--- trunk/tengDissertation/Methodology.tex	2006/06/06 19:47:27	2804
+++ trunk/tengDissertation/Methodology.tex	2006/06/11 02:53:15	2854
@@ -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
@@ -278,10 +278,7 @@ 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
@@ -299,9 +296,9 @@ To carry out isobaric-isothermal ensemble calculations
 \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{Melchionna1993}
+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), \\
@@ -356,11 +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:
 
@@ -401,12 +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)
@@ -448,10 +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,
@@ -473,10 +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)}
 
@@ -550,8 +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:
@@ -590,13 +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(
-\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}
+\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}
 \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
@@ -606,8 +592,6 @@ 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}
-
 \subsubsection{\label{methodSection:NPAT}NPAT Ensemble}
 
 A comprehensive understanding of structure�Cfunction relations of
@@ -631,7 +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
@@ -758,477 +743,3 @@ F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\part
 F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}=
     -k_{\text{Harmonic}}(z(t)-z_{\text{cons}}).
 \end{equation}
-
-
-\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies}
-
-%\subsection{\label{methodSection:temperature}Temperature Control}
-
-%\subsection{\label{methodSection:pressureControl}Pressure Control}
-
-%\section{\label{methodSection:hydrodynamics}Hydrodynamics}
-
-%applications of langevin dynamics
-As an excellent alternative to newtonian dynamics, Langevin
-dynamics, which mimics a simple heat bath with stochastic and
-dissipative forces, has been applied in a variety of studies. The
-stochastic treatment of the solvent enables us to carry out
-substantially longer time simulation. Implicit solvent Langevin
-dynamics simulation of met-enkephalin not only outperforms explicit
-solvent simulation on computation efficiency, but also agrees very
-well with explicit solvent simulation on dynamics
-properties\cite{Shen2002}. Recently, applying Langevin dynamics with
-UNRES model, Liow and his coworkers suggest that protein folding
-pathways can be possibly exploited within a reasonable amount of
-time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics
-also enhances the sampling of the system and increases the
-probability of crossing energy barrier\cite{Banerjee2004, Cui2003}.
-Combining Langevin dynamics with Kramers's theory, Klimov and
-Thirumalai identified the free-energy barrier by studying the
-viscosity dependence of the protein folding rates\cite{Klimov1997}.
-In order to account for solvent induced interactions missing from
-implicit solvent model, Kaya incorporated desolvation free energy
-barrier into implicit coarse-grained solvent model in protein
-folding/unfolding study and discovered a higher free energy barrier
-between the native and denatured states. Because of its stability
-against noise, Langevin dynamics is very suitable for studying
-remagnetization processes in various
-systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For
-instance, the oscillation power spectrum of nanoparticles from
-Langevin dynamics simulation has the same peak frequencies for
-different wave vectors,which recovers the property of magnetic
-excitations in small finite structures\cite{Berkov2005a}. In an
-attempt to reduce the computational cost of simulation, multiple
-time stepping (MTS) methods have been introduced and have been of
-great interest to macromolecule and protein
-community\cite{Tuckerman1992}. Relying on the observation that
-forces between distant atoms generally demonstrate slower
-fluctuations than forces between close atoms, MTS method are
-generally implemented by evaluating the slowly fluctuating forces
-less frequently than the fast ones. Unfortunately, nonlinear
-instability resulting from increasing timestep in MTS simulation
-have became a critical obstruction preventing the long time
-simulation. Due to the coupling to the heat bath, Langevin dynamics
-has been shown to be able to damp out the resonance artifact more
-efficiently\cite{Sandu1999}.
-
-%review rigid body dynamics
-Rigid bodies are frequently involved in the modeling of different
-areas, from engineering, physics, to chemistry. For example,
-missiles and vehicle are usually modeled by rigid bodies.  The
-movement of the objects in 3D gaming engine or other physics
-simulator is governed by the rigid body dynamics. In molecular
-simulation, rigid body is used to simplify the model in
-protein-protein docking study\cite{Gray2003}.
-
-It is very important to develop stable and efficient methods to
-integrate the equations of motion of orientational degrees of
-freedom. Euler angles are the nature choice to describe the
-rotational degrees of freedom. However, due to its singularity, the
-numerical integration of corresponding equations of motion is very
-inefficient and inaccurate. Although an alternative integrator using
-different sets of Euler angles can overcome this
-difficulty\cite{Ryckaert1977, Andersen1983}, the computational
-penalty and the lost of angular momentum conservation still remain.
-In 1977, a singularity free representation utilizing quaternions was
-developed by Evans\cite{Evans1977}. Unfortunately, this approach
-suffer from the nonseparable Hamiltonian resulted from quaternion
-representation, which prevents the symplectic algorithm to be
-utilized. Another different approach is to apply holonomic
-constraints to the atoms belonging to the rigid
-body\cite{Barojas1973}. Each atom moves independently under the
-normal forces deriving from potential energy and constraint forces
-which are used to guarantee the rigidness. However, due to their
-iterative nature, SHAKE and Rattle algorithm converge very slowly
-when the number of constraint increases.
-
-The break through in geometric literature suggests that, in order to
-develop a long-term integration scheme, one should preserve the
-geometric structure of the flow. 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. 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
-is RSHAKE\cite{Kol1997}, in which a conjugate momentum to rotation
-matrix is introduced to re-formulate the Hamiltonian's equation and
-the Hamiltonian is evolved in a constraint manifold by iteratively
-satisfying the orthogonality constraint. However, RSHAKE is
-inefficient because of the iterative procedure. An extremely
-efficient integration scheme in rotation matrix representation,
-which also preserves the same structural properties of the
-Hamiltonian flow as Miller's integrator, is proposed by Dullweber,
-Leimkuhler and McLachlan (DLM)\cite{Dullweber1997}.
-
-%review langevin/browninan dynamics for arbitrarily shaped rigid body
-Combining Langevin or Brownian dynamics with rigid body dynamics,
-one can study the slow processes in biomolecular systems. Modeling
-the DNA as a chain of rigid spheres beads, which subject to harmonic
-potentials as well as excluded volume potentials, Mielke and his
-coworkers discover rapid superhelical stress generations from the
-stochastic simulation of twin supercoiling DNA with response to
-induced torques\cite{Mielke2004}. Membrane fusion is another key
-biological process which controls a variety of physiological
-functions, such as release of neurotransmitters \textit{etc}. A
-typical fusion event happens on the time scale of millisecond, which
-is impracticable to study using all atomistic model with newtonian
-mechanics. With the help of coarse-grained rigid body model and
-stochastic dynamics, the fusion pathways were exploited by many
-researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the
-difficulty of numerical integration of anisotropy rotation, most of
-the rigid body models are simply modeled by sphere, cylinder,
-ellipsoid or other regular shapes in stochastic simulations. In an
-effort to account for the diffusion anisotropy of the arbitrary
-particles, Fernandes and de la Torre improved the original Brownian
-dynamics simulation algorithm\cite{Ermak1978,Allison1991} by
-incorporating a generalized $6\times6$ diffusion tensor and
-introducing a simple rotation evolution scheme consisting of three
-consecutive rotations\cite{Fernandes2002}. Unfortunately, unexpected
-error and bias are introduced into the system due to the arbitrary
-order of applying the noncommuting rotation
-operators\cite{Beard2003}. Based on the observation the momentum
-relaxation time is much less than the time step, one may ignore the
-inertia in Brownian dynamics. However, assumption of the zero
-average acceleration is not always true for cooperative motion which
-is common in protein motion. An inertial Brownian dynamics (IBD) was
-proposed to address this issue by adding an inertial correction
-term\cite{Beard2001}. As a complement to IBD which has a lower bound
-in time step because of the inertial relaxation time, long-time-step
-inertial dynamics (LTID) can be used to investigate the inertial
-behavior of the polymer segments in low friction
-regime\cite{Beard2001}. LTID can also deal with the rotational
-dynamics for nonskew bodies without translation-rotation coupling by
-separating the translation and rotation motion and taking advantage
-of the analytical solution of hydrodynamics properties. However,
-typical nonskew bodies like cylinder and ellipsoid are inadequate to
-represent most complex macromolecule assemblies. These intricate
-molecules have been represented by a set of beads and their
-hydrodynamics properties can be calculated using variant
-hydrodynamic interaction tensors.
-
-The goal of the present work is to develop a Langevin dynamics
-algorithm for arbitrary rigid particles by integrating the accurate
-estimation of friction tensor from hydrodynamics theory into the
-sophisticated rigid body dynamics.
-
-
-\subsection{Friction Tensor}
-
-For an arbitrary rigid body moves in a fluid, it may experience
-friction force $f_r$ or friction torque $\tau _r$ along the opposite
-direction of the velocity $v$ or angular velocity $\omega$ at
-arbitrary origin $P$,
-\begin{equation}
-\left( \begin{array}{l}
- f_r  \\
- \tau _r  \\
- \end{array} \right) =  - \left( {\begin{array}{*{20}c}
-   {\Xi _{P,t} } & {\Xi _{P,c}^T }  \\
-   {\Xi _{P,c} } & {\Xi _{P,r} }  \\
-\end{array}} \right)\left( \begin{array}{l}
- \nu  \\
- \omega  \\
- \end{array} \right)
-\end{equation}
-where $\Xi _{P,t}t$ is the translation friction tensor, $\Xi _{P,r}$
-is the rotational friction tensor and $\Xi _{P,c}$ is the
-translation-rotation coupling tensor. The procedure of calculating
-friction tensor using hydrodynamic tensor and comparison between
-bead model and shell model were elaborated by Carrasco \textit{et
-al}\cite{Carrasco1999}. An important property of the friction tensor
-is that the translational friction tensor is independent of origin
-while the rotational and coupling are sensitive to the choice of the
-origin \cite{Brenner1967}, which can be described by
-\begin{equation}
-\begin{array}{c}
- \Xi _{P,t}  = \Xi _{O,t}  = \Xi _t  \\
- \Xi _{P,c}  = \Xi _{O,c}  - r_{OP}  \times \Xi _t  \\
- \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  \\
- \end{array}
-\end{equation}
-Where $O$ is another origin and $r_{OP}$ is the vector joining $O$
-and $P$. It is also worthy of mention that both of translational and
-rotational frictional tensors are always symmetric. In contrast,
-coupling tensor is only symmetric at center of reaction:
-\begin{equation}
-\Xi _{R,c}  = \Xi _{R,c}^T
-\end{equation}
-The proper location for applying friction force is the center of
-reaction, at which the trace of rotational resistance tensor reaches
-minimum.
-
-\subsection{Rigid body dynamics}
-
-The Hamiltonian of rigid body can be separated in terms of potential
-energy $V(r,A)$ and kinetic energy $T(p,\pi)$,
-\[
-H = V(r,A) + T(v,\pi )
-\]
-A second-order symplectic method is now obtained by the composition
-of the flow maps,
-\[
-\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
-_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
-\]
-Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
-sub-flows which corresponding to force and torque respectively,
-\[
-\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
-_{\Delta t/2,\tau }.
-\]
-Since the associated operators of $\varphi _{\Delta t/2,F} $ and
-$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition
-order inside $\varphi _{\Delta t/2,V}$ does not matter.
-
-Furthermore, kinetic potential can be separated to translational
-kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$,
-\begin{equation}
-T(p,\pi ) =T^t (p) + T^r (\pi ).
-\end{equation}
-where $ T^t (p) = \frac{1}{2}v^T m v $ and $T^r (\pi )$ is defined
-by \ref{introEquation:rotationalKineticRB}. Therefore, the
-corresponding flow maps are given by
-\[
-\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
-_{\Delta t,T^r }.
-\]
-The free rigid body is an example of Lie-Poisson system with
-Hamiltonian function
-\begin{equation}
-T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
-\label{introEquation:rotationalKineticRB}
-\end{equation}
-where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
-Lie-Poisson structure matrix,
-\begin{equation}
-J(\pi ) = \left( {\begin{array}{*{20}c}
-   0 & {\pi _3 } & { - \pi _2 }  \\
-   { - \pi _3 } & 0 & {\pi _1 }  \\
-   {\pi _2 } & { - \pi _1 } & 0  \\
-\end{array}} \right)
-\end{equation}
-Thus, the dynamics of free rigid body is governed by
-\begin{equation}
-\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
-\end{equation}
-One may notice that each $T_i^r$ in Equation
-\ref{introEquation:rotationalKineticRB} can be solved exactly. For
-instance, the equations of motion due to $T_1^r$ are given by
-\begin{equation}
-\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}A = AR_1
-\label{introEqaution:RBMotionSingleTerm}
-\end{equation}
-where
-\[ R_1  = \left( {\begin{array}{*{20}c}
-   0 & 0 & 0  \\
-   0 & 0 & {\pi _1 }  \\
-   0 & { - \pi _1 } & 0  \\
-\end{array}} \right).
-\]
-The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is
-\[
-\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),A(\Delta t) =
-A(0)e^{\Delta tR_1 }
-\]
-with
-\[
-e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
-   0 & 0 & 0  \\
-   0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
-   0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
-\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
-\]
-To reduce the cost of computing expensive functions in $e^{\Delta
-tR_1 }$, we can use Cayley transformation,
-\[
-e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
-)
-\]
-The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
-manner.
-
-In order to construct a second-order symplectic method, we split the
-angular kinetic Hamiltonian function into five terms
-\[
-T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
-) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
-(\pi _1 )
-\].
-Concatenating flows corresponding to these five terms, we can obtain
-the flow map for free rigid body,
-\[
-\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
-\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
-\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
-_1 }.
-\]
-
-The equations of motion corresponding to potential energy and
-kinetic energy are listed in the below table,
-\begin{center}
-\begin{tabular}{|l|l|}
-  \hline
-  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
-  Potential & Kinetic \\
-  $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
-  $\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
-  $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
-  $ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
-  \hline
-\end{tabular}
-\end{center}
-
-Finally, we obtain the overall symplectic flow maps for free moving
-rigid body
-\begin{align*}
- \varphi _{\Delta t}  = &\varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau } \circ  \\
-  &\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  \\
-  &\varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
-\label{introEquation:overallRBFlowMaps}
-\end{align*}
-
-\subsection{Langevin dynamics for rigid particles of arbitrary shape}
-
-Consider a Langevin equation of motions in generalized coordinates
-\begin{equation}
-M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)}  + F_{r,i} (t)
-\label{LDGeneralizedForm}
-\end{equation}
-where $M_i$ is a $6\times6$ generalized diagonal mass (include mass
-and moment of inertial) matrix and $V_i$ is a generalized velocity,
-$V_i = V_i(v_i,\omega _i)$. The right side of Eq.
-(\ref{LDGeneralizedForm}) consists of three generalized forces in
-lab-fixed frame, systematic force $F_{s,i}$, dissipative force
-$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the
-system in Newtownian mechanics typically refers to lab-fixed frame,
-it is also convenient to handle the rotation of rigid body in
-body-fixed frame. Thus the friction and random forces are calculated
-in body-fixed frame and converted back to lab-fixed frame by:
-\[
-\begin{array}{l}
- F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\
- F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\
- \end{array}.
-\]
-Here, the body-fixed friction force $F_{r,i}^b$ is proportional to
-the body-fixed velocity at center of resistance $v_{R,i}^b$ and
-angular velocity $\omega _i$,
-\begin{equation}
-F_{r,i}^b (t) = \left( \begin{array}{l}
- f_{r,i}^b (t) \\
- \tau _{r,i}^b (t) \\
- \end{array} \right) =  - \left( {\begin{array}{*{20}c}
-   {\Xi _{R,t} } & {\Xi _{R,c}^T }  \\
-   {\Xi _{R,c} } & {\Xi _{R,r} }  \\
-\end{array}} \right)\left( \begin{array}{l}
- v_{R,i}^b (t) \\
- \omega _i (t) \\
- \end{array} \right),
-\end{equation}
-while the random force $F_{r,i}^l$ is a Gaussian stochastic variable
-with zero mean and variance
-\begin{equation}
-\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle  =
-\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle  =
-2k_B T\Xi _R \delta (t - t').
-\end{equation}
-The equation of motion for $v_i$ can be written as
-\begin{equation}
-m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) +
-f_{r,i}^l (t)
-\end{equation}
-Since the frictional force is applied at the center of resistance
-which generally does not coincide with the center of mass, an extra
-torque is exerted at the center of mass. Thus, the net body-fixed
-frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is
-given by
-\begin{equation}
-\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b
-\end{equation}
-where $r_{MR}$ is the vector from the center of mass to the center
-of the resistance. Instead of integrating angular velocity in
-lab-fixed frame, we consider the equation of motion of angular
-momentum in body-fixed frame
-\begin{equation}
-\dot \pi _i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b
-(t) + \tau _{r,i}^b(t)
-\end{equation}
-
-Embedding the friction terms into force and torque, one can
-integrate the langevin equations of motion for rigid body of
-arbitrary shape in a velocity-Verlet style 2-part algorithm, where
-$h= \delta t$:
-
-{\tt part one:}
-\begin{align*}
- v_i (t + h/2) &\leftarrow v_i (t) + \frac{{hf_{t,i}^l (t)}}{{2m_i }} \\
- \pi _i (t + h/2) &\leftarrow \pi _i (t) + \frac{{h\tau _{t,i}^b (t)}}{2} \\
- r_i (t + h) &\leftarrow r_i (t) + hv_i (t + h/2) \\
- A_i (t + h) &\leftarrow rotate\left( {h\pi _i (t + h/2)I^{ - 1} } \right) \\
-\end{align*}
-In this context, the $\mathrm{rotate}$ function is the reversible
-product of five consecutive body-fixed rotations,
-\begin{equation}
-\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot
-\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y
-/ 2) \cdot \mathsf{G}_x(a_x /2),
-\end{equation}
-where each rotational propagator, $\mathsf{G}_\alpha(\theta)$,
-rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed
-angular momentum ($\pi$) by an angle $\theta$ around body-fixed axis
-$\alpha$,
-\begin{equation}
-\mathsf{G}_\alpha( \theta ) = \left\{
-\begin{array}{lcl}
-\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\
-{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf
-j}(0).
-\end{array}
-\right.
-\end{equation}
-$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis
-rotation matrix.  For example, in the small-angle limit, the
-rotation matrix around the body-fixed x-axis can be approximated as
-\begin{equation}
-\mathsf{R}_x(\theta) \approx \left(
-\begin{array}{ccc}
-1 & 0 & 0 \\
-0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4}  & -\frac{\theta}{1+
-\theta^2 / 4} \\
-0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 +
-\theta^2 / 4}
-\end{array}
-\right).
-\end{equation}
-All other rotations follow in a straightforward manner.
-
-After the first part of the propagation, the friction and random
-forces are generated at the center of resistance in body-fixed frame
-and converted back into lab-fixed frame
-\[
-f_{t,i}^l (t + h) =  - \left( {\frac{{\partial V}}{{\partial r_i }}}
-\right)_{r_i (t + h)}  + A_i^T (t + h)[F_{f,i}^b (t + h) + F_{r,i}^b
-(t + h)],
-\]
-while the system torque in lab-fixed frame is transformed into
-body-fixed frame,
-\[
-\tau _{t,i}^b (t + h) = A\tau _{s,i}^l (t) + \tau _{n,i}^b (t) +
-\tau _{r,i}^b (t).
-\]
-Once the forces and torques have been obtained at the new time step,
-the velocities can be advanced to the same time value.
-
-{\tt part two:}
-\begin{align*}
- v_i (t) &\leftarrow v_i (t + h/2) + \frac{{hf_{t,i}^l (t + h)}}{{2m_i }} \\
- \pi _i (t) &\leftarrow \pi _i (t + h/2) + \frac{{h\tau _{t,i}^b (t + h)}}{2} \\
-\end{align*}
-
-\subsection{Results and discussion}
