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\chapter{\label{chapt:methodology}METHODOLOGY} |
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\section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} |
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In order to mimic the experiments, which are usually performed under |
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constant temperature and/or pressure, extended Hamiltonian system |
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methods have been developed to generate statistical ensembles, such |
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as canonical ensemble and isobaric-isothermal ensemble \textit{etc}. |
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In addition to the standard ensemble, specific ensembles have been |
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developed to account for the anisotropy between the lateral and |
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normal directions of membranes. The $NPAT$ ensemble, in which the |
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normal pressure and the lateral surface area of the membrane are |
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kept constant, and the $NP\gamma T$ ensemble, in which the normal |
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pressure and the lateral surface tension are kept constant were |
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proposed to address this issue. |
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Integration schemes for rotational motion of the rigid molecules in |
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microcanonical ensemble have been extensively studied in the last |
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two decades. Matubayasi and Nakahara developed a time-reversible |
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integrator for rigid bodies in quaternion representation. Although |
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it is not symplectic, this integrator still demonstrates a better |
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long-time energy conservation than traditional methods because of |
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the time-reversible nature. Extending Trotter-Suzuki to general |
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system with a flat phase space, Miller and his colleagues devised an |
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novel symplectic, time-reversible and volume-preserving integrator |
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in quaternion representation, which was shown to be superior to the |
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time-reversible integrator of Matubayasi and Nakahara. However, all |
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of the integrators in quaternion representation suffer from the |
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computational penalty of constructing a rotation matrix from |
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quaternions to evolve coordinates and velocities at every time step. |
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An alternative integration scheme utilizing rotation matrix directly |
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proposed by Dullweber, Leimkuhler and McLachlan (DLM) also preserved |
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the same structural properties of the Hamiltonian flow. In this |
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section, the integration scheme of DLM method will be reviewed and |
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extended to other ensembles. |
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\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
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DLM method} |
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The DLM method uses a Trotter factorization of the orientational |
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propagator. This has three effects: |
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\begin{enumerate} |
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\item the integrator is area-preserving in phase space (i.e. it is |
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{\it symplectic}), |
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\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
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Monte Carlo applications, and |
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\item the error for a single time step is of order $\mathcal{O}\left(h^4\right)$ |
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for timesteps of length $h$. |
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\end{enumerate} |
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The integration of the equations of motion is carried out in a |
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velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
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{\tt moveA:} |
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\begin{align*} |
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{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
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+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
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% |
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{\bf r}(t + h) &\leftarrow {\bf r}(t) |
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+ h {\bf v}\left(t + h / 2 \right), \\ |
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% |
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{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
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+ \frac{h}{2} {\bf \tau}^b(t), \\ |
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% |
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\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
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(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
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\end{align*} |
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In this context, the $\mathrm{rotate}$ function is the reversible |
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product of the three body-fixed rotations, |
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\begin{equation} |
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\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
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\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
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/ 2) \cdot \mathsf{G}_x(a_x /2), |
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\end{equation} |
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where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
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rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
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angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
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axis $\alpha$, |
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\begin{equation} |
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\mathsf{G}_\alpha( \theta ) = \left\{ |
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\begin{array}{lcl} |
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\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
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{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
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j}(0). |
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\end{array} |
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\right. |
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\end{equation} |
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$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
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rotation matrix. For example, in the small-angle limit, the |
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rotation matrix around the body-fixed x-axis can be approximated as |
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\begin{equation} |
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\mathsf{R}_x(\theta) \approx \left( |
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\begin{array}{ccc} |
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1 & 0 & 0 \\ |
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0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
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\theta^2 / 4} \\ |
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0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
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\theta^2 / 4} |
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\end{array} |
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\right). |
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\end{equation} |
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All other rotations follow in a straightforward manner. |
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After the first part of the propagation, the forces and body-fixed |
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torques are calculated at the new positions and orientations |
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{\tt doForces:} |
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\begin{align*} |
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{\bf f}(t + h) &\leftarrow |
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- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
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% |
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{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
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\times \frac{\partial V}{\partial {\bf u}}, \\ |
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% |
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{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
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\cdot {\bf \tau}^s(t + h). |
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\end{align*} |
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{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
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$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
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torques have been obtained at the new time step, the velocities can |
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be advanced to the same time value. |
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{\tt moveB:} |
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\begin{align*} |
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{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
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\right) |
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+ \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\ |
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% |
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{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 |
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\right) |
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+ \frac{h}{2} {\bf \tau}^b(t + h) . |
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\end{align*} |
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The matrix rotations used in the DLM method end up being more costly |
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computationally than the simpler arithmetic quaternion propagation. |
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With the same time step, a 1000-molecule water simulation shows an |
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average 7\% increase in computation time using the DLM method in |
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place of quaternions. This cost is more than justified when |
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comparing the energy conservation of the two methods as illustrated |
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in Fig.~\ref{methodFig:timestep}. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{timeStep.eps} |
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\caption[Energy conservation for quaternion versus DLM |
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dynamics]{Energy conservation using quaternion based integration |
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versus the method proposed by Dullweber \emph{et al.} with |
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increasing time step. For each time step, the dotted line is total |
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energy using the DLM integrator, and the solid line comes from the |
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quaternion integrator. The larger time step plots are shifted up |
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from the true energy baseline for clarity.} |
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\label{methodFig:timestep} |
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\end{figure} |
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In Fig.~\ref{methodFig:timestep}, the resulting energy drift at |
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various time steps for both the DLM and quaternion integration |
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schemes is compared. All of the 1000 molecule water simulations |
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started with the same configuration, and the only difference was the |
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method for handling rotational motion. At time steps of 0.1 and 0.5 |
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fs, both methods for propagating molecule rotation conserve energy |
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fairly well, with the quaternion method showing a slight energy |
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drift over time in the 0.5 fs time step simulation. At time steps of |
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1 and 2 fs, the energy conservation benefits of the DLM method are |
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clearly demonstrated. Thus, while maintaining the same degree of |
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energy conservation, one can take considerably longer time steps, |
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leading to an overall reduction in computation time. |
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\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
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The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
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\begin{eqnarray} |
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\dot{{\bf r}} & = & {\bf v}, \\ |
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\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
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\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
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\mbox{ skew}\left(\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j}\right), \\ |
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\dot{{\bf j}} & = & {\bf j} \times \left( |
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\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j} \right) - \mbox{ |
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rot}\left(\mathsf{A}^{T} \cdot \frac{\partial V}{\partial |
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\mathsf{A}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
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\end{eqnarray} |
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$\chi$ is an ``extra'' variable included in the extended system, and |
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it is propagated using the first order equation of motion |
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\begin{equation} |
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\dot{\chi} = \frac{1}{\tau_{T}^2} \left( |
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\frac{T}{T_{\mathrm{target}}} - 1 \right). \label{eq:nosehooverext} |
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\end{equation} |
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The instantaneous temperature $T$ is proportional to the total |
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kinetic energy (both translational and orientational) and is given |
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by |
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\begin{equation} |
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T = \frac{2 K}{f k_B} |
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\end{equation} |
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Here, $f$ is the total number of degrees of freedom in the system, |
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\begin{equation} |
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f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
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\end{equation} |
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and $K$ is the total kinetic energy, |
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\begin{equation} |
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K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
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\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
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\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
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\end{equation} |
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In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
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relaxation of the temperature to the target value. To set values |
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for $\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use |
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the {\tt tauThermostat} and {\tt targetTemperature} keywords in the |
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{\tt .bass} file. The units for {\tt tauThermostat} are fs, and the |
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units for the {\tt targetTemperature} are degrees K. The |
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integration of the equations of motion is carried out in a |
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velocity-Verlet style 2 part algorithm: |
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{\tt moveA:} |
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\begin{align*} |
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T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
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% |
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{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
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+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
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\chi(t)\right), \\ |
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% |
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{\bf r}(t + h) &\leftarrow {\bf r}(t) |
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+ h {\bf v}\left(t + h / 2 \right) ,\\ |
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% |
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{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
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+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
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\chi(t) \right) ,\\ |
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% |
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\mathsf{A}(t + h) &\leftarrow \mathrm{rotate} |
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\left(h * {\bf j}(t + h / 2) |
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\overleftrightarrow{\mathsf{I}}^{-1} \right) ,\\ |
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% |
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\chi\left(t + h / 2 \right) &\leftarrow \chi(t) |
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+ \frac{h}{2 \tau_T^2} \left( \frac{T(t)} |
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{T_{\mathrm{target}}} - 1 \right) . |
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\end{align*} |
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Here $\mathrm{rotate}(h * {\bf j} |
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\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic |
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Trotter factorization of the three rotation operations that was |
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discussed in the section on the DLM integrator. Note that this |
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operation modifies both the rotation matrix $\mathsf{A}$ and the |
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angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
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half time step, and positional degrees of freedom by a full time |
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step. The new positions (and orientations) are then used to |
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calculate a new set of forces and torques in exactly the same way |
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they are calculated in the {\tt doForces} portion of the DLM |
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integrator. |
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Once the forces and torques have been obtained at the new time step, |
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the temperature, velocities, and the extended system variable can be |
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advanced to the same time value. |
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{\tt moveB:} |
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\begin{align*} |
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T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
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\left\{{\bf j}(t + h)\right\}, \\ |
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% |
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\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
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2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
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{T_{\mathrm{target}}} - 1 \right), \\ |
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% |
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{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
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+ h / 2 \right) + \frac{h}{2} \left( |
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\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
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\chi(t h)\right) ,\\ |
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% |
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{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
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+ h / 2 \right) + \frac{h}{2} |
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\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
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\chi(t + h) \right) . |
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\end{align*} |
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Since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required to |
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caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
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depend on their own values at time $t + h$. {\tt moveB} is |
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therefore done in an iterative fashion until $\chi(t + h)$ becomes |
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self-consistent. The relative tolerance for the self-consistency |
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check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will |
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terminate the iteration after 4 loops even if the consistency check |
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has not been satisfied. |
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The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
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the extended system that is, to within a constant, identical to the |
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Helmholtz free energy,\cite{Melchionna1993} |
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\begin{equation} |
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H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
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\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
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dt^\prime \right). |
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\end{equation} |
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Poor choices of $h$ or $\tau_T$ can result in non-conservation of |
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$H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
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last column of the {\tt .stat} file to allow checks on the quality |
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of the integration. |
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\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
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isotropic box deformations (NPTi)} |
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To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
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implements the Melchionna modifications to the |
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Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
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\begin{eqnarray} |
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\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
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\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v}, \\ |
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\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
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\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right),\\ |
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\dot{{\bf j}} & = & {\bf j} \times \left( |
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\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
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rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
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V}{\partial \mathsf{A}} \right) - \chi {\bf j}, \\ |
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\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
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\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
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\dot{\eta} & = & \frac{1}{\tau_{B}^2 f k_B T_{\mathrm{target}}} V |
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\left( P - |
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P_{\mathrm{target}} \right), \\ |
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\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . \label{eq:melchionna1} |
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\end{eqnarray} |
322 |
|
|
|
323 |
|
|
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the |
324 |
|
|
extended system. $\chi$ is a thermostat, and it has the same |
325 |
|
|
function as it does in the Nos\'e-Hoover NVT integrator. $\eta$ is |
326 |
|
|
a barostat which controls changes to the volume of the simulation |
327 |
|
|
box. ${\bf R}_0$ is the location of the center of mass for the |
328 |
|
|
entire system, and $\mathcal{V}$ is the volume of the simulation |
329 |
|
|
box. At any time, the volume can be calculated from the determinant |
330 |
|
|
of the matrix which describes the box shape: |
331 |
|
|
\begin{equation} |
332 |
|
|
\mathcal{V} = \det(\mathsf{H}). |
333 |
|
|
\end{equation} |
334 |
|
|
|
335 |
|
|
The NPTi integrator requires an instantaneous pressure. This |
336 |
|
|
quantity is calculated via the pressure tensor, |
337 |
|
|
\begin{equation} |
338 |
|
|
\overleftrightarrow{\mathsf{P}}(t) = \frac{1}{\mathcal{V}(t)} \left( |
339 |
|
|
\sum_{i=1}^{N} m_i {\bf v}_i(t) \otimes {\bf v}_i(t) \right) + |
340 |
|
|
\overleftrightarrow{\mathsf{W}}(t). |
341 |
|
|
\end{equation} |
342 |
|
|
The kinetic contribution to the pressure tensor utilizes the {\it |
343 |
|
|
outer} product of the velocities denoted by the $\otimes$ symbol. |
344 |
|
|
The stress tensor is calculated from another outer product of the |
345 |
|
|
inter-atomic separation vectors (${\bf r}_{ij} = {\bf r}_j - {\bf |
346 |
|
|
r}_i$) with the forces between the same two atoms, |
347 |
|
|
\begin{equation} |
348 |
|
|
\overleftrightarrow{\mathsf{W}}(t) = \sum_{i} \sum_{j>i} {\bf |
349 |
|
|
r}_{ij}(t) \otimes {\bf f}_{ij}(t). |
350 |
|
|
\end{equation} |
351 |
|
|
The instantaneous pressure is then simply obtained from the trace of |
352 |
|
|
the Pressure tensor, |
353 |
|
|
\begin{equation} |
354 |
|
|
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
355 |
|
|
\overleftrightarrow{\mathsf{P}}(t). \right) |
356 |
|
|
\end{equation} |
357 |
|
|
|
358 |
|
|
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
359 |
|
|
relaxation of the pressure to the target value. To set values for |
360 |
|
|
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
361 |
|
|
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt |
362 |
|
|
.bass} file. The units for {\tt tauBarostat} are fs, and the units |
363 |
|
|
for the {\tt targetPressure} are atmospheres. Like in the NVT |
364 |
|
|
integrator, the integration of the equations of motion is carried |
365 |
|
|
out in a velocity-Verlet style 2 part algorithm: |
366 |
|
|
|
367 |
|
|
{\tt moveA:} |
368 |
|
|
\begin{align*} |
369 |
|
|
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
370 |
|
|
% |
371 |
|
|
P(t) &\leftarrow \left\{{\bf r}(t)\right\}, \left\{{\bf v}(t)\right\} ,\\ |
372 |
|
|
% |
373 |
|
|
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
374 |
|
|
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
375 |
|
|
\left(\chi(t) + \eta(t) \right) \right), \\ |
376 |
|
|
% |
377 |
|
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
378 |
|
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
379 |
|
|
\chi(t) \right), \\ |
380 |
|
|
% |
381 |
|
|
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
382 |
|
|
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
383 |
|
|
\right) ,\\ |
384 |
|
|
% |
385 |
|
|
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
386 |
|
|
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
387 |
|
|
\right) ,\\ |
388 |
|
|
% |
389 |
|
|
\eta(t + h / 2) &\leftarrow \eta(t) + \frac{h |
390 |
|
|
\mathcal{V}(t)}{2 N k_B T(t) \tau_B^2} \left( P(t) |
391 |
|
|
- P_{\mathrm{target}} \right), \\ |
392 |
|
|
% |
393 |
|
|
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h |
394 |
|
|
\left\{ {\bf v}\left(t + h / 2 \right) |
395 |
|
|
+ \eta(t + h / 2)\left[ {\bf r}(t + h) |
396 |
|
|
- {\bf R}_0 \right] \right\} ,\\ |
397 |
|
|
% |
398 |
|
|
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
399 |
|
|
\mathsf{H}(t). |
400 |
|
|
\end{align*} |
401 |
|
|
|
402 |
|
|
Most of these equations are identical to their counterparts in the |
403 |
|
|
NVT integrator, but the propagation of positions to time $t + h$ |
404 |
|
|
depends on the positions at the same time. {\sc oopse} carries out |
405 |
|
|
this step iteratively (with a limit of 5 passes through the |
406 |
|
|
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
407 |
|
|
uniformly for one full time step by an exponential factor that |
408 |
|
|
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
409 |
|
|
box uniformly also scales the volume of the box by |
410 |
|
|
\begin{equation} |
411 |
|
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
412 |
|
|
\mathcal{V}(t) |
413 |
|
|
\end{equation} |
414 |
|
|
|
415 |
|
|
The {\tt doForces} step for the NPTi integrator is exactly the same |
416 |
|
|
as in both the DLM and NVT integrators. Once the forces and torques |
417 |
|
|
have been obtained at the new time step, the velocities can be |
418 |
|
|
advanced to the same time value. |
419 |
|
|
|
420 |
|
|
{\tt moveB:} |
421 |
|
|
\begin{align*} |
422 |
|
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
423 |
|
|
\left\{{\bf j}(t + h)\right\} ,\\ |
424 |
|
|
% |
425 |
|
|
P(t + h) &\leftarrow \left\{{\bf r}(t + h)\right\}, |
426 |
|
|
\left\{{\bf v}(t + h)\right\}, \\ |
427 |
|
|
% |
428 |
|
|
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
429 |
|
|
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
430 |
|
|
{T_{\mathrm{target}}} - 1 \right), \\ |
431 |
|
|
% |
432 |
|
|
\eta(t + h) &\leftarrow \eta(t + h / 2) + |
433 |
|
|
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
434 |
|
|
\tau_B^2} \left( P(t + h) - P_{\mathrm{target}} \right), \\ |
435 |
|
|
% |
436 |
|
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
437 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( |
438 |
|
|
\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
439 |
|
|
(\chi(t + h) + \eta(t + h)) \right) ,\\ |
440 |
|
|
% |
441 |
|
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
442 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( {\bf |
443 |
|
|
\tau}^b(t + h) - {\bf j}(t + h) |
444 |
|
|
\chi(t + h) \right) . |
445 |
|
|
\end{align*} |
446 |
|
|
|
447 |
|
|
Once again, since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required |
448 |
|
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
449 |
|
|
h)$, they indirectly depend on their own values at time $t + h$. |
450 |
|
|
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
451 |
|
|
+ h)$ and $\eta(t + h)$ become self-consistent. The relative |
452 |
|
|
tolerance for the self-consistency check defaults to a value of |
453 |
|
|
$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after |
454 |
|
|
4 loops even if the consistency check has not been satisfied. |
455 |
|
|
|
456 |
|
|
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
457 |
|
|
is known to conserve a Hamiltonian for the extended system that is, |
458 |
|
|
to within a constant, identical to the Gibbs free energy, |
459 |
|
|
\begin{equation} |
460 |
|
|
H_{\mathrm{NPTi}} = V + K + f k_B T_{\mathrm{target}} \left( |
461 |
|
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
462 |
|
|
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t). |
463 |
|
|
\end{equation} |
464 |
|
|
Poor choices of $\delta t$, $\tau_T$, or $\tau_B$ can result in |
465 |
|
|
non-conservation of $H_{\mathrm{NPTi}}$, so the conserved quantity |
466 |
|
|
is maintained in the last column of the {\tt .stat} file to allow |
467 |
|
|
checks on the quality of the integration. It is also known that |
468 |
|
|
this algorithm samples the equilibrium distribution for the enthalpy |
469 |
|
|
(including contributions for the thermostat and barostat), |
470 |
|
|
\begin{equation} |
471 |
|
|
H_{\mathrm{NPTi}} = V + K + \frac{f k_B T_{\mathrm{target}}}{2} |
472 |
|
|
\left( \chi^2 \tau_T^2 + \eta^2 \tau_B^2 \right) + |
473 |
|
|
P_{\mathrm{target}} \mathcal{V}(t). |
474 |
|
|
\end{equation} |
475 |
|
|
|
476 |
|
|
Bond constraints are applied at the end of both the {\tt moveA} and |
477 |
tim |
2807 |
{\tt moveB} portions of the algorithm. |
478 |
tim |
2729 |
|
479 |
|
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
480 |
|
|
flexible box (NPTf)} |
481 |
|
|
|
482 |
|
|
There is a relatively simple generalization of the |
483 |
|
|
Nos\'e-Hoover-Andersen method to include changes in the simulation |
484 |
|
|
box {\it shape} as well as in the volume of the box. This method |
485 |
|
|
utilizes the full $3 \times 3$ pressure tensor and introduces a |
486 |
|
|
tensor of extended variables ($\overleftrightarrow{\eta}$) to |
487 |
|
|
control changes to the box shape. The equations of motion for this |
488 |
|
|
method are |
489 |
|
|
\begin{eqnarray} |
490 |
|
|
\dot{{\bf r}} & = & {\bf v} + \overleftrightarrow{\eta} \cdot \left( {\bf r} - {\bf R}_0 \right), \\ |
491 |
|
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\overleftrightarrow{\eta} + |
492 |
|
|
\chi \cdot \mathsf{1}) {\bf v}, \\ |
493 |
|
|
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
494 |
|
|
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) ,\\ |
495 |
|
|
\dot{{\bf j}} & = & {\bf j} \times \left( |
496 |
|
|
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
497 |
|
|
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial |
498 |
|
|
V}{\partial \mathsf{A}} \right) - \chi {\bf j} ,\\ |
499 |
|
|
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
500 |
|
|
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
501 |
|
|
\dot{\overleftrightarrow{\eta}} & = & \frac{1}{\tau_{B}^2 f k_B |
502 |
|
|
T_{\mathrm{target}}} V \left( \overleftrightarrow{\mathsf{P}} - P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
503 |
|
|
\dot{\mathsf{H}} & = & \overleftrightarrow{\eta} \cdot \mathsf{H} . |
504 |
|
|
\label{eq:melchionna2} |
505 |
|
|
\end{eqnarray} |
506 |
|
|
|
507 |
|
|
Here, $\mathsf{1}$ is the unit matrix and |
508 |
|
|
$\overleftrightarrow{\mathsf{P}}$ is the pressure tensor. Again, |
509 |
|
|
the volume, $\mathcal{V} = \det \mathsf{H}$. |
510 |
|
|
|
511 |
|
|
The propagation of the equations of motion is nearly identical to |
512 |
|
|
the NPTi integration: |
513 |
|
|
|
514 |
|
|
{\tt moveA:} |
515 |
|
|
\begin{align*} |
516 |
|
|
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
517 |
|
|
% |
518 |
|
|
\overleftrightarrow{\mathsf{P}}(t) &\leftarrow \left\{{\bf |
519 |
|
|
r}(t)\right\}, |
520 |
|
|
\left\{{\bf v}(t)\right\} ,\\ |
521 |
|
|
% |
522 |
|
|
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
523 |
|
|
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - |
524 |
|
|
\left(\chi(t)\mathsf{1} + \overleftrightarrow{\eta}(t) \right) \cdot |
525 |
|
|
{\bf v}(t) \right), \\ |
526 |
|
|
% |
527 |
|
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
528 |
|
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
529 |
|
|
\chi(t) \right), \\ |
530 |
|
|
% |
531 |
|
|
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
532 |
|
|
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
533 |
|
|
\right), \\ |
534 |
|
|
% |
535 |
|
|
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
536 |
|
|
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} |
537 |
|
|
- 1 \right), \\ |
538 |
|
|
% |
539 |
|
|
\overleftrightarrow{\eta}(t + h / 2) &\leftarrow |
540 |
|
|
\overleftrightarrow{\eta}(t) + \frac{h \mathcal{V}(t)}{2 N k_B |
541 |
|
|
T(t) \tau_B^2} \left( \overleftrightarrow{\mathsf{P}}(t) |
542 |
|
|
- P_{\mathrm{target}}\mathsf{1} \right), \\ |
543 |
|
|
% |
544 |
|
|
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h \left\{ {\bf v} |
545 |
|
|
\left(t + h / 2 \right) + \overleftrightarrow{\eta}(t + |
546 |
|
|
h / 2) \cdot \left[ {\bf r}(t + h) |
547 |
|
|
- {\bf R}_0 \right] \right\}, \\ |
548 |
|
|
% |
549 |
|
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
550 |
|
|
\overleftrightarrow{\eta}(t + h / 2)} . |
551 |
|
|
\end{align*} |
552 |
|
|
{\sc oopse} uses a power series expansion truncated at second order |
553 |
|
|
for the exponential operation which scales the simulation box. |
554 |
|
|
|
555 |
|
|
The {\tt moveB} portion of the algorithm is largely unchanged from |
556 |
|
|
the NPTi integrator: |
557 |
|
|
|
558 |
|
|
{\tt moveB:} |
559 |
|
|
\begin{align*} |
560 |
|
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
561 |
|
|
\left\{{\bf j}(t + h)\right\}, \\ |
562 |
|
|
% |
563 |
|
|
\overleftrightarrow{\mathsf{P}}(t + h) &\leftarrow \left\{{\bf r} |
564 |
|
|
(t + h)\right\}, \left\{{\bf v}(t |
565 |
|
|
+ h)\right\}, \left\{{\bf f}(t + h)\right\} ,\\ |
566 |
|
|
% |
567 |
|
|
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
568 |
|
|
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+ |
569 |
|
|
h)}{T_{\mathrm{target}}} - 1 \right), \\ |
570 |
|
|
% |
571 |
|
|
\overleftrightarrow{\eta}(t + h) &\leftarrow |
572 |
|
|
\overleftrightarrow{\eta}(t + h / 2) + |
573 |
|
|
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
574 |
|
|
\tau_B^2} \left( \overleftrightarrow{P}(t + h) |
575 |
|
|
- P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
576 |
|
|
% |
577 |
|
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
578 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( |
579 |
|
|
\frac{{\bf f}(t + h)}{m} - |
580 |
|
|
(\chi(t + h)\mathsf{1} + \overleftrightarrow{\eta}(t |
581 |
|
|
+ h)) \right) \cdot {\bf v}(t + h), \\ |
582 |
|
|
% |
583 |
|
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
584 |
|
|
+ h / 2 \right) + \frac{h}{2} \left( {\bf \tau}^b(t |
585 |
|
|
+ h) - {\bf j}(t + h) \chi(t + h) \right) . |
586 |
|
|
\end{align*} |
587 |
|
|
|
588 |
|
|
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
589 |
|
|
identical to those described for the NPTi integrator. |
590 |
|
|
|
591 |
|
|
The NPTf integrator is known to conserve the following Hamiltonian: |
592 |
|
|
\begin{equation} |
593 |
|
|
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
594 |
|
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
595 |
|
|
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
596 |
|
|
T_{\mathrm{target}}}{2} |
597 |
|
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
598 |
|
|
\end{equation} |
599 |
|
|
|
600 |
|
|
This integrator must be used with care, particularly in liquid |
601 |
|
|
simulations. Liquids have very small restoring forces in the |
602 |
|
|
off-diagonal directions, and the simulation box can very quickly |
603 |
|
|
form elongated and sheared geometries which become smaller than the |
604 |
|
|
electrostatic or Lennard-Jones cutoff radii. The NPTf integrator |
605 |
|
|
finds most use in simulating crystals or liquid crystals which |
606 |
|
|
assume non-orthorhombic geometries. |
607 |
|
|
|
608 |
tim |
2739 |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
609 |
tim |
2729 |
|
610 |
tim |
2776 |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
611 |
tim |
2729 |
|
612 |
tim |
2739 |
A comprehensive understanding of structure¨Cfunction relations of |
613 |
|
|
biological membrane system ultimately relies on structure and |
614 |
|
|
dynamics of lipid bilayer, which are strongly affected by the |
615 |
|
|
interfacial interaction between lipid molecules and surrounding |
616 |
|
|
media. One quantity to describe the interfacial interaction is so |
617 |
|
|
called the average surface area per lipid. Constat area and constant |
618 |
|
|
lateral pressure simulation can be achieved by extending the |
619 |
|
|
standard NPT ensemble with a different pressure control strategy |
620 |
tim |
2798 |
|
621 |
tim |
2729 |
\begin{equation} |
622 |
tim |
2799 |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
623 |
tim |
2798 |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
624 |
tim |
2799 |
& \mbox{if $ \alpha = \beta = z$}\\ |
625 |
tim |
2798 |
0 & \mbox{otherwise}\\ |
626 |
|
|
\end{array} |
627 |
|
|
\right. |
628 |
tim |
2729 |
\end{equation} |
629 |
tim |
2798 |
|
630 |
tim |
2739 |
Note that the iterative schemes for NPAT are identical to those |
631 |
|
|
described for the NPTi integrator. |
632 |
tim |
2729 |
|
633 |
tim |
2776 |
\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble} |
634 |
tim |
2729 |
|
635 |
tim |
2739 |
Theoretically, the surface tension $\gamma$ of a stress free |
636 |
|
|
membrane system should be zero since its surface free energy $G$ is |
637 |
|
|
minimum with respect to surface area $A$ |
638 |
|
|
\[ |
639 |
|
|
\gamma = \frac{{\partial G}}{{\partial A}}. |
640 |
|
|
\] |
641 |
|
|
However, a surface tension of zero is not appropriate for relatively |
642 |
|
|
small patches of membrane. In order to eliminate the edge effect of |
643 |
tim |
2776 |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
644 |
|
|
proposed to maintain the lateral surface tension and normal |
645 |
|
|
pressure. The equation of motion for cell size control tensor, |
646 |
tim |
2778 |
$\eta$, in $NP\gamma T$ is |
647 |
tim |
2729 |
\begin{equation} |
648 |
tim |
2799 |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
649 |
tim |
2798 |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
650 |
|
|
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
651 |
|
|
0 & \mbox{$\alpha \ne \beta$} \\ |
652 |
tim |
2799 |
\end{array} |
653 |
tim |
2798 |
\right. |
654 |
tim |
2729 |
\end{equation} |
655 |
tim |
2739 |
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
656 |
|
|
the instantaneous surface tensor $\gamma _\alpha$ is given by |
657 |
tim |
2729 |
\begin{equation} |
658 |
tim |
2800 |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
659 |
|
|
- P_{{\rm{target}}} ) |
660 |
tim |
2739 |
\label{methodEquation:instantaneousSurfaceTensor} |
661 |
tim |
2729 |
\end{equation} |
662 |
|
|
|
663 |
tim |
2739 |
There is one additional extended system integrator (NPTxyz), in |
664 |
|
|
which each attempt to preserve the target pressure along the box |
665 |
|
|
walls perpendicular to that particular axis. The lengths of the box |
666 |
|
|
axes are allowed to fluctuate independently, but the angle between |
667 |
|
|
the box axes does not change. It should be noted that the NPTxyz |
668 |
|
|
integrator is a special case of $NP\gamma T$ if the surface tension |
669 |
|
|
$\gamma$ is set to zero. |
670 |
tim |
2729 |
|
671 |
tim |
2804 |
\section{\label{methodSection:zcons}Z-Constraint Method} |
672 |
tim |
2776 |
|
673 |
tim |
2804 |
Based on the fluctuation-dissipation theorem, a force |
674 |
|
|
auto-correlation method was developed by Roux and Karplus to |
675 |
|
|
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
676 |
|
|
The time-dependent friction coefficient can be calculated from the |
677 |
|
|
deviation of the instantaneous force from its mean force. |
678 |
|
|
\begin{equation} |
679 |
|
|
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
680 |
|
|
\end{equation} |
681 |
|
|
where% |
682 |
|
|
\begin{equation} |
683 |
|
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
684 |
|
|
\end{equation} |
685 |
tim |
2776 |
|
686 |
tim |
2804 |
If the time-dependent friction decays rapidly, the static friction |
687 |
|
|
coefficient can be approximated by |
688 |
|
|
\begin{equation} |
689 |
|
|
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
690 |
|
|
F(z,0)\rangle dt. |
691 |
|
|
\end{equation} |
692 |
|
|
Allowing diffusion constant to then be calculated through the |
693 |
|
|
Einstein relation:\cite{Marrink1994} |
694 |
|
|
\begin{equation} |
695 |
|
|
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
696 |
|
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
697 |
|
|
\end{equation} |
698 |
tim |
2776 |
|
699 |
tim |
2804 |
The Z-Constraint method, which fixes the z coordinates of the |
700 |
|
|
molecules with respect to the center of the mass of the system, has |
701 |
|
|
been a method suggested to obtain the forces required for the force |
702 |
|
|
auto-correlation calculation.\cite{Marrink1994} However, simply |
703 |
|
|
resetting the coordinate will move the center of the mass of the |
704 |
|
|
whole system. To avoid this problem, we reset the forces of |
705 |
|
|
z-constrained molecules as well as subtract the total constraint |
706 |
|
|
forces from the rest of the system after the force calculation at |
707 |
|
|
each time step instead of resetting the coordinate. |
708 |
|
|
|
709 |
|
|
After the force calculation, define $G_\alpha$ as |
710 |
|
|
\begin{equation} |
711 |
|
|
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
712 |
|
|
\end{equation} |
713 |
|
|
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
714 |
|
|
z-constrained molecule $\alpha$. The forces of the z constrained |
715 |
|
|
molecule are then set to: |
716 |
|
|
\begin{equation} |
717 |
|
|
F_{\alpha i} = F_{\alpha i} - |
718 |
|
|
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
719 |
|
|
\end{equation} |
720 |
|
|
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
721 |
|
|
molecule. Having rescaled the forces, the velocities must also be |
722 |
|
|
rescaled to subtract out any center of mass velocity in the z |
723 |
|
|
direction. |
724 |
|
|
\begin{equation} |
725 |
|
|
v_{\alpha i} = v_{\alpha i} - |
726 |
|
|
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
727 |
|
|
\end{equation} |
728 |
|
|
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
729 |
|
|
Lastly, all of the accumulated z constrained forces must be |
730 |
|
|
subtracted from the system to keep the system center of mass from |
731 |
|
|
drifting. |
732 |
|
|
\begin{equation} |
733 |
|
|
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
734 |
|
|
G_{\alpha}} |
735 |
|
|
{\sum_{\beta}\sum_i m_{\beta i}}, |
736 |
|
|
\end{equation} |
737 |
|
|
where $\beta$ are all of the unconstrained molecules in the system. |
738 |
|
|
Similarly, the velocities of the unconstrained molecules must also |
739 |
|
|
be scaled. |
740 |
|
|
\begin{equation} |
741 |
|
|
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
742 |
|
|
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
743 |
|
|
\end{equation} |
744 |
|
|
|
745 |
|
|
At the very beginning of the simulation, the molecules may not be at |
746 |
|
|
their constrained positions. To move a z-constrained molecule to its |
747 |
|
|
specified position, a simple harmonic potential is used |
748 |
|
|
\begin{equation} |
749 |
|
|
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
750 |
|
|
\end{equation} |
751 |
|
|
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
752 |
|
|
is the current $z$ coordinate of the center of mass of the |
753 |
|
|
constrained molecule, and $z_{\text{cons}}$ is the constrained |
754 |
|
|
position. The harmonic force operating on the z-constrained molecule |
755 |
|
|
at time $t$ can be calculated by |
756 |
|
|
\begin{equation} |
757 |
|
|
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
758 |
|
|
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
759 |
|
|
\end{equation} |
760 |
|
|
|
761 |
|
|
|
762 |
tim |
2685 |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
763 |
|
|
|
764 |
tim |
2804 |
%\subsection{\label{methodSection:temperature}Temperature Control} |
765 |
tim |
2685 |
|
766 |
tim |
2804 |
%\subsection{\label{methodSection:pressureControl}Pressure Control} |
767 |
tim |
2729 |
|
768 |
tim |
2804 |
%\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
769 |
tim |
2729 |
|
770 |
tim |
2804 |
%applications of langevin dynamics |
771 |
|
|
As an excellent alternative to newtonian dynamics, Langevin |
772 |
|
|
dynamics, which mimics a simple heat bath with stochastic and |
773 |
|
|
dissipative forces, has been applied in a variety of studies. The |
774 |
|
|
stochastic treatment of the solvent enables us to carry out |
775 |
|
|
substantially longer time simulation. Implicit solvent Langevin |
776 |
|
|
dynamics simulation of met-enkephalin not only outperforms explicit |
777 |
|
|
solvent simulation on computation efficiency, but also agrees very |
778 |
|
|
well with explicit solvent simulation on dynamics |
779 |
|
|
properties\cite{Shen2002}. Recently, applying Langevin dynamics with |
780 |
|
|
UNRES model, Liow and his coworkers suggest that protein folding |
781 |
|
|
pathways can be possibly exploited within a reasonable amount of |
782 |
|
|
time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics |
783 |
|
|
also enhances the sampling of the system and increases the |
784 |
|
|
probability of crossing energy barrier\cite{Banerjee2004, Cui2003}. |
785 |
|
|
Combining Langevin dynamics with Kramers's theory, Klimov and |
786 |
|
|
Thirumalai identified the free-energy barrier by studying the |
787 |
|
|
viscosity dependence of the protein folding rates\cite{Klimov1997}. |
788 |
|
|
In order to account for solvent induced interactions missing from |
789 |
|
|
implicit solvent model, Kaya incorporated desolvation free energy |
790 |
|
|
barrier into implicit coarse-grained solvent model in protein |
791 |
|
|
folding/unfolding study and discovered a higher free energy barrier |
792 |
|
|
between the native and denatured states. Because of its stability |
793 |
|
|
against noise, Langevin dynamics is very suitable for studying |
794 |
|
|
remagnetization processes in various |
795 |
tim |
2807 |
systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
796 |
|
|
oscillation power spectrum of nanoparticles from Langevin dynamics |
797 |
|
|
simulation has the same peak frequencies for different wave |
798 |
|
|
vectors,which recovers the property of magnetic excitations in small |
799 |
|
|
finite structures\cite{Berkov2005a}. In an attempt to reduce the |
800 |
|
|
computational cost of simulation, multiple time stepping (MTS) |
801 |
|
|
methods have been introduced and have been of great interest to |
802 |
|
|
macromolecule and protein community\cite{Tuckerman1992}. Relying on |
803 |
|
|
the observation that forces between distant atoms generally |
804 |
|
|
demonstrate slower fluctuations than forces between close atoms, MTS |
805 |
|
|
method are generally implemented by evaluating the slowly |
806 |
|
|
fluctuating forces less frequently than the fast ones. |
807 |
|
|
Unfortunately, nonlinear instability resulting from increasing |
808 |
|
|
timestep in MTS simulation have became a critical obstruction |
809 |
|
|
preventing the long time simulation. Due to the coupling to the heat |
810 |
|
|
bath, Langevin dynamics has been shown to be able to damp out the |
811 |
|
|
resonance artifact more efficiently\cite{Sandu1999}. |
812 |
tim |
2729 |
|
813 |
tim |
2804 |
%review rigid body dynamics |
814 |
|
|
Rigid bodies are frequently involved in the modeling of different |
815 |
|
|
areas, from engineering, physics, to chemistry. For example, |
816 |
|
|
missiles and vehicle are usually modeled by rigid bodies. The |
817 |
|
|
movement of the objects in 3D gaming engine or other physics |
818 |
|
|
simulator is governed by the rigid body dynamics. In molecular |
819 |
|
|
simulation, rigid body is used to simplify the model in |
820 |
|
|
protein-protein docking study\cite{Gray2003}. |
821 |
|
|
|
822 |
|
|
It is very important to develop stable and efficient methods to |
823 |
|
|
integrate the equations of motion of orientational degrees of |
824 |
|
|
freedom. Euler angles are the nature choice to describe the |
825 |
|
|
rotational degrees of freedom. However, due to its singularity, the |
826 |
|
|
numerical integration of corresponding equations of motion is very |
827 |
|
|
inefficient and inaccurate. Although an alternative integrator using |
828 |
|
|
different sets of Euler angles can overcome this |
829 |
|
|
difficulty\cite{Ryckaert1977, Andersen1983}, the computational |
830 |
|
|
penalty and the lost of angular momentum conservation still remain. |
831 |
|
|
In 1977, a singularity free representation utilizing quaternions was |
832 |
|
|
developed by Evans\cite{Evans1977}. Unfortunately, this approach |
833 |
|
|
suffer from the nonseparable Hamiltonian resulted from quaternion |
834 |
|
|
representation, which prevents the symplectic algorithm to be |
835 |
|
|
utilized. Another different approach is to apply holonomic |
836 |
|
|
constraints to the atoms belonging to the rigid |
837 |
|
|
body\cite{Barojas1973}. Each atom moves independently under the |
838 |
|
|
normal forces deriving from potential energy and constraint forces |
839 |
|
|
which are used to guarantee the rigidness. However, due to their |
840 |
|
|
iterative nature, SHAKE and Rattle algorithm converge very slowly |
841 |
|
|
when the number of constraint increases. |
842 |
|
|
|
843 |
|
|
The break through in geometric literature suggests that, in order to |
844 |
|
|
develop a long-term integration scheme, one should preserve the |
845 |
|
|
geometric structure of the flow. Matubayasi and Nakahara developed a |
846 |
|
|
time-reversible integrator for rigid bodies in quaternion |
847 |
|
|
representation. Although it is not symplectic, this integrator still |
848 |
|
|
demonstrates a better long-time energy conservation than traditional |
849 |
|
|
methods because of the time-reversible nature. Extending |
850 |
|
|
Trotter-Suzuki to general system with a flat phase space, Miller and |
851 |
|
|
his colleagues devised an novel symplectic, time-reversible and |
852 |
|
|
volume-preserving integrator in quaternion representation. However, |
853 |
|
|
all of the integrators in quaternion representation suffer from the |
854 |
|
|
computational penalty of constructing a rotation matrix from |
855 |
|
|
quaternions to evolve coordinates and velocities at every time step. |
856 |
|
|
An alternative integration scheme utilizing rotation matrix directly |
857 |
|
|
is RSHAKE\cite{Kol1997}, in which a conjugate momentum to rotation |
858 |
|
|
matrix is introduced to re-formulate the Hamiltonian's equation and |
859 |
|
|
the Hamiltonian is evolved in a constraint manifold by iteratively |
860 |
|
|
satisfying the orthogonality constraint. However, RSHAKE is |
861 |
|
|
inefficient because of the iterative procedure. An extremely |
862 |
|
|
efficient integration scheme in rotation matrix representation, |
863 |
|
|
which also preserves the same structural properties of the |
864 |
|
|
Hamiltonian flow as Miller's integrator, is proposed by Dullweber, |
865 |
|
|
Leimkuhler and McLachlan (DLM)\cite{Dullweber1997}. |
866 |
|
|
|
867 |
|
|
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
868 |
|
|
Combining Langevin or Brownian dynamics with rigid body dynamics, |
869 |
|
|
one can study the slow processes in biomolecular systems. Modeling |
870 |
|
|
the DNA as a chain of rigid spheres beads, which subject to harmonic |
871 |
|
|
potentials as well as excluded volume potentials, Mielke and his |
872 |
|
|
coworkers discover rapid superhelical stress generations from the |
873 |
|
|
stochastic simulation of twin supercoiling DNA with response to |
874 |
|
|
induced torques\cite{Mielke2004}. Membrane fusion is another key |
875 |
|
|
biological process which controls a variety of physiological |
876 |
|
|
functions, such as release of neurotransmitters \textit{etc}. A |
877 |
|
|
typical fusion event happens on the time scale of millisecond, which |
878 |
|
|
is impracticable to study using all atomistic model with newtonian |
879 |
|
|
mechanics. With the help of coarse-grained rigid body model and |
880 |
|
|
stochastic dynamics, the fusion pathways were exploited by many |
881 |
|
|
researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the |
882 |
|
|
difficulty of numerical integration of anisotropy rotation, most of |
883 |
|
|
the rigid body models are simply modeled by sphere, cylinder, |
884 |
|
|
ellipsoid or other regular shapes in stochastic simulations. In an |
885 |
|
|
effort to account for the diffusion anisotropy of the arbitrary |
886 |
|
|
particles, Fernandes and de la Torre improved the original Brownian |
887 |
|
|
dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
888 |
|
|
incorporating a generalized $6\times6$ diffusion tensor and |
889 |
|
|
introducing a simple rotation evolution scheme consisting of three |
890 |
|
|
consecutive rotations\cite{Fernandes2002}. Unfortunately, unexpected |
891 |
|
|
error and bias are introduced into the system due to the arbitrary |
892 |
|
|
order of applying the noncommuting rotation |
893 |
|
|
operators\cite{Beard2003}. Based on the observation the momentum |
894 |
|
|
relaxation time is much less than the time step, one may ignore the |
895 |
|
|
inertia in Brownian dynamics. However, assumption of the zero |
896 |
|
|
average acceleration is not always true for cooperative motion which |
897 |
|
|
is common in protein motion. An inertial Brownian dynamics (IBD) was |
898 |
|
|
proposed to address this issue by adding an inertial correction |
899 |
tim |
2807 |
term\cite{Beard2003}. As a complement to IBD which has a lower bound |
900 |
tim |
2804 |
in time step because of the inertial relaxation time, long-time-step |
901 |
|
|
inertial dynamics (LTID) can be used to investigate the inertial |
902 |
|
|
behavior of the polymer segments in low friction |
903 |
|
|
regime\cite{Beard2001}. LTID can also deal with the rotational |
904 |
|
|
dynamics for nonskew bodies without translation-rotation coupling by |
905 |
|
|
separating the translation and rotation motion and taking advantage |
906 |
|
|
of the analytical solution of hydrodynamics properties. However, |
907 |
|
|
typical nonskew bodies like cylinder and ellipsoid are inadequate to |
908 |
|
|
represent most complex macromolecule assemblies. These intricate |
909 |
|
|
molecules have been represented by a set of beads and their |
910 |
|
|
hydrodynamics properties can be calculated using variant |
911 |
|
|
hydrodynamic interaction tensors. |
912 |
|
|
|
913 |
|
|
The goal of the present work is to develop a Langevin dynamics |
914 |
|
|
algorithm for arbitrary rigid particles by integrating the accurate |
915 |
|
|
estimation of friction tensor from hydrodynamics theory into the |
916 |
|
|
sophisticated rigid body dynamics. |
917 |
|
|
|
918 |
|
|
|
919 |
|
|
\subsection{Friction Tensor} |
920 |
|
|
|
921 |
|
|
For an arbitrary rigid body moves in a fluid, it may experience |
922 |
|
|
friction force $f_r$ or friction torque $\tau _r$ along the opposite |
923 |
|
|
direction of the velocity $v$ or angular velocity $\omega$ at |
924 |
|
|
arbitrary origin $P$, |
925 |
|
|
\begin{equation} |
926 |
|
|
\left( \begin{array}{l} |
927 |
|
|
f_r \\ |
928 |
|
|
\tau _r \\ |
929 |
|
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
930 |
|
|
{\Xi _{P,t} } & {\Xi _{P,c}^T } \\ |
931 |
|
|
{\Xi _{P,c} } & {\Xi _{P,r} } \\ |
932 |
|
|
\end{array}} \right)\left( \begin{array}{l} |
933 |
|
|
\nu \\ |
934 |
|
|
\omega \\ |
935 |
|
|
\end{array} \right) |
936 |
|
|
\end{equation} |
937 |
|
|
where $\Xi _{P,t}t$ is the translation friction tensor, $\Xi _{P,r}$ |
938 |
|
|
is the rotational friction tensor and $\Xi _{P,c}$ is the |
939 |
|
|
translation-rotation coupling tensor. The procedure of calculating |
940 |
|
|
friction tensor using hydrodynamic tensor and comparison between |
941 |
|
|
bead model and shell model were elaborated by Carrasco \textit{et |
942 |
|
|
al}\cite{Carrasco1999}. An important property of the friction tensor |
943 |
|
|
is that the translational friction tensor is independent of origin |
944 |
|
|
while the rotational and coupling are sensitive to the choice of the |
945 |
|
|
origin \cite{Brenner1967}, which can be described by |
946 |
|
|
\begin{equation} |
947 |
|
|
\begin{array}{c} |
948 |
|
|
\Xi _{P,t} = \Xi _{O,t} = \Xi _t \\ |
949 |
|
|
\Xi _{P,c} = \Xi _{O,c} - r_{OP} \times \Xi _t \\ |
950 |
|
|
\Xi _{P,r} = \Xi _{O,r} - r_{OP} \times \Xi _t \times r_{OP} + \Xi _{O,c} \times r_{OP} - r_{OP} \times \Xi _{O,c}^T \\ |
951 |
|
|
\end{array} |
952 |
|
|
\end{equation} |
953 |
|
|
Where $O$ is another origin and $r_{OP}$ is the vector joining $O$ |
954 |
|
|
and $P$. It is also worthy of mention that both of translational and |
955 |
|
|
rotational frictional tensors are always symmetric. In contrast, |
956 |
|
|
coupling tensor is only symmetric at center of reaction: |
957 |
|
|
\begin{equation} |
958 |
|
|
\Xi _{R,c} = \Xi _{R,c}^T |
959 |
|
|
\end{equation} |
960 |
|
|
The proper location for applying friction force is the center of |
961 |
|
|
reaction, at which the trace of rotational resistance tensor reaches |
962 |
|
|
minimum. |
963 |
|
|
|
964 |
|
|
\subsection{Rigid body dynamics} |
965 |
|
|
|
966 |
|
|
The Hamiltonian of rigid body can be separated in terms of potential |
967 |
|
|
energy $V(r,A)$ and kinetic energy $T(p,\pi)$, |
968 |
|
|
\[ |
969 |
|
|
H = V(r,A) + T(v,\pi ) |
970 |
|
|
\] |
971 |
|
|
A second-order symplectic method is now obtained by the composition |
972 |
|
|
of the flow maps, |
973 |
|
|
\[ |
974 |
|
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
975 |
|
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
976 |
|
|
\] |
977 |
|
|
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
978 |
|
|
sub-flows which corresponding to force and torque respectively, |
979 |
|
|
\[ |
980 |
|
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
981 |
|
|
_{\Delta t/2,\tau }. |
982 |
|
|
\] |
983 |
|
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
984 |
|
|
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
985 |
|
|
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
986 |
|
|
|
987 |
|
|
Furthermore, kinetic potential can be separated to translational |
988 |
|
|
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
989 |
|
|
\begin{equation} |
990 |
|
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
991 |
|
|
\end{equation} |
992 |
|
|
where $ T^t (p) = \frac{1}{2}v^T m v $ and $T^r (\pi )$ is defined |
993 |
|
|
by \ref{introEquation:rotationalKineticRB}. Therefore, the |
994 |
|
|
corresponding flow maps are given by |
995 |
|
|
\[ |
996 |
|
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
997 |
|
|
_{\Delta t,T^r }. |
998 |
|
|
\] |
999 |
|
|
The free rigid body is an example of Lie-Poisson system with |
1000 |
|
|
Hamiltonian function |
1001 |
|
|
\begin{equation} |
1002 |
|
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1003 |
|
|
\label{introEquation:rotationalKineticRB} |
1004 |
|
|
\end{equation} |
1005 |
|
|
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
1006 |
|
|
Lie-Poisson structure matrix, |
1007 |
|
|
\begin{equation} |
1008 |
|
|
J(\pi ) = \left( {\begin{array}{*{20}c} |
1009 |
|
|
0 & {\pi _3 } & { - \pi _2 } \\ |
1010 |
|
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
1011 |
|
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
1012 |
|
|
\end{array}} \right) |
1013 |
|
|
\end{equation} |
1014 |
|
|
Thus, the dynamics of free rigid body is governed by |
1015 |
|
|
\begin{equation} |
1016 |
|
|
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
1017 |
|
|
\end{equation} |
1018 |
|
|
One may notice that each $T_i^r$ in Equation |
1019 |
|
|
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1020 |
|
|
instance, the equations of motion due to $T_1^r$ are given by |
1021 |
|
|
\begin{equation} |
1022 |
|
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}A = AR_1 |
1023 |
|
|
\label{introEqaution:RBMotionSingleTerm} |
1024 |
|
|
\end{equation} |
1025 |
|
|
where |
1026 |
|
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
1027 |
|
|
0 & 0 & 0 \\ |
1028 |
|
|
0 & 0 & {\pi _1 } \\ |
1029 |
|
|
0 & { - \pi _1 } & 0 \\ |
1030 |
|
|
\end{array}} \right). |
1031 |
|
|
\] |
1032 |
|
|
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1033 |
|
|
\[ |
1034 |
|
|
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),A(\Delta t) = |
1035 |
|
|
A(0)e^{\Delta tR_1 } |
1036 |
|
|
\] |
1037 |
|
|
with |
1038 |
|
|
\[ |
1039 |
|
|
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
1040 |
|
|
0 & 0 & 0 \\ |
1041 |
|
|
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
1042 |
|
|
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
1043 |
|
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1044 |
|
|
\] |
1045 |
|
|
To reduce the cost of computing expensive functions in $e^{\Delta |
1046 |
|
|
tR_1 }$, we can use Cayley transformation, |
1047 |
|
|
\[ |
1048 |
|
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1049 |
|
|
) |
1050 |
|
|
\] |
1051 |
|
|
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1052 |
|
|
manner. |
1053 |
|
|
|
1054 |
|
|
In order to construct a second-order symplectic method, we split the |
1055 |
|
|
angular kinetic Hamiltonian function into five terms |
1056 |
|
|
\[ |
1057 |
|
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1058 |
|
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1059 |
|
|
(\pi _1 ) |
1060 |
|
|
\]. |
1061 |
|
|
Concatenating flows corresponding to these five terms, we can obtain |
1062 |
|
|
the flow map for free rigid body, |
1063 |
|
|
\[ |
1064 |
|
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
1065 |
|
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1066 |
|
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1067 |
|
|
_1 }. |
1068 |
|
|
\] |
1069 |
|
|
|
1070 |
|
|
The equations of motion corresponding to potential energy and |
1071 |
|
|
kinetic energy are listed in the below table, |
1072 |
|
|
\begin{center} |
1073 |
|
|
\begin{tabular}{|l|l|} |
1074 |
|
|
\hline |
1075 |
|
|
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
1076 |
|
|
Potential & Kinetic \\ |
1077 |
|
|
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
1078 |
|
|
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
1079 |
|
|
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
1080 |
|
|
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
1081 |
|
|
\hline |
1082 |
|
|
\end{tabular} |
1083 |
|
|
\end{center} |
1084 |
|
|
|
1085 |
|
|
Finally, we obtain the overall symplectic flow maps for free moving |
1086 |
|
|
rigid body |
1087 |
|
|
\begin{align*} |
1088 |
|
|
\varphi _{\Delta t} = &\varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \circ \\ |
1089 |
|
|
&\varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \circ \\ |
1090 |
|
|
&\varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1091 |
|
|
\label{introEquation:overallRBFlowMaps} |
1092 |
|
|
\end{align*} |
1093 |
|
|
|
1094 |
|
|
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
1095 |
|
|
|
1096 |
|
|
Consider a Langevin equation of motions in generalized coordinates |
1097 |
|
|
\begin{equation} |
1098 |
|
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
1099 |
|
|
\label{LDGeneralizedForm} |
1100 |
|
|
\end{equation} |
1101 |
|
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
1102 |
|
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
1103 |
|
|
$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
1104 |
|
|
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
1105 |
|
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
1106 |
|
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
1107 |
|
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
1108 |
|
|
it is also convenient to handle the rotation of rigid body in |
1109 |
|
|
body-fixed frame. Thus the friction and random forces are calculated |
1110 |
|
|
in body-fixed frame and converted back to lab-fixed frame by: |
1111 |
|
|
\[ |
1112 |
|
|
\begin{array}{l} |
1113 |
|
|
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
1114 |
|
|
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
1115 |
|
|
\end{array}. |
1116 |
|
|
\] |
1117 |
|
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
1118 |
|
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
1119 |
|
|
angular velocity $\omega _i$, |
1120 |
|
|
\begin{equation} |
1121 |
|
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
1122 |
|
|
f_{r,i}^b (t) \\ |
1123 |
|
|
\tau _{r,i}^b (t) \\ |
1124 |
|
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
1125 |
|
|
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
1126 |
|
|
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
1127 |
|
|
\end{array}} \right)\left( \begin{array}{l} |
1128 |
|
|
v_{R,i}^b (t) \\ |
1129 |
|
|
\omega _i (t) \\ |
1130 |
|
|
\end{array} \right), |
1131 |
|
|
\end{equation} |
1132 |
|
|
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
1133 |
|
|
with zero mean and variance |
1134 |
|
|
\begin{equation} |
1135 |
|
|
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
1136 |
|
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
1137 |
|
|
2k_B T\Xi _R \delta (t - t'). |
1138 |
|
|
\end{equation} |
1139 |
|
|
The equation of motion for $v_i$ can be written as |
1140 |
|
|
\begin{equation} |
1141 |
|
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
1142 |
|
|
f_{r,i}^l (t) |
1143 |
|
|
\end{equation} |
1144 |
|
|
Since the frictional force is applied at the center of resistance |
1145 |
|
|
which generally does not coincide with the center of mass, an extra |
1146 |
|
|
torque is exerted at the center of mass. Thus, the net body-fixed |
1147 |
|
|
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
1148 |
|
|
given by |
1149 |
|
|
\begin{equation} |
1150 |
|
|
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
1151 |
|
|
\end{equation} |
1152 |
|
|
where $r_{MR}$ is the vector from the center of mass to the center |
1153 |
|
|
of the resistance. Instead of integrating angular velocity in |
1154 |
|
|
lab-fixed frame, we consider the equation of motion of angular |
1155 |
|
|
momentum in body-fixed frame |
1156 |
|
|
\begin{equation} |
1157 |
|
|
\dot \pi _i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b |
1158 |
|
|
(t) + \tau _{r,i}^b(t) |
1159 |
|
|
\end{equation} |
1160 |
|
|
|
1161 |
|
|
Embedding the friction terms into force and torque, one can |
1162 |
|
|
integrate the langevin equations of motion for rigid body of |
1163 |
|
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
1164 |
|
|
$h= \delta t$: |
1165 |
|
|
|
1166 |
|
|
{\tt part one:} |
1167 |
|
|
\begin{align*} |
1168 |
|
|
v_i (t + h/2) &\leftarrow v_i (t) + \frac{{hf_{t,i}^l (t)}}{{2m_i }} \\ |
1169 |
|
|
\pi _i (t + h/2) &\leftarrow \pi _i (t) + \frac{{h\tau _{t,i}^b (t)}}{2} \\ |
1170 |
|
|
r_i (t + h) &\leftarrow r_i (t) + hv_i (t + h/2) \\ |
1171 |
|
|
A_i (t + h) &\leftarrow rotate\left( {h\pi _i (t + h/2)I^{ - 1} } \right) \\ |
1172 |
|
|
\end{align*} |
1173 |
|
|
In this context, the $\mathrm{rotate}$ function is the reversible |
1174 |
|
|
product of five consecutive body-fixed rotations, |
1175 |
|
|
\begin{equation} |
1176 |
|
|
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
1177 |
|
|
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
1178 |
|
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
1179 |
|
|
\end{equation} |
1180 |
|
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
1181 |
|
|
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
1182 |
|
|
angular momentum ($\pi$) by an angle $\theta$ around body-fixed axis |
1183 |
|
|
$\alpha$, |
1184 |
|
|
\begin{equation} |
1185 |
|
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
1186 |
|
|
\begin{array}{lcl} |
1187 |
|
|
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
1188 |
|
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
1189 |
|
|
j}(0). |
1190 |
|
|
\end{array} |
1191 |
|
|
\right. |
1192 |
|
|
\end{equation} |
1193 |
|
|
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
1194 |
|
|
rotation matrix. For example, in the small-angle limit, the |
1195 |
|
|
rotation matrix around the body-fixed x-axis can be approximated as |
1196 |
|
|
\begin{equation} |
1197 |
|
|
\mathsf{R}_x(\theta) \approx \left( |
1198 |
|
|
\begin{array}{ccc} |
1199 |
|
|
1 & 0 & 0 \\ |
1200 |
|
|
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
1201 |
|
|
\theta^2 / 4} \\ |
1202 |
|
|
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
1203 |
|
|
\theta^2 / 4} |
1204 |
|
|
\end{array} |
1205 |
|
|
\right). |
1206 |
|
|
\end{equation} |
1207 |
|
|
All other rotations follow in a straightforward manner. |
1208 |
|
|
|
1209 |
|
|
After the first part of the propagation, the friction and random |
1210 |
|
|
forces are generated at the center of resistance in body-fixed frame |
1211 |
|
|
and converted back into lab-fixed frame |
1212 |
|
|
\[ |
1213 |
|
|
f_{t,i}^l (t + h) = - \left( {\frac{{\partial V}}{{\partial r_i }}} |
1214 |
|
|
\right)_{r_i (t + h)} + A_i^T (t + h)[F_{f,i}^b (t + h) + F_{r,i}^b |
1215 |
|
|
(t + h)], |
1216 |
|
|
\] |
1217 |
|
|
while the system torque in lab-fixed frame is transformed into |
1218 |
|
|
body-fixed frame, |
1219 |
|
|
\[ |
1220 |
|
|
\tau _{t,i}^b (t + h) = A\tau _{s,i}^l (t) + \tau _{n,i}^b (t) + |
1221 |
|
|
\tau _{r,i}^b (t). |
1222 |
|
|
\] |
1223 |
|
|
Once the forces and torques have been obtained at the new time step, |
1224 |
|
|
the velocities can be advanced to the same time value. |
1225 |
|
|
|
1226 |
|
|
{\tt part two:} |
1227 |
|
|
\begin{align*} |
1228 |
|
|
v_i (t) &\leftarrow v_i (t + h/2) + \frac{{hf_{t,i}^l (t + h)}}{{2m_i }} \\ |
1229 |
|
|
\pi _i (t) &\leftarrow \pi _i (t + h/2) + \frac{{h\tau _{t,i}^b (t + h)}}{2} \\ |
1230 |
|
|
\end{align*} |
1231 |
|
|
|
1232 |
|
|
\subsection{Results and discussion} |