--- trunk/tengDissertation/Methodology.tex	2006/04/24 18:49:04	2729
+++ trunk/tengDissertation/Methodology.tex	2006/04/26 01:27:56	2739
@@ -4,7 +4,7 @@ methods have been developed to generate statistical en
 
 In order to mimic the experiments, which are usually performed under
 constant temperature and/or pressure, extended Hamiltonian system
-methods have been developed to generate statistical ensemble, such
+methods have been developed to generate statistical ensembles, such
 as canonical ensemble and isobaric-isothermal ensemble \textit{etc}.
 In addition to the standard ensemble, specific ensembles have been
 developed to account for the anisotropy between the lateral and
@@ -608,123 +608,71 @@ assume non-orthorhombic geometries.
 electrostatic or Lennard-Jones cutoff radii.  The NPTf integrator
 finds most use in simulating crystals or liquid crystals which
 assume non-orthorhombic geometries.
-
-\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)}
-
-\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) }
 
-\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)}
-
-There is one additional extended system integrator which is somewhat
-simpler than the NPTf method described above.  In this case, the
-three axes have independent barostats which each attempt to preserve
-the target pressure along the box walls perpendicular to that
-particular axis.  The lengths of the box axes are allowed to
-fluctuate independently, but the angle between the box axes does not
-change. The equations of motion are identical to those described
-above, but only the {\it diagonal} elements of
-$\overleftrightarrow{\eta}$ are computed.  The off-diagonal elements
-are set to zero (even when the pressure tensor has non-zero
-off-diagonal elements). It should be noted that the NPTxyz
-integrator is a special case of $NP\gamma T$ if the surface tension
-$\gamma$ is set to zero.
+\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles}
 
+\subsubsection{\label{methodSection:NPAT}Constant Normal  Pressure, Constant Lateral Surface Area and Constant Temperature (NPAT) Ensemble}
 
-\section{\label{methodSection:constraintMethods}Constraint Methods}
-
-\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond
-    Constraints}
-
-\subsection{\label{methodSection:zcons}Z-Constraint Method}
-
-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{Roux91}
-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%
+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
+interfacial interaction between lipid molecules and surrounding
+media. One quantity to describe the interfacial interaction is so
+called the average surface area per lipid. Constat area and constant
+lateral pressure simulation can be achieved by extending the
+standard NPT ensemble with a different pressure control strategy
 \begin{equation}
-\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle.
+\dot
+\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}
+\over \eta } _{\alpha \beta }  = \left\{ \begin{array}{l}
+ \frac{{V(P_{\alpha \beta }  - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{   }}(\alpha  = \beta  = z), \\
+ 0{\rm{                        }}(\alpha  \ne z{\rm{ }}or{\rm{ }}\beta  \ne z) \\
+ \end{array} \right.
+\label{methodEquation:NPATeta}
 \end{equation}
+Note that the iterative schemes for NPAT are identical to those
+described for the NPTi integrator.
 
-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{Marrink94}
-\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}
+\subsubsection{\label{methodSection:NPrT}Constant Normal Pressure, Constant Lateral Surface Tension and Constant Temperature (NP\gamma T) Ensemble }
 
-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{Marrink94} However, simply
-resetting the coordinate will move the center of the mass of the
-whole system. To avoid this problem, a new method was used in {\sc
-oopse}. Instead of resetting the coordinate, 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.
-
-After the force calculation, define $G_\alpha$ as
+Theoretically, the surface tension $\gamma$ of a stress free
+membrane system should be zero since its surface free energy $G$ is
+minimum with respect to surface area $A$
+\[
+\gamma  = \frac{{\partial G}}{{\partial A}}.
+\]
+However, a surface tension of zero is not appropriate for relatively
+small patches of membrane. In order to eliminate the edge effect of
+the membrane simulation, a special ensemble, NP\gamma T, is proposed
+to maintain the lateral surface tension and normal pressure. The
+equation of motion for cell size control tensor, $\eta$, in NP\gamma
+T is
 \begin{equation}
-G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1}
+\dot
+\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}
+\over \eta } _{\alpha \beta }  = \left\{ \begin{array}{l}
+  - A_{xy} (\gamma _\alpha   - \gamma _{{\rm{target}}} ){\rm{ (}}\alpha  = \beta  = x{\rm{ or }} = y{\rm{)}} \\
+ \frac{{V(P_{\alpha \beta }  - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{   }}(\alpha  = \beta  = z) \\
+ 0{\rm{                         }}(\alpha  \ne \beta ) \\
+ \end{array} \right.
+\label{methodEquation:NPrTeta}
 \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:
+where $ \gamma _{{\rm{target}}}$ is the external surface tension and
+the instantaneous surface tensor $\gamma _\alpha$ is given by
 \begin{equation}
-F_{\alpha i} = F_{\alpha i} -
-    \frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}.
+\gamma _\alpha   =  - h_z
+(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}
+\over P} _{\alpha \alpha }  - P_{{\rm{target}}} )
+\label{methodEquation:instantaneousSurfaceTensor}
 \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}
 
-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}
+There is one additional extended system integrator (NPTxyz), in
+which each attempt to preserve the target pressure along the box
+walls perpendicular to that particular axis.  The lengths of the box
+axes are allowed to fluctuate independently, but the angle between
+the box axes does not change. It should be noted that the NPTxyz
+integrator is a special case of $NP\gamma T$ if the surface tension
+$\gamma$ is set to zero.
 
 \section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies}
 
