--- trunk/tengDissertation/Methodology.tex	2006/06/06 14:12:59	2798
+++ trunk/tengDissertation/Methodology.tex	2006/06/11 02:55:13	2855
@@ -16,16 +16,16 @@ two decades. Matubayasi and Nakahara developed a time-
 
 Integration schemes for rotational motion of the rigid molecules in
 microcanonical ensemble have been extensively studied in the last
-two decades. Matubayasi and Nakahara developed a time-reversible
-integrator for rigid bodies in quaternion representation. Although
-it is not symplectic, this integrator still demonstrates a better
-long-time energy conservation than traditional methods because of
-the time-reversible nature. Extending Trotter-Suzuki to general
-system with a flat phase space, Miller and his colleagues devised an
-novel symplectic, time-reversible and volume-preserving integrator
-in quaternion representation, which was shown to be superior to the
-time-reversible integrator of Matubayasi and Nakahara. However, all
-of the integrators in quaternion representation suffer from the
+two decades. Matubayasi developed a time-reversible integrator for
+rigid bodies in quaternion representation. Although it is not
+symplectic, this integrator still demonstrates a better long-time
+energy conservation than traditional methods because of the
+time-reversible nature. Extending Trotter-Suzuki to general system
+with a flat phase space, Miller and his colleagues devised an novel
+symplectic, time-reversible and volume-preserving integrator in
+quaternion representation, which was shown to be superior to the
+Matubayasi's time-reversible integrator. However, all of the
+integrators in quaternion representation suffer from the
 computational penalty of constructing a rotation matrix from
 quaternions to evolve coordinates and velocities at every time step.
 An alternative integration scheme utilizing rotation matrix directly
@@ -117,7 +117,7 @@ torques are calculated at the new positions and orient
     \cdot {\bf \tau}^s(t + h).
 \end{align*}
 
-{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix
+${\bf u}$ will be automatically updated when the rotation matrix
 $\mathsf{A}$ is calculated in {\tt moveA}.  Once the forces and
 torques have been obtained at the new time step, the velocities can
 be advanced to the same time value.
@@ -139,7 +139,7 @@ in Fig.~\ref{timestep}.
 average 7\% increase in computation time using the DLM method in
 place of quaternions. This cost is more than justified when
 comparing the energy conservation of the two methods as illustrated
-in Fig.~\ref{timestep}.
+in Fig.~\ref{methodFig:timestep}.
 
 \begin{figure}
 \centering
@@ -150,25 +150,26 @@ from the true energy baseline for clarity.} \label{tim
 increasing time step. For each time step, the dotted line is total
 energy using the DLM integrator, and the solid line comes from the
 quaternion integrator. The larger time step plots are shifted up
-from the true energy baseline for clarity.} \label{timestep}
+from the true energy baseline for clarity.}
+\label{methodFig:timestep}
 \end{figure}
 
-In Fig.~\ref{timestep}, the resulting energy drift at various time
-steps for both the DLM and quaternion integration schemes is
-compared. All of the 1000 molecule water simulations started with
-the same configuration, and the only difference was the method for
-handling rotational motion. At time steps of 0.1 and 0.5 fs, both
-methods for propagating molecule rotation conserve energy fairly
-well, with the quaternion method showing a slight energy drift over
-time in the 0.5 fs time step simulation. At time steps of 1 and 2
-fs, the energy conservation benefits of the DLM method are clearly
-demonstrated. Thus, while maintaining the same degree of energy
-conservation, one can take considerably longer time steps, leading
-to an overall reduction in computation time.
+In Fig.~\ref{methodFig:timestep}, the resulting energy drift at
+various time steps for both the DLM and quaternion integration
+schemes is compared. All of the 1000 molecule water simulations
+started with the same configuration, and the only difference was the
+method for handling rotational motion. At time steps of 0.1 and 0.5
+fs, both methods for propagating molecule rotation conserve energy
+fairly well, with the quaternion method showing a slight energy
+drift over time in the 0.5 fs time step simulation. At time steps of
+1 and 2 fs, the energy conservation benefits of the DLM method are
+clearly demonstrated. Thus, while maintaining the same degree of
+energy conservation, one can take considerably longer time steps,
+leading to an overall reduction in computation time.
 
 \subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting}
 
-The Nos\'e-Hoover equations of motion are given by\cite{Hoover85}
+The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985}
 \begin{eqnarray}
 \dot{{\bf r}} & = & {\bf v}, \\
 \dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\
@@ -277,14 +278,11 @@ self-consistent.  The relative tolerance for the self-
 caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly
 depend on their own values at time $t + h$.  {\tt moveB} is
 therefore done in an iterative fashion until $\chi(t + h)$ becomes
-self-consistent.  The relative tolerance for the self-consistency
-check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will
-terminate the iteration after 4 loops even if the consistency check
-has not been satisfied.
+self-consistent.
 
 The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for
 the extended system that is, to within a constant, identical to the
-Helmholtz free energy,\cite{melchionna93}
+Helmholtz free energy,\cite{Melchionna1993}
 \begin{equation}
 H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left(
 \frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime)
@@ -294,17 +292,13 @@ of the integration.
 $H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the
 last column of the {\tt .stat} file to allow checks on the quality
 of the integration.
-
-Bond constraints are applied at the end of both the {\tt moveA} and
-{\tt moveB} portions of the algorithm.  Details on the constraint
-algorithms are given in section \ref{oopseSec:rattle}.
 
 \subsection{\label{methodSection:NPTi}Constant-pressure integration with
 isotropic box deformations (NPTi)}
 
-To carry out isobaric-isothermal ensemble calculations {\sc oopse}
-implements the Melchionna modifications to the
-Nos\'e-Hoover-Andersen equations of motion,\cite{melchionna93}
+Isobaric-isothermal ensemble integrator is implemented using the
+Melchionna modifications to the Nos\'e-Hoover-Andersen equations of
+motion,\cite{Melchionna1993}
 
 \begin{eqnarray}
 \dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\
@@ -359,11 +353,7 @@ relaxation of the pressure to the target value.  To se
 \end{equation}
 
 In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for
-relaxation of the pressure to the target value.  To set values for
-$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the
-{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt
-.bass} file.  The units for {\tt tauBarostat} are fs, and the units
-for the {\tt targetPressure} are atmospheres.  Like in the NVT
+relaxation of the pressure to the target value. Like in the NVT
 integrator, the integration of the equations of motion is carried
 out in a velocity-Verlet style 2 part algorithm:
 
@@ -404,12 +394,11 @@ depends on the positions at the same time.  {\sc oopse
 
 Most of these equations are identical to their counterparts in the
 NVT integrator, but the propagation of positions to time $t + h$
-depends on the positions at the same time.  {\sc oopse} carries out
-this step iteratively (with a limit of 5 passes through the
-iterative loop).  Also, the simulation box $\mathsf{H}$ is scaled
-uniformly for one full time step by an exponential factor that
-depends on the value of $\eta$ at time $t + h / 2$.  Reshaping the
-box uniformly also scales the volume of the box by
+depends on the positions at the same time. The simulation box
+$\mathsf{H}$ is scaled uniformly for one full time step by an
+exponential factor that depends on the value of $\eta$ at time $t +
+h / 2$.  Reshaping the box uniformly also scales the volume of the
+box by
 \begin{equation}
 \mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}.
 \mathcal{V}(t)
@@ -451,10 +440,7 @@ + h)$ and $\eta(t + h)$ become self-consistent.  The r
 to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t +
 h)$, they indirectly depend on their own values at time $t + h$.
 {\tt moveB} is therefore done in an iterative fashion until $\chi(t
-+ h)$ and $\eta(t + h)$ become self-consistent.  The relative
-tolerance for the self-consistency check defaults to a value of
-$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after
-4 loops even if the consistency check has not been satisfied.
++ h)$ and $\eta(t + h)$ become self-consistent.
 
 The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm
 is known to conserve a Hamiltonian for the extended system that is,
@@ -476,10 +462,6 @@ Bond constraints are applied at the end of both the {\
 P_{\mathrm{target}} \mathcal{V}(t).
 \end{equation}
 
-Bond constraints are applied at the end of both the {\tt moveA} and
-{\tt moveB} portions of the algorithm.  Details on the constraint
-algorithms are given in section \ref{oopseSec:rattle}.
-
 \subsection{\label{methodSection:NPTf}Constant-pressure integration with a
 flexible box (NPTf)}
 
@@ -553,8 +535,8 @@ r}(t)\right\},
 \mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h
     \overleftrightarrow{\eta}(t + h / 2)} .
 \end{align*}
-{\sc oopse} uses a power series expansion truncated at second order
-for the exponential operation which scales the simulation box.
+Here, a power series expansion truncated at second order for the
+exponential operation is used to scale the simulation box.
 
 The {\tt moveB} portion of the algorithm is largely unchanged from
 the NPTi integrator:
@@ -593,13 +575,14 @@ The NPTf integrator is known to conserve the following
 identical to those described for the NPTi integrator.
 
 The NPTf integrator is known to conserve the following Hamiltonian:
-\begin{equation}
-H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left(
+\begin{eqnarray*}
+H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left(
 \frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime)
-dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B
-T_{\mathrm{target}}}{2}
+dt^\prime \right) \\
++ P_{\mathrm{target}} \mathcal{V}(t) + \frac{f
+k_B T_{\mathrm{target}}}{2}
 \mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2.
-\end{equation}
+\end{eqnarray*}
 
 This integrator must be used with care, particularly in liquid
 simulations.  Liquids have very small restoring forces in the
@@ -609,10 +592,8 @@ assume non-orthorhombic geometries.
 finds most use in simulating crystals or liquid crystals which
 assume non-orthorhombic geometries.
 
-\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles}
+\subsection{\label{methodSection:NPAT}NPAT Ensemble}
 
-\subsubsection{\label{methodSection:NPAT}NPAT Ensemble}
-
 A comprehensive understanding of structure�Cfunction relations of
 biological membrane system ultimately relies on structure and
 dynamics of lipid bilayer, which are strongly affected by the
@@ -623,9 +604,9 @@ standard NPT ensemble with a different pressure contro
 standard NPT ensemble with a different pressure control strategy
 
 \begin{equation}
- \.{\overleftrightarrow{{\eta _{\alpha \beta}}}}=\left\{\begin{array}{ll}
+ \dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll}
                   \frac{{V(P_{\alpha \beta }  - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}
-                  & \mbox{if \[ \alpha = \beta  = z)$}\\
+                  & \mbox{if $ \alpha = \beta  = z$}\\
                   0 & \mbox{otherwise}\\
            \end{array}
     \right.
@@ -634,7 +615,8 @@ described for the NPTi integrator.
 Note that the iterative schemes for NPAT are identical to those
 described for the NPTi integrator.
 
-\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble}
+\subsection{\label{methodSection:NPrT}NP$\gamma$T
+Ensemble}
 
 Theoretically, the surface tension $\gamma$ of a stress free
 membrane system should be zero since its surface free energy $G$ is
@@ -649,18 +631,18 @@ $\eta$, in $NP\gamma T$ is
 pressure. The equation of motion for cell size control tensor,
 $\eta$, in $NP\gamma T$ is
 \begin{equation}
- \.{\overleftrightarrow{{\eta _{\alpha \beta}}}}=\left\{\begin{array}{ll}
+ \dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll}
     - A_{xy} (\gamma _\alpha   - \gamma _{{\rm{target}}} ) & \mbox{$\alpha  = \beta  = x$ or $\alpha  = \beta  = y$}\\
     \frac{{V(P_{\alpha \beta }  - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha  = \beta  = z$} \\
     0 & \mbox{$\alpha  \ne \beta$} \\
+       \end{array}
     \right.
 \end{equation}
 where $ \gamma _{{\rm{target}}}$ is the external surface tension and
 the instantaneous surface tensor $\gamma _\alpha$ is given by
 \begin{equation}
-\gamma _\alpha   =  - h_z
-(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}
-\over P} _{\alpha \alpha }  - P_{{\rm{target}}} )
+\gamma _\alpha   =  - h_z( \overleftrightarrow{P} _{\alpha \alpha }
+- P_{{\rm{target}}} )
 \label{methodEquation:instantaneousSurfaceTensor}
 \end{equation}
 
@@ -672,20 +654,92 @@ $\gamma$ is set to zero.
 integrator is a special case of $NP\gamma T$ if the surface tension
 $\gamma$ is set to zero.
 
-%\section{\label{methodSection:constraintMethod}Constraint Method}
+\section{\label{methodSection:zcons}Z-Constraint Method}
 
-%\subsection{\label{methodSection:bondConstraint}Bond Constraint for Rigid Body}
+Based on the fluctuation-dissipation theorem, a force
+auto-correlation method was developed by Roux and Karplus to
+investigate the dynamics of ions inside ion channels\cite{Roux1991}.
+The time-dependent friction coefficient can be calculated from the
+deviation of the instantaneous force from its mean force.
+\begin{equation}
+\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T,
+\end{equation}
+where%
+\begin{equation}
+\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle.
+\end{equation}
 
-%\subsection{\label{methodSection:zcons}Z-constraint Method}
+If the time-dependent friction decays rapidly, the static friction
+coefficient can be approximated by
+\begin{equation}
+\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta
+F(z,0)\rangle dt.
+\end{equation}
+Allowing diffusion constant to then be calculated through the
+Einstein relation:\cite{Marrink1994}
+\begin{equation}
+D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty
+}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.%
+\end{equation}
 
-\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies}
+The Z-Constraint method, which fixes the z coordinates of the
+molecules with respect to the center of the mass of the system, has
+been a method suggested to obtain the forces required for the force
+auto-correlation calculation.\cite{Marrink1994} However, simply
+resetting the coordinate will move the center of the mass of the
+whole system. To avoid this problem, we reset the forces of
+z-constrained molecules as well as subtract the total constraint
+forces from the rest of the system after the force calculation at
+each time step instead of resetting the coordinate.
 
-\subsection{\label{methodSection:temperature}Temperature Control}
+After the force calculation, define $G_\alpha$ as
+\begin{equation}
+G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1}
+\end{equation}
+where $F_{\alpha i}$ is the force in the z direction of atom $i$ in
+z-constrained molecule $\alpha$. The forces of the z constrained
+molecule are then set to:
+\begin{equation}
+F_{\alpha i} = F_{\alpha i} -
+    \frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}.
+\end{equation}
+Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained
+molecule. Having rescaled the forces, the velocities must also be
+rescaled to subtract out any center of mass velocity in the z
+direction.
+\begin{equation}
+v_{\alpha i} = v_{\alpha i} -
+    \frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}},
+\end{equation}
+where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction.
+Lastly, all of the accumulated z constrained forces must be
+subtracted from the system to keep the system center of mass from
+drifting.
+\begin{equation}
+F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha}
+G_{\alpha}}
+    {\sum_{\beta}\sum_i m_{\beta i}},
+\end{equation}
+where $\beta$ are all of the unconstrained molecules in the system.
+Similarly, the velocities of the unconstrained molecules must also
+be scaled.
+\begin{equation}
+v_{\beta i} = v_{\beta i} + \sum_{\alpha}
+    \frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}.
+\end{equation}
 
-\subsection{\label{methodSection:pressureControl}Pressure Control}
-
-\section{\label{methodSection:hydrodynamics}Hydrodynamics}
-
-%\section{\label{methodSection:coarseGrained}Coarse-Grained Modeling}
-
-%\section{\label{methodSection:moleculeScale}Molecular-Scale Modeling}
+At the very beginning of the simulation, the molecules may not be at
+their constrained positions. To move a z-constrained molecule to its
+specified position, a simple harmonic potential is used
+\begin{equation}
+U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},%
+\end{equation}
+where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$
+is the current $z$ coordinate of the center of mass of the
+constrained molecule, and $z_{\text{cons}}$ is the constrained
+position. The harmonic force operating on the z-constrained molecule
+at time $t$ can be calculated by
+\begin{equation}
+F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}=
+    -k_{\text{Harmonic}}(z(t)-z_{\text{cons}}).
+\end{equation}
