16 |
|
|
17 |
|
Integration schemes for rotational motion of the rigid molecules in |
18 |
|
microcanonical ensemble have been extensively studied in the last |
19 |
< |
two decades. Matubayasi and Nakahara developed a time-reversible |
20 |
< |
integrator for rigid bodies in quaternion representation. Although |
21 |
< |
it is not symplectic, this integrator still demonstrates a better |
22 |
< |
long-time energy conservation than traditional methods because of |
23 |
< |
the time-reversible nature. Extending Trotter-Suzuki to general |
24 |
< |
system with a flat phase space, Miller and his colleagues devised an |
25 |
< |
novel symplectic, time-reversible and volume-preserving integrator |
26 |
< |
in quaternion representation, which was shown to be superior to the |
27 |
< |
time-reversible integrator of Matubayasi and Nakahara. However, all |
28 |
< |
of the integrators in quaternion representation suffer from the |
19 |
> |
two decades. Matubayasi developed a time-reversible integrator for |
20 |
> |
rigid bodies in quaternion representation. Although it is not |
21 |
> |
symplectic, this integrator still demonstrates a better long-time |
22 |
> |
energy conservation than traditional methods because of the |
23 |
> |
time-reversible nature. Extending Trotter-Suzuki to general system |
24 |
> |
with a flat phase space, Miller and his colleagues devised an novel |
25 |
> |
symplectic, time-reversible and volume-preserving integrator in |
26 |
> |
quaternion representation, which was shown to be superior to the |
27 |
> |
Matubayasi's time-reversible integrator. However, all of the |
28 |
> |
integrators in quaternion representation suffer from the |
29 |
|
computational penalty of constructing a rotation matrix from |
30 |
|
quaternions to evolve coordinates and velocities at every time step. |
31 |
|
An alternative integration scheme utilizing rotation matrix directly |
117 |
|
\cdot {\bf \tau}^s(t + h). |
118 |
|
\end{align*} |
119 |
|
|
120 |
< |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
120 |
> |
${\bf u}$ will be automatically updated when the rotation matrix |
121 |
|
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
122 |
|
torques have been obtained at the new time step, the velocities can |
123 |
|
be advanced to the same time value. |
278 |
|
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
279 |
|
depend on their own values at time $t + h$. {\tt moveB} is |
280 |
|
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
281 |
< |
self-consistent. The relative tolerance for the self-consistency |
282 |
< |
check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will |
283 |
< |
terminate the iteration after 4 loops even if the consistency check |
284 |
< |
has not been satisfied. |
281 |
> |
self-consistent. |
282 |
|
|
283 |
|
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
284 |
|
the extended system that is, to within a constant, identical to the |
296 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
297 |
|
isotropic box deformations (NPTi)} |
298 |
|
|
299 |
< |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
300 |
< |
implements the Melchionna modifications to the |
301 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
299 |
> |
Isobaric-isothermal ensemble integrator is implemented using the |
300 |
> |
Melchionna modifications to the Nos\'e-Hoover-Andersen equations of |
301 |
> |
motion,\cite{Melchionna1993} |
302 |
|
|
303 |
|
\begin{eqnarray} |
304 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
353 |
|
\end{equation} |
354 |
|
|
355 |
|
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
356 |
< |
relaxation of the pressure to the target value. To set values for |
360 |
< |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
361 |
< |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt |
362 |
< |
.bass} file. The units for {\tt tauBarostat} are fs, and the units |
363 |
< |
for the {\tt targetPressure} are atmospheres. Like in the NVT |
356 |
> |
relaxation of the pressure to the target value. Like in the NVT |
357 |
|
integrator, the integration of the equations of motion is carried |
358 |
|
out in a velocity-Verlet style 2 part algorithm: |
359 |
|
|
394 |
|
|
395 |
|
Most of these equations are identical to their counterparts in the |
396 |
|
NVT integrator, but the propagation of positions to time $t + h$ |
397 |
< |
depends on the positions at the same time. {\sc oopse} carries out |
398 |
< |
this step iteratively (with a limit of 5 passes through the |
399 |
< |
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
400 |
< |
uniformly for one full time step by an exponential factor that |
401 |
< |
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
409 |
< |
box uniformly also scales the volume of the box by |
397 |
> |
depends on the positions at the same time. The simulation box |
398 |
> |
$\mathsf{H}$ is scaled uniformly for one full time step by an |
399 |
> |
exponential factor that depends on the value of $\eta$ at time $t + |
400 |
> |
h / 2$. Reshaping the box uniformly also scales the volume of the |
401 |
> |
box by |
402 |
|
\begin{equation} |
403 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
404 |
|
\mathcal{V}(t) |
440 |
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
441 |
|
h)$, they indirectly depend on their own values at time $t + h$. |
442 |
|
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
443 |
< |
+ h)$ and $\eta(t + h)$ become self-consistent. The relative |
452 |
< |
tolerance for the self-consistency check defaults to a value of |
453 |
< |
$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after |
454 |
< |
4 loops even if the consistency check has not been satisfied. |
443 |
> |
+ h)$ and $\eta(t + h)$ become self-consistent. |
444 |
|
|
445 |
|
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
446 |
|
is known to conserve a Hamiltonian for the extended system that is, |
462 |
|
P_{\mathrm{target}} \mathcal{V}(t). |
463 |
|
\end{equation} |
464 |
|
|
476 |
– |
Bond constraints are applied at the end of both the {\tt moveA} and |
477 |
– |
{\tt moveB} portions of the algorithm. |
478 |
– |
|
465 |
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
466 |
|
flexible box (NPTf)} |
467 |
|
|
535 |
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
536 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
537 |
|
\end{align*} |
538 |
< |
{\sc oopse} uses a power series expansion truncated at second order |
539 |
< |
for the exponential operation which scales the simulation box. |
538 |
> |
Here, a power series expansion truncated at second order for the |
539 |
> |
exponential operation is used to scale the simulation box. |
540 |
|
|
541 |
|
The {\tt moveB} portion of the algorithm is largely unchanged from |
542 |
|
the NPTi integrator: |
575 |
|
identical to those described for the NPTi integrator. |
576 |
|
|
577 |
|
The NPTf integrator is known to conserve the following Hamiltonian: |
578 |
< |
\begin{equation} |
579 |
< |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
578 |
> |
\begin{eqnarray*} |
579 |
> |
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
580 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
581 |
< |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
582 |
< |
T_{\mathrm{target}}}{2} |
581 |
> |
dt^\prime \right) \\ |
582 |
> |
+ P_{\mathrm{target}} \mathcal{V}(t) + \frac{f |
583 |
> |
k_B T_{\mathrm{target}}}{2} |
584 |
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
585 |
< |
\end{equation} |
585 |
> |
\end{eqnarray*} |
586 |
|
|
587 |
|
This integrator must be used with care, particularly in liquid |
588 |
|
simulations. Liquids have very small restoring forces in the |
592 |
|
finds most use in simulating crystals or liquid crystals which |
593 |
|
assume non-orthorhombic geometries. |
594 |
|
|
608 |
– |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
609 |
– |
|
595 |
|
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
596 |
|
|
597 |
|
A comprehensive understanding of structure¨Cfunction relations of |
615 |
|
Note that the iterative schemes for NPAT are identical to those |
616 |
|
described for the NPTi integrator. |
617 |
|
|
618 |
< |
\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble} |
618 |
> |
\subsection{\label{methodSection:NPrT}NP$\gamma$T |
619 |
> |
Ensemble} |
620 |
|
|
621 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
622 |
|
membrane system should be zero since its surface free energy $G$ is |
743 |
|
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
744 |
|
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
745 |
|
\end{equation} |
760 |
– |
|
761 |
– |
|
762 |
– |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
763 |
– |
|
764 |
– |
%\subsection{\label{methodSection:temperature}Temperature Control} |
765 |
– |
|
766 |
– |
%\subsection{\label{methodSection:pressureControl}Pressure Control} |
767 |
– |
|
768 |
– |
%\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
769 |
– |
|
770 |
– |
%applications of langevin dynamics |
771 |
– |
As an excellent alternative to newtonian dynamics, Langevin |
772 |
– |
dynamics, which mimics a simple heat bath with stochastic and |
773 |
– |
dissipative forces, has been applied in a variety of studies. The |
774 |
– |
stochastic treatment of the solvent enables us to carry out |
775 |
– |
substantially longer time simulation. Implicit solvent Langevin |
776 |
– |
dynamics simulation of met-enkephalin not only outperforms explicit |
777 |
– |
solvent simulation on computation efficiency, but also agrees very |
778 |
– |
well with explicit solvent simulation on dynamics |
779 |
– |
properties\cite{Shen2002}. Recently, applying Langevin dynamics with |
780 |
– |
UNRES model, Liow and his coworkers suggest that protein folding |
781 |
– |
pathways can be possibly exploited within a reasonable amount of |
782 |
– |
time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics |
783 |
– |
also enhances the sampling of the system and increases the |
784 |
– |
probability of crossing energy barrier\cite{Banerjee2004, Cui2003}. |
785 |
– |
Combining Langevin dynamics with Kramers's theory, Klimov and |
786 |
– |
Thirumalai identified the free-energy barrier by studying the |
787 |
– |
viscosity dependence of the protein folding rates\cite{Klimov1997}. |
788 |
– |
In order to account for solvent induced interactions missing from |
789 |
– |
implicit solvent model, Kaya incorporated desolvation free energy |
790 |
– |
barrier into implicit coarse-grained solvent model in protein |
791 |
– |
folding/unfolding study and discovered a higher free energy barrier |
792 |
– |
between the native and denatured states. Because of its stability |
793 |
– |
against noise, Langevin dynamics is very suitable for studying |
794 |
– |
remagnetization processes in various |
795 |
– |
systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
796 |
– |
oscillation power spectrum of nanoparticles from Langevin dynamics |
797 |
– |
simulation has the same peak frequencies for different wave |
798 |
– |
vectors,which recovers the property of magnetic excitations in small |
799 |
– |
finite structures\cite{Berkov2005a}. In an attempt to reduce the |
800 |
– |
computational cost of simulation, multiple time stepping (MTS) |
801 |
– |
methods have been introduced and have been of great interest to |
802 |
– |
macromolecule and protein community\cite{Tuckerman1992}. Relying on |
803 |
– |
the observation that forces between distant atoms generally |
804 |
– |
demonstrate slower fluctuations than forces between close atoms, MTS |
805 |
– |
method are generally implemented by evaluating the slowly |
806 |
– |
fluctuating forces less frequently than the fast ones. |
807 |
– |
Unfortunately, nonlinear instability resulting from increasing |
808 |
– |
timestep in MTS simulation have became a critical obstruction |
809 |
– |
preventing the long time simulation. Due to the coupling to the heat |
810 |
– |
bath, Langevin dynamics has been shown to be able to damp out the |
811 |
– |
resonance artifact more efficiently\cite{Sandu1999}. |
812 |
– |
|
813 |
– |
%review rigid body dynamics |
814 |
– |
Rigid bodies are frequently involved in the modeling of different |
815 |
– |
areas, from engineering, physics, to chemistry. For example, |
816 |
– |
missiles and vehicle are usually modeled by rigid bodies. The |
817 |
– |
movement of the objects in 3D gaming engine or other physics |
818 |
– |
simulator is governed by the rigid body dynamics. In molecular |
819 |
– |
simulation, rigid body is used to simplify the model in |
820 |
– |
protein-protein docking study\cite{Gray2003}. |
821 |
– |
|
822 |
– |
It is very important to develop stable and efficient methods to |
823 |
– |
integrate the equations of motion of orientational degrees of |
824 |
– |
freedom. Euler angles are the nature choice to describe the |
825 |
– |
rotational degrees of freedom. However, due to its singularity, the |
826 |
– |
numerical integration of corresponding equations of motion is very |
827 |
– |
inefficient and inaccurate. Although an alternative integrator using |
828 |
– |
different sets of Euler angles can overcome this |
829 |
– |
difficulty\cite{Ryckaert1977, Andersen1983}, the computational |
830 |
– |
penalty and the lost of angular momentum conservation still remain. |
831 |
– |
In 1977, a singularity free representation utilizing quaternions was |
832 |
– |
developed by Evans\cite{Evans1977}. Unfortunately, this approach |
833 |
– |
suffer from the nonseparable Hamiltonian resulted from quaternion |
834 |
– |
representation, which prevents the symplectic algorithm to be |
835 |
– |
utilized. Another different approach is to apply holonomic |
836 |
– |
constraints to the atoms belonging to the rigid |
837 |
– |
body\cite{Barojas1973}. Each atom moves independently under the |
838 |
– |
normal forces deriving from potential energy and constraint forces |
839 |
– |
which are used to guarantee the rigidness. However, due to their |
840 |
– |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
841 |
– |
when the number of constraint increases. |
842 |
– |
|
843 |
– |
The break through in geometric literature suggests that, in order to |
844 |
– |
develop a long-term integration scheme, one should preserve the |
845 |
– |
geometric structure of the flow. Matubayasi and Nakahara developed a |
846 |
– |
time-reversible integrator for rigid bodies in quaternion |
847 |
– |
representation. Although it is not symplectic, this integrator still |
848 |
– |
demonstrates a better long-time energy conservation than traditional |
849 |
– |
methods because of the time-reversible nature. Extending |
850 |
– |
Trotter-Suzuki to general system with a flat phase space, Miller and |
851 |
– |
his colleagues devised an novel symplectic, time-reversible and |
852 |
– |
volume-preserving integrator in quaternion representation. However, |
853 |
– |
all of the integrators in quaternion representation suffer from the |
854 |
– |
computational penalty of constructing a rotation matrix from |
855 |
– |
quaternions to evolve coordinates and velocities at every time step. |
856 |
– |
An alternative integration scheme utilizing rotation matrix directly |
857 |
– |
is RSHAKE\cite{Kol1997}, in which a conjugate momentum to rotation |
858 |
– |
matrix is introduced to re-formulate the Hamiltonian's equation and |
859 |
– |
the Hamiltonian is evolved in a constraint manifold by iteratively |
860 |
– |
satisfying the orthogonality constraint. However, RSHAKE is |
861 |
– |
inefficient because of the iterative procedure. An extremely |
862 |
– |
efficient integration scheme in rotation matrix representation, |
863 |
– |
which also preserves the same structural properties of the |
864 |
– |
Hamiltonian flow as Miller's integrator, is proposed by Dullweber, |
865 |
– |
Leimkuhler and McLachlan (DLM)\cite{Dullweber1997}. |
866 |
– |
|
867 |
– |
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
868 |
– |
Combining Langevin or Brownian dynamics with rigid body dynamics, |
869 |
– |
one can study the slow processes in biomolecular systems. Modeling |
870 |
– |
the DNA as a chain of rigid spheres beads, which subject to harmonic |
871 |
– |
potentials as well as excluded volume potentials, Mielke and his |
872 |
– |
coworkers discover rapid superhelical stress generations from the |
873 |
– |
stochastic simulation of twin supercoiling DNA with response to |
874 |
– |
induced torques\cite{Mielke2004}. Membrane fusion is another key |
875 |
– |
biological process which controls a variety of physiological |
876 |
– |
functions, such as release of neurotransmitters \textit{etc}. A |
877 |
– |
typical fusion event happens on the time scale of millisecond, which |
878 |
– |
is impracticable to study using all atomistic model with newtonian |
879 |
– |
mechanics. With the help of coarse-grained rigid body model and |
880 |
– |
stochastic dynamics, the fusion pathways were exploited by many |
881 |
– |
researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the |
882 |
– |
difficulty of numerical integration of anisotropy rotation, most of |
883 |
– |
the rigid body models are simply modeled by sphere, cylinder, |
884 |
– |
ellipsoid or other regular shapes in stochastic simulations. In an |
885 |
– |
effort to account for the diffusion anisotropy of the arbitrary |
886 |
– |
particles, Fernandes and de la Torre improved the original Brownian |
887 |
– |
dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
888 |
– |
incorporating a generalized $6\times6$ diffusion tensor and |
889 |
– |
introducing a simple rotation evolution scheme consisting of three |
890 |
– |
consecutive rotations\cite{Fernandes2002}. Unfortunately, unexpected |
891 |
– |
error and bias are introduced into the system due to the arbitrary |
892 |
– |
order of applying the noncommuting rotation |
893 |
– |
operators\cite{Beard2003}. Based on the observation the momentum |
894 |
– |
relaxation time is much less than the time step, one may ignore the |
895 |
– |
inertia in Brownian dynamics. However, assumption of the zero |
896 |
– |
average acceleration is not always true for cooperative motion which |
897 |
– |
is common in protein motion. An inertial Brownian dynamics (IBD) was |
898 |
– |
proposed to address this issue by adding an inertial correction |
899 |
– |
term\cite{Beard2003}. As a complement to IBD which has a lower bound |
900 |
– |
in time step because of the inertial relaxation time, long-time-step |
901 |
– |
inertial dynamics (LTID) can be used to investigate the inertial |
902 |
– |
behavior of the polymer segments in low friction |
903 |
– |
regime\cite{Beard2001}. LTID can also deal with the rotational |
904 |
– |
dynamics for nonskew bodies without translation-rotation coupling by |
905 |
– |
separating the translation and rotation motion and taking advantage |
906 |
– |
of the analytical solution of hydrodynamics properties. However, |
907 |
– |
typical nonskew bodies like cylinder and ellipsoid are inadequate to |
908 |
– |
represent most complex macromolecule assemblies. These intricate |
909 |
– |
molecules have been represented by a set of beads and their |
910 |
– |
hydrodynamics properties can be calculated using variant |
911 |
– |
hydrodynamic interaction tensors. |
912 |
– |
|
913 |
– |
The goal of the present work is to develop a Langevin dynamics |
914 |
– |
algorithm for arbitrary rigid particles by integrating the accurate |
915 |
– |
estimation of friction tensor from hydrodynamics theory into the |
916 |
– |
sophisticated rigid body dynamics. |
917 |
– |
|
918 |
– |
|
919 |
– |
\subsection{Friction Tensor} |
920 |
– |
|
921 |
– |
For an arbitrary rigid body moves in a fluid, it may experience |
922 |
– |
friction force $f_r$ or friction torque $\tau _r$ along the opposite |
923 |
– |
direction of the velocity $v$ or angular velocity $\omega$ at |
924 |
– |
arbitrary origin $P$, |
925 |
– |
\begin{equation} |
926 |
– |
\left( \begin{array}{l} |
927 |
– |
f_r \\ |
928 |
– |
\tau _r \\ |
929 |
– |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
930 |
– |
{\Xi _{P,t} } & {\Xi _{P,c}^T } \\ |
931 |
– |
{\Xi _{P,c} } & {\Xi _{P,r} } \\ |
932 |
– |
\end{array}} \right)\left( \begin{array}{l} |
933 |
– |
\nu \\ |
934 |
– |
\omega \\ |
935 |
– |
\end{array} \right) |
936 |
– |
\end{equation} |
937 |
– |
where $\Xi _{P,t}t$ is the translation friction tensor, $\Xi _{P,r}$ |
938 |
– |
is the rotational friction tensor and $\Xi _{P,c}$ is the |
939 |
– |
translation-rotation coupling tensor. The procedure of calculating |
940 |
– |
friction tensor using hydrodynamic tensor and comparison between |
941 |
– |
bead model and shell model were elaborated by Carrasco \textit{et |
942 |
– |
al}\cite{Carrasco1999}. An important property of the friction tensor |
943 |
– |
is that the translational friction tensor is independent of origin |
944 |
– |
while the rotational and coupling are sensitive to the choice of the |
945 |
– |
origin \cite{Brenner1967}, which can be described by |
946 |
– |
\begin{equation} |
947 |
– |
\begin{array}{c} |
948 |
– |
\Xi _{P,t} = \Xi _{O,t} = \Xi _t \\ |
949 |
– |
\Xi _{P,c} = \Xi _{O,c} - r_{OP} \times \Xi _t \\ |
950 |
– |
\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 \\ |
951 |
– |
\end{array} |
952 |
– |
\end{equation} |
953 |
– |
Where $O$ is another origin and $r_{OP}$ is the vector joining $O$ |
954 |
– |
and $P$. It is also worthy of mention that both of translational and |
955 |
– |
rotational frictional tensors are always symmetric. In contrast, |
956 |
– |
coupling tensor is only symmetric at center of reaction: |
957 |
– |
\begin{equation} |
958 |
– |
\Xi _{R,c} = \Xi _{R,c}^T |
959 |
– |
\end{equation} |
960 |
– |
The proper location for applying friction force is the center of |
961 |
– |
reaction, at which the trace of rotational resistance tensor reaches |
962 |
– |
minimum. |
963 |
– |
|
964 |
– |
\subsection{Rigid body dynamics} |
965 |
– |
|
966 |
– |
The Hamiltonian of rigid body can be separated in terms of potential |
967 |
– |
energy $V(r,A)$ and kinetic energy $T(p,\pi)$, |
968 |
– |
\[ |
969 |
– |
H = V(r,A) + T(v,\pi ) |
970 |
– |
\] |
971 |
– |
A second-order symplectic method is now obtained by the composition |
972 |
– |
of the flow maps, |
973 |
– |
\[ |
974 |
– |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
975 |
– |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
976 |
– |
\] |
977 |
– |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
978 |
– |
sub-flows which corresponding to force and torque respectively, |
979 |
– |
\[ |
980 |
– |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
981 |
– |
_{\Delta t/2,\tau }. |
982 |
– |
\] |
983 |
– |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
984 |
– |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
985 |
– |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
986 |
– |
|
987 |
– |
Furthermore, kinetic potential can be separated to translational |
988 |
– |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
989 |
– |
\begin{equation} |
990 |
– |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
991 |
– |
\end{equation} |
992 |
– |
where $ T^t (p) = \frac{1}{2}v^T m v $ and $T^r (\pi )$ is defined |
993 |
– |
by \ref{introEquation:rotationalKineticRB}. Therefore, the |
994 |
– |
corresponding flow maps are given by |
995 |
– |
\[ |
996 |
– |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
997 |
– |
_{\Delta t,T^r }. |
998 |
– |
\] |
999 |
– |
The free rigid body is an example of Lie-Poisson system with |
1000 |
– |
Hamiltonian function |
1001 |
– |
\begin{equation} |
1002 |
– |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1003 |
– |
\label{introEquation:rotationalKineticRB} |
1004 |
– |
\end{equation} |
1005 |
– |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
1006 |
– |
Lie-Poisson structure matrix, |
1007 |
– |
\begin{equation} |
1008 |
– |
J(\pi ) = \left( {\begin{array}{*{20}c} |
1009 |
– |
0 & {\pi _3 } & { - \pi _2 } \\ |
1010 |
– |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
1011 |
– |
{\pi _2 } & { - \pi _1 } & 0 \\ |
1012 |
– |
\end{array}} \right) |
1013 |
– |
\end{equation} |
1014 |
– |
Thus, the dynamics of free rigid body is governed by |
1015 |
– |
\begin{equation} |
1016 |
– |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
1017 |
– |
\end{equation} |
1018 |
– |
One may notice that each $T_i^r$ in Equation |
1019 |
– |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1020 |
– |
instance, the equations of motion due to $T_1^r$ are given by |
1021 |
– |
\begin{equation} |
1022 |
– |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}A = AR_1 |
1023 |
– |
\label{introEqaution:RBMotionSingleTerm} |
1024 |
– |
\end{equation} |
1025 |
– |
where |
1026 |
– |
\[ R_1 = \left( {\begin{array}{*{20}c} |
1027 |
– |
0 & 0 & 0 \\ |
1028 |
– |
0 & 0 & {\pi _1 } \\ |
1029 |
– |
0 & { - \pi _1 } & 0 \\ |
1030 |
– |
\end{array}} \right). |
1031 |
– |
\] |
1032 |
– |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1033 |
– |
\[ |
1034 |
– |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),A(\Delta t) = |
1035 |
– |
A(0)e^{\Delta tR_1 } |
1036 |
– |
\] |
1037 |
– |
with |
1038 |
– |
\[ |
1039 |
– |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
1040 |
– |
0 & 0 & 0 \\ |
1041 |
– |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
1042 |
– |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
1043 |
– |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1044 |
– |
\] |
1045 |
– |
To reduce the cost of computing expensive functions in $e^{\Delta |
1046 |
– |
tR_1 }$, we can use Cayley transformation, |
1047 |
– |
\[ |
1048 |
– |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1049 |
– |
) |
1050 |
– |
\] |
1051 |
– |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1052 |
– |
manner. |
1053 |
– |
|
1054 |
– |
In order to construct a second-order symplectic method, we split the |
1055 |
– |
angular kinetic Hamiltonian function into five terms |
1056 |
– |
\[ |
1057 |
– |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1058 |
– |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1059 |
– |
(\pi _1 ) |
1060 |
– |
\]. |
1061 |
– |
Concatenating flows corresponding to these five terms, we can obtain |
1062 |
– |
the flow map for free rigid body, |
1063 |
– |
\[ |
1064 |
– |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
1065 |
– |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1066 |
– |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1067 |
– |
_1 }. |
1068 |
– |
\] |
1069 |
– |
|
1070 |
– |
The equations of motion corresponding to potential energy and |
1071 |
– |
kinetic energy are listed in the below table, |
1072 |
– |
\begin{center} |
1073 |
– |
\begin{tabular}{|l|l|} |
1074 |
– |
\hline |
1075 |
– |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
1076 |
– |
Potential & Kinetic \\ |
1077 |
– |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
1078 |
– |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
1079 |
– |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
1080 |
– |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
1081 |
– |
\hline |
1082 |
– |
\end{tabular} |
1083 |
– |
\end{center} |
1084 |
– |
|
1085 |
– |
Finally, we obtain the overall symplectic flow maps for free moving |
1086 |
– |
rigid body |
1087 |
– |
\begin{align*} |
1088 |
– |
\varphi _{\Delta t} = &\varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \circ \\ |
1089 |
– |
&\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 \\ |
1090 |
– |
&\varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1091 |
– |
\label{introEquation:overallRBFlowMaps} |
1092 |
– |
\end{align*} |
1093 |
– |
|
1094 |
– |
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
1095 |
– |
|
1096 |
– |
Consider a Langevin equation of motions in generalized coordinates |
1097 |
– |
\begin{equation} |
1098 |
– |
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
1099 |
– |
\label{LDGeneralizedForm} |
1100 |
– |
\end{equation} |
1101 |
– |
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
1102 |
– |
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
1103 |
– |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
1104 |
– |
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
1105 |
– |
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
1106 |
– |
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
1107 |
– |
system in Newtownian mechanics typically refers to lab-fixed frame, |
1108 |
– |
it is also convenient to handle the rotation of rigid body in |
1109 |
– |
body-fixed frame. Thus the friction and random forces are calculated |
1110 |
– |
in body-fixed frame and converted back to lab-fixed frame by: |
1111 |
– |
\[ |
1112 |
– |
\begin{array}{l} |
1113 |
– |
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
1114 |
– |
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
1115 |
– |
\end{array}. |
1116 |
– |
\] |
1117 |
– |
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
1118 |
– |
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
1119 |
– |
angular velocity $\omega _i$, |
1120 |
– |
\begin{equation} |
1121 |
– |
F_{r,i}^b (t) = \left( \begin{array}{l} |
1122 |
– |
f_{r,i}^b (t) \\ |
1123 |
– |
\tau _{r,i}^b (t) \\ |
1124 |
– |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
1125 |
– |
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
1126 |
– |
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
1127 |
– |
\end{array}} \right)\left( \begin{array}{l} |
1128 |
– |
v_{R,i}^b (t) \\ |
1129 |
– |
\omega _i (t) \\ |
1130 |
– |
\end{array} \right), |
1131 |
– |
\end{equation} |
1132 |
– |
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
1133 |
– |
with zero mean and variance |
1134 |
– |
\begin{equation} |
1135 |
– |
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
1136 |
– |
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
1137 |
– |
2k_B T\Xi _R \delta (t - t'). |
1138 |
– |
\end{equation} |
1139 |
– |
The equation of motion for $v_i$ can be written as |
1140 |
– |
\begin{equation} |
1141 |
– |
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
1142 |
– |
f_{r,i}^l (t) |
1143 |
– |
\end{equation} |
1144 |
– |
Since the frictional force is applied at the center of resistance |
1145 |
– |
which generally does not coincide with the center of mass, an extra |
1146 |
– |
torque is exerted at the center of mass. Thus, the net body-fixed |
1147 |
– |
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
1148 |
– |
given by |
1149 |
– |
\begin{equation} |
1150 |
– |
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
1151 |
– |
\end{equation} |
1152 |
– |
where $r_{MR}$ is the vector from the center of mass to the center |
1153 |
– |
of the resistance. Instead of integrating angular velocity in |
1154 |
– |
lab-fixed frame, we consider the equation of motion of angular |
1155 |
– |
momentum in body-fixed frame |
1156 |
– |
\begin{equation} |
1157 |
– |
\dot \pi _i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b |
1158 |
– |
(t) + \tau _{r,i}^b(t) |
1159 |
– |
\end{equation} |
1160 |
– |
|
1161 |
– |
Embedding the friction terms into force and torque, one can |
1162 |
– |
integrate the langevin equations of motion for rigid body of |
1163 |
– |
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
1164 |
– |
$h= \delta t$: |
1165 |
– |
|
1166 |
– |
{\tt part one:} |
1167 |
– |
\begin{align*} |
1168 |
– |
v_i (t + h/2) &\leftarrow v_i (t) + \frac{{hf_{t,i}^l (t)}}{{2m_i }} \\ |
1169 |
– |
\pi _i (t + h/2) &\leftarrow \pi _i (t) + \frac{{h\tau _{t,i}^b (t)}}{2} \\ |
1170 |
– |
r_i (t + h) &\leftarrow r_i (t) + hv_i (t + h/2) \\ |
1171 |
– |
A_i (t + h) &\leftarrow rotate\left( {h\pi _i (t + h/2)I^{ - 1} } \right) \\ |
1172 |
– |
\end{align*} |
1173 |
– |
In this context, the $\mathrm{rotate}$ function is the reversible |
1174 |
– |
product of five consecutive body-fixed rotations, |
1175 |
– |
\begin{equation} |
1176 |
– |
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
1177 |
– |
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
1178 |
– |
/ 2) \cdot \mathsf{G}_x(a_x /2), |
1179 |
– |
\end{equation} |
1180 |
– |
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
1181 |
– |
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
1182 |
– |
angular momentum ($\pi$) by an angle $\theta$ around body-fixed axis |
1183 |
– |
$\alpha$, |
1184 |
– |
\begin{equation} |
1185 |
– |
\mathsf{G}_\alpha( \theta ) = \left\{ |
1186 |
– |
\begin{array}{lcl} |
1187 |
– |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
1188 |
– |
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
1189 |
– |
j}(0). |
1190 |
– |
\end{array} |
1191 |
– |
\right. |
1192 |
– |
\end{equation} |
1193 |
– |
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
1194 |
– |
rotation matrix. For example, in the small-angle limit, the |
1195 |
– |
rotation matrix around the body-fixed x-axis can be approximated as |
1196 |
– |
\begin{equation} |
1197 |
– |
\mathsf{R}_x(\theta) \approx \left( |
1198 |
– |
\begin{array}{ccc} |
1199 |
– |
1 & 0 & 0 \\ |
1200 |
– |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
1201 |
– |
\theta^2 / 4} \\ |
1202 |
– |
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
1203 |
– |
\theta^2 / 4} |
1204 |
– |
\end{array} |
1205 |
– |
\right). |
1206 |
– |
\end{equation} |
1207 |
– |
All other rotations follow in a straightforward manner. |
1208 |
– |
|
1209 |
– |
After the first part of the propagation, the friction and random |
1210 |
– |
forces are generated at the center of resistance in body-fixed frame |
1211 |
– |
and converted back into lab-fixed frame |
1212 |
– |
\[ |
1213 |
– |
f_{t,i}^l (t + h) = - \left( {\frac{{\partial V}}{{\partial r_i }}} |
1214 |
– |
\right)_{r_i (t + h)} + A_i^T (t + h)[F_{f,i}^b (t + h) + F_{r,i}^b |
1215 |
– |
(t + h)], |
1216 |
– |
\] |
1217 |
– |
while the system torque in lab-fixed frame is transformed into |
1218 |
– |
body-fixed frame, |
1219 |
– |
\[ |
1220 |
– |
\tau _{t,i}^b (t + h) = A\tau _{s,i}^l (t) + \tau _{n,i}^b (t) + |
1221 |
– |
\tau _{r,i}^b (t). |
1222 |
– |
\] |
1223 |
– |
Once the forces and torques have been obtained at the new time step, |
1224 |
– |
the velocities can be advanced to the same time value. |
1225 |
– |
|
1226 |
– |
{\tt part two:} |
1227 |
– |
\begin{align*} |
1228 |
– |
v_i (t) &\leftarrow v_i (t + h/2) + \frac{{hf_{t,i}^l (t + h)}}{{2m_i }} \\ |
1229 |
– |
\pi _i (t) &\leftarrow \pi _i (t + h/2) + \frac{{h\tau _{t,i}^b (t + h)}}{2} \\ |
1230 |
– |
\end{align*} |
1231 |
– |
|
1232 |
– |
\subsection{Results and discussion} |