--- trunk/tengDissertation/Methodology.tex 2006/06/06 19:47:27 2804 +++ trunk/tengDissertation/Methodology.tex 2006/06/23 21:33:52 2882 @@ -2,37 +2,38 @@ In order to mimic the experiments, which are usually p \section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} -In order to mimic the experiments, which are usually performed under +In order to mimic experiments which are usually performed under constant temperature and/or pressure, extended Hamiltonian system 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 -normal directions of membranes. The $NPAT$ ensemble, in which the -normal pressure and the lateral surface area of the membrane are -kept constant, and the $NP\gamma T$ ensemble, in which the normal -pressure and the lateral surface tension are kept constant were -proposed to address this issue. +as the canonical and isobaric-isothermal ensembles. In addition to +the standard ensemble, specific ensembles have been developed to +account for the anisotropy between the lateral and normal directions +of membranes. The $NPAT$ ensemble, in which the normal pressure and +the lateral surface area of the membrane are kept constant, and the +$NP\gamma T$ ensemble, in which the normal pressure and the lateral +surface tension are kept constant were proposed to address the +issues. -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 +Integration schemes for the rotational motion of the rigid molecules +in the microcanonical ensemble have been extensively studied over +the last 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 -time-reversible integrator of Matubayasi and Nakahara. 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 -proposed by Dullweber, Leimkuhler and McLachlan (DLM) also preserved -the same structural properties of the Hamiltonian flow. In this -section, the integration scheme of DLM method will be reviewed and -extended to other ensembles. +long-time energy conservation than Euler angle methods because of +the time-reversible nature. Extending the Trotter-Suzuki +factorization to general system with a flat phase space, Miller and +his colleagues devised a novel symplectic, time-reversible and +volume-preserving integrator in the quaternion representation, which +was shown to be superior to the Matubayasi's time-reversible +integrator. However, all of the integrators in the 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 the rotation matrix directly proposed by Dullweber, +Leimkuhler and McLachlan (DLM) also preserved the same structural +properties of the Hamiltonian flow. In this section, the integration +scheme of DLM method will be reviewed and extended to other +ensembles. \subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the DLM method} @@ -111,13 +112,13 @@ torques are calculated at the new positions and orient - \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ % {\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) - \times \frac{\partial V}{\partial {\bf u}}, \\ + \times (\frac{\partial V}{\partial {\bf u}})_{u(t+h)}, \\ % {\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) \cdot {\bf \tau}^s(t + h). \end{align*} -{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix +${\bf u}$ is 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 +140,7 @@ 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 @@ -198,7 +199,9 @@ and $K$ is the total kinetic energy, \begin{equation} f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, \end{equation} -and $K$ is the total kinetic energy, +where $N_{\mathrm{orient}}$ is the number of molecules with +orientational degrees of freedom, and $K$ is the total kinetic +energy, \begin{equation} K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + \sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot @@ -206,13 +209,9 @@ relaxation of the temperature to the target value. To \end{equation} In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for -relaxation of the temperature to the target value. To set values -for $\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use -the {\tt tauThermostat} and {\tt targetTemperature} keywords in the -{\tt .bass} file. The units for {\tt tauThermostat} are fs, and the -units for the {\tt targetTemperature} are degrees K. The -integration of the equations of motion is carried out in a -velocity-Verlet style 2 part algorithm: +relaxation of the temperature to the target value. The integration +of the equations of motion is carried out in a velocity-Verlet style +2 part algorithm: {\tt moveA:} \begin{align*} @@ -278,10 +277,7 @@ 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 @@ -299,8 +295,8 @@ 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 +We can used an isobaric-isothermal ensemble integrator which is +implemented using the Melchionna modifications to the Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} \begin{eqnarray} @@ -356,11 +352,7 @@ 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 +393,11 @@ 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 +439,7 @@ + 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 +461,6 @@ 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 +534,8 @@ 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 +574,14 @@ 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 +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 @@ -606,19 +591,18 @@ 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 relations between structures and +functions in biological membrane system ultimately relies on +structure and dynamics of lipid bilayers, 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. +Constant area and constant lateral pressure simulations can be +achieved by extending the standard NPT ensemble with a different +pressure control strategy -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} \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}}} }} @@ -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 @@ -641,9 +626,9 @@ the membrane simulation, a special ensemble, NP$\gamma \] 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 +the membrane simulation, a special ensemble, NP$\gamma$T, has been proposed to maintain the lateral surface tension and normal -pressure. The equation of motion for cell size control tensor, +pressure. The equation of motion for the cell size control tensor, $\eta$, in $NP\gamma T$ is \begin{equation} \dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} @@ -667,9 +652,9 @@ $\gamma$ is set to zero. 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. +$\gamma$ is set to zero, and if $x$ and $y$ can move independently. -\section{\label{methodSection:zcons}Z-Constraint Method} +\section{\label{methodSection:zcons}The Z-Constraint Method} Based on the fluctuation-dissipation theorem, a force auto-correlation method was developed by Roux and Karplus to @@ -707,7 +692,7 @@ After the force calculation, define $G_\alpha$ as forces from the rest of the system after the force calculation at each time step instead of resetting the coordinate. -After the force calculation, define $G_\alpha$ as +After the force calculation, we define $G_\alpha$ as \begin{equation} G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} \end{equation} @@ -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}