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# Line 18 | Line 18 | representation\cite{Matubayasi1999}. Although it is no
18   in the microcanonical ensemble have been extensively studied over
19   the last two decades. Matubayasi developed a
20   time-reversible integrator for rigid bodies in quaternion
21 < representation\cite{Matubayasi1999}. Although it is not symplectic, this integrator still
21 > representation.\cite{Matubayasi1999} Although it is not symplectic, this integrator still
22   demonstrates a better long-time energy conservation than Euler angle
23   methods because of the time-reversible nature. Extending the
24   Trotter-Suzuki factorization to general system with a flat phase
# Line 32 | Line 32 | propagator\cite{Dullweber1997}. In this section, the i
32   An alternative integration scheme utilizing the rotation matrix
33   directly proposed by Dullweber, Leimkuhler and McLachlan (DLM) also
34   preserved the same structural properties of the Hamiltonian
35 < propagator\cite{Dullweber1997}. In this section, the integration
35 > propagator.\cite{Dullweber1997} In this section, the integration
36   scheme of DLM method will be reviewed and extended to other
37   ensembles.
38  
39 < \subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the
40 < DLM method}
39 > \subsection{\label{methodSection:DLM}Integrating the Equations of Motion: The
40 > DLM Method}
41  
42   The DLM method uses a Trotter factorization of the orientational
43   propagator.  This has three effects:
# Line 46 | Line 46 | Monte Carlo applications, and
46   {\it symplectic}),
47   \item the integrator is time-{\it reversible}, making it suitable for Hybrid
48   Monte Carlo applications, and
49 < \item the error for a single time step is of order $\mathcal{O}\left(h^4\right)$
49 > \item the error for a single time step is of order $\mathcal{O}\left(h^3\right)$
50   for timesteps of length $h$.
51   \end{enumerate}
52   The integration of the equations of motion is carried out in a
# Line 136 | Line 136 | in Fig.~\ref{methodFig:timestep} where the resulting e
136   average 7\% increase in computation time using the DLM method in
137   place of quaternions. This cost is more than justified when
138   comparing the energy conservation of the two methods as illustrated
139 < in Fig.~\ref{methodFig:timestep} where the resulting energy drift at
139 > in Fig.~\ref{methodFig:timestep} where the resulting energy drifts at
140   various time steps for both the DLM and quaternion integration
141 < schemes is compared. All of the 1000 molecule water simulations
141 > schemes are compared. All of the 1000 molecule water simulations
142   started with the same configuration, and the only difference was the
143   method for handling rotational motion. At time steps of 0.1 and 0.5
144   fs, both methods for propagating molecule rotation conserve energy
# Line 211 | Line 211 | T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\b
211      \chi(t) \right) ,\\
212   %
213   \mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}
214 <    \left(h * {\bf j}(t + h / 2)
214 >    \left(h {\bf j}(t + h / 2)
215      \overleftrightarrow{\mathsf{I}}^{-1} \right) ,\\
216   %
217   \chi\left(t + h / 2 \right) &\leftarrow \chi(t)
# Line 318 | Line 318 | the Pressure tensor,
318   r}_{ij}(t) \otimes {\bf f}_{ij}(t).
319   \end{equation}
320   The instantaneous pressure is then simply obtained from the trace of
321 < the Pressure tensor,
321 > the pressure tensor,
322   \begin{equation}
323   P(t) = \frac{1}{3} \mathrm{Tr} \left(
324   \overleftrightarrow{\mathsf{P}}(t) \right) .
# Line 573 | Line 573 | pressure control strategy
573             \end{array}
574      \right.
575   \end{equation}
576 –
576   Note that the iterative schemes for NPAT are identical to those
577   described for the NPTi integrator.
578  
# Line 582 | Line 581 | minimum with respect to surface area $A$, $\gamma  = \
581  
582   Theoretically, the surface tension $\gamma$ of a stress free
583   membrane system should be zero since its surface free energy $G$ is
584 < minimum with respect to surface area $A$, $\gamma  = \frac{{\partial
585 < G}}{{\partial A}}.$ However, a surface tension of zero is not
584 > minimum with respect to surface area $A$,
585 > \begin{equation}
586 > \gamma  = \frac{{\partial G}}{{\partial A}}=0.
587 > \end{equation}
588 > However, a surface tension of zero is not
589   appropriate for relatively small patches of membrane. In order to
590   eliminate the edge effect of membrane simulations, a special
591 < ensemble, NP$\gamma$T, has been proposed to maintain the lateral
591 > ensemble NP$\gamma$T has been proposed to maintain the lateral
592   surface tension and normal pressure. The equation of motion for the
593   cell size control tensor, $\eta$, in $NP\gamma T$ is
594   \begin{equation}
# Line 616 | Line 618 | investigate the dynamics of ions inside ion channels\c
618  
619   Based on the fluctuation-dissipation theorem, a force
620   auto-correlation method was developed by Roux and Karplus to
621 < investigate the dynamics of ions inside ion channels\cite{Roux1991}.
621 > investigate the dynamics of ions inside ion channels.\cite{Roux1991}
622   The time-dependent friction coefficient can be calculated from the
623 < deviation of the instantaneous force from its mean force.
623 > deviation of the instantaneous force from its mean force:
624   \begin{equation}
625   \xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T,
626   \end{equation}

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