy,\cite{Melchionna1993} |
258 |
> |
self-consistent. The Nos\'e-Hoover algorithm is known to conserve a |
259 |
> |
Hamiltonian for the extended system that is, to within a constant, |
260 |
> |
identical to the Helmholtz free energy,\cite{Melchionna1993} |
261 |
|
\begin{equation} |
262 |
|
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
263 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
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 |
293 |
< |
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)} |
272 |
|
|
273 |
|
We can used an isobaric-isothermal ensemble integrator which is |
274 |
|
implemented using the Melchionna modifications to the |
275 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
301 |
< |
|
275 |
> |
Nos\'e-Hoover-Andersen equations of motion\cite{Melchionna1993} |
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 |
289 |
|
P_{\mathrm{target}} \right), \\ |
290 |
|
\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . \label{eq:melchionna1} |
291 |
|
\end{eqnarray} |
318 |
– |
|
292 |
|
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the |
293 |
|
extended system. $\chi$ is a thermostat, and it has the same |
294 |
|
function as it does in the Nos\'e-Hoover NVT integrator. $\eta$ is |
321 |
|
the Pressure tensor, |
322 |
|
\begin{equation} |
323 |
|
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
324 |
< |
\overleftrightarrow{\mathsf{P}}(t). \right) |
324 |
> |
\overleftrightarrow{\mathsf{P}}(t) \right) . |
325 |
|
\end{equation} |
326 |
< |
|
354 |
< |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
326 |
> |
In Eq.~\ref{eq:melchionna1}, $\tau_B$ is the time constant for |
327 |
|
relaxation of the pressure to the target value. Like in the NVT |
328 |
|
integrator, the integration of the equations of motion is carried |
329 |
|
out in a velocity-Verlet style 2 part algorithm: |
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 |
|
% |
362 |
|
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
363 |
|
\mathsf{H}(t). |
364 |
|
\end{align*} |
393 |
– |
|
365 |
|
Most of these equations are identical to their counterparts in the |
366 |
|
NVT integrator, but the propagation of positions to time $t + h$ |
367 |
|
depends on the positions at the same time. The simulation box |
373 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
374 |
|
\mathcal{V}(t) |
375 |
|
\end{equation} |
405 |
– |
|
376 |
|
The {\tt doForces} step for the NPTi integrator is exactly the same |
377 |
|
as in both the DLM and NVT integrators. Once the forces and torques |
378 |
|
have been obtained at the new time step, the velocities can be |
404 |
|
\tau}^b(t + h) - {\bf j}(t + h) |
405 |
|
\chi(t + h) \right) . |
406 |
|
\end{align*} |
437 |
– |
|
407 |
|
Once again, since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required |
408 |
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
409 |
|
h)$, they indirectly depend on their own values at time $t + h$. |
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 |
455 |
< |
checks on the quality of the integration. It is also known that |
456 |
< |
this algorithm samples the equilibrium distribution for the enthalpy |
457 |
< |
(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 |
|
% |
501 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
502 |
|
\end{align*} |
503 |
|
Here, a power series expansion truncated at second order for the |
504 |
< |
exponential operation is used to scale the simulation box. |
504 |
> |
exponential operation is used to scale the simulation box. The {\tt |
505 |
> |
moveB} portion of the algorithm is largely unchanged from the NPTi |
506 |
> |
integrator: |
507 |
|
|
540 |
– |
The {\tt moveB} portion of the algorithm is largely unchanged from |
541 |
– |
the NPTi integrator: |
542 |
– |
|
508 |
|
{\tt moveB:} |
509 |
|
\begin{align*} |
510 |
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
534 |
|
+ h / 2 \right) + \frac{h}{2} \left( {\bf \tau}^b(t |
535 |
|
+ h) - {\bf j}(t + h) \chi(t + h) \right) . |
536 |
|
\end{align*} |
572 |
– |
|
537 |
|
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
538 |
< |
identical to those described for the NPTi integrator. |
539 |
< |
|
576 |
< |
The NPTf integrator is known to conserve the following Hamiltonian: |
538 |
> |
identical to those described for the NPTi integrator. The NPTf |
539 |
> |
integrator is known to conserve the following Hamiltonian: |
540 |
|
\begin{eqnarray*} |
541 |
|
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
542 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
545 |
|
T_{\mathrm{target}}}{2} |
546 |
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
547 |
|
\end{eqnarray*} |
585 |
– |
|
548 |
|
This integrator must be used with care, particularly in liquid |
549 |
|
simulations. Liquids have very small restoring forces in the |
550 |
|
off-diagonal directions, and the simulation box can very quickly |
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 |
573 |
|
\end{array} |
574 |
|
\right. |
575 |
|
\end{equation} |
614 |
– |
|
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 $A$ |
585 |
< |
\[ |
584 |
> |
minimum with respect to surface area $A$, |
585 |
> |
\begin{equation} |
586 |
|
\gamma = \frac{{\partial G}}{{\partial A}}. |
587 |
< |
\] |
588 |
< |
However, a surface tension of zero is not appropriate for relatively |
589 |
< |
small patches of membrane. In order to eliminate the edge effect of |
590 |
< |
the membrane simulation, a special ensemble, NP$\gamma$T, has been |
591 |
< |
proposed to maintain the lateral surface tension and normal |
592 |
< |
pressure. The equation of motion for the cell size control tensor, |
593 |
< |
$\eta$, in $NP\gamma T$ is |
587 |
> |
\end{equation}0 |
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 |
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} |
595 |
|
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
596 |
|
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
606 |
|
- P_{{\rm{target}}} ) |
607 |
|
\label{methodEquation:instantaneousSurfaceTensor} |
608 |
|
\end{equation} |
648 |
– |
|
609 |
|
There is one additional extended system integrator (NPTxyz), in |
610 |
|
which each attempt to preserve the target pressure along the box |
611 |
|
walls perpendicular to that particular axis. The lengths of the box |
628 |
|
\begin{equation} |
629 |
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
630 |
|
\end{equation} |
671 |
– |
|
631 |
|
If the time-dependent friction decays rapidly, the static friction |
632 |
|
coefficient can be approximated by |
633 |
|
\begin{equation} |
640 |
|
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
641 |
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
642 |
|
\end{equation} |
684 |
– |
|
643 |
|
The Z-Constraint method, which fixes the z coordinates of the |
644 |
|
molecules with respect to the center of the mass of the system, has |
645 |
|
been a method suggested to obtain the forces required for the force |
648 |
|
whole system. To avoid this problem, we reset the forces of |
649 |
|
z-constrained molecules as well as subtract the total constraint |
650 |
|
forces from the rest of the system after the force calculation at |
651 |
< |
each time step instead of resetting the coordinate. |
652 |
< |
|
695 |
< |
After the force calculation, we define $G_\alpha$ as |
651 |
> |
each time step instead of resetting the coordinate. After the force |
652 |
> |
calculation, we define $G_\alpha$ as |
653 |
|
\begin{equation} |
654 |
|
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
655 |
|
\end{equation} |
684 |
|
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
685 |
|
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
686 |
|
\end{equation} |
730 |
– |
|
687 |
|
At the very beginning of the simulation, the molecules may not be at |
688 |
|
their constrained positions. To move a z-constrained molecule to its |
689 |
|
specified position, a simple harmonic potential is used |