| 668 |
|
the box axes does not change. It should be noted that the NPTxyz |
| 669 |
|
integrator is a special case of $NP\gamma T$ if the surface tension |
| 670 |
|
$\gamma$ is set to zero. |
| 671 |
– |
|
| 672 |
– |
%\section{\label{methodSection:constraintMethod}Constraint Method} |
| 671 |
|
|
| 672 |
< |
%\subsection{\label{methodSection:bondConstraint}Bond Constraint for Rigid Body} |
| 672 |
> |
\section{\label{methodSection:zcons}Z-Constraint Method} |
| 673 |
|
|
| 674 |
< |
%\subsection{\label{methodSection:zcons}Z-constraint Method} |
| 674 |
> |
Based on the fluctuation-dissipation theorem, a force |
| 675 |
> |
auto-correlation method was developed by Roux and Karplus to |
| 676 |
> |
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
| 677 |
> |
The time-dependent friction coefficient can be calculated from the |
| 678 |
> |
deviation of the instantaneous force from its mean force. |
| 679 |
> |
\begin{equation} |
| 680 |
> |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
| 681 |
> |
\end{equation} |
| 682 |
> |
where% |
| 683 |
> |
\begin{equation} |
| 684 |
> |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 685 |
> |
\end{equation} |
| 686 |
|
|
| 687 |
< |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
| 687 |
> |
If the time-dependent friction decays rapidly, the static friction |
| 688 |
> |
coefficient can be approximated by |
| 689 |
> |
\begin{equation} |
| 690 |
> |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
| 691 |
> |
F(z,0)\rangle dt. |
| 692 |
> |
\end{equation} |
| 693 |
> |
Allowing diffusion constant to then be calculated through the |
| 694 |
> |
Einstein relation:\cite{Marrink1994} |
| 695 |
> |
\begin{equation} |
| 696 |
> |
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 697 |
> |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 698 |
> |
\end{equation} |
| 699 |
|
|
| 700 |
< |
\subsection{\label{methodSection:temperature}Temperature Control} |
| 701 |
< |
|
| 702 |
< |
\subsection{\label{methodSection:pressureControl}Pressure Control} |
| 700 |
> |
The Z-Constraint method, which fixes the z coordinates of the |
| 701 |
> |
molecules with respect to the center of the mass of the system, has |
| 702 |
> |
been a method suggested to obtain the forces required for the force |
| 703 |
> |
auto-correlation calculation.\cite{Marrink1994} However, simply |
| 704 |
> |
resetting the coordinate will move the center of the mass of the |
| 705 |
> |
whole system. To avoid this problem, we reset the forces of |
| 706 |
> |
z-constrained molecules as well as subtract the total constraint |
| 707 |
> |
forces from the rest of the system after the force calculation at |
| 708 |
> |
each time step instead of resetting the coordinate. |
| 709 |
|
|
| 710 |
< |
\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
| 710 |
> |
After the force calculation, define $G_\alpha$ as |
| 711 |
> |
\begin{equation} |
| 712 |
> |
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
| 713 |
> |
\end{equation} |
| 714 |
> |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
| 715 |
> |
z-constrained molecule $\alpha$. The forces of the z constrained |
| 716 |
> |
molecule are then set to: |
| 717 |
> |
\begin{equation} |
| 718 |
> |
F_{\alpha i} = F_{\alpha i} - |
| 719 |
> |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
| 720 |
> |
\end{equation} |
| 721 |
> |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
| 722 |
> |
molecule. Having rescaled the forces, the velocities must also be |
| 723 |
> |
rescaled to subtract out any center of mass velocity in the z |
| 724 |
> |
direction. |
| 725 |
> |
\begin{equation} |
| 726 |
> |
v_{\alpha i} = v_{\alpha i} - |
| 727 |
> |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
| 728 |
> |
\end{equation} |
| 729 |
> |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
| 730 |
> |
Lastly, all of the accumulated z constrained forces must be |
| 731 |
> |
subtracted from the system to keep the system center of mass from |
| 732 |
> |
drifting. |
| 733 |
> |
\begin{equation} |
| 734 |
> |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
| 735 |
> |
G_{\alpha}} |
| 736 |
> |
{\sum_{\beta}\sum_i m_{\beta i}}, |
| 737 |
> |
\end{equation} |
| 738 |
> |
where $\beta$ are all of the unconstrained molecules in the system. |
| 739 |
> |
Similarly, the velocities of the unconstrained molecules must also |
| 740 |
> |
be scaled. |
| 741 |
> |
\begin{equation} |
| 742 |
> |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 743 |
> |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 744 |
> |
\end{equation} |
| 745 |
|
|
| 746 |
< |
%\section{\label{methodSection:coarseGrained}Coarse-Grained Modeling} |
| 746 |
> |
At the very beginning of the simulation, the molecules may not be at |
| 747 |
> |
their constrained positions. To move a z-constrained molecule to its |
| 748 |
> |
specified position, a simple harmonic potential is used |
| 749 |
> |
\begin{equation} |
| 750 |
> |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
| 751 |
> |
\end{equation} |
| 752 |
> |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
| 753 |
> |
is the current $z$ coordinate of the center of mass of the |
| 754 |
> |
constrained molecule, and $z_{\text{cons}}$ is the constrained |
| 755 |
> |
position. The harmonic force operating on the z-constrained molecule |
| 756 |
> |
at time $t$ can be calculated by |
| 757 |
> |
\begin{equation} |
| 758 |
> |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 759 |
> |
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
| 760 |
> |
\end{equation} |
| 761 |
> |
|
| 762 |
> |
|
| 763 |
> |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
| 764 |
> |
|
| 765 |
> |
%\subsection{\label{methodSection:temperature}Temperature Control} |
| 766 |
> |
|
| 767 |
> |
%\subsection{\label{methodSection:pressureControl}Pressure Control} |
| 768 |
> |
|
| 769 |
> |
%\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
| 770 |
> |
|
| 771 |
> |
%applications of langevin dynamics |
| 772 |
> |
As an excellent alternative to newtonian dynamics, Langevin |
| 773 |
> |
dynamics, which mimics a simple heat bath with stochastic and |
| 774 |
> |
dissipative forces, has been applied in a variety of studies. The |
| 775 |
> |
stochastic treatment of the solvent enables us to carry out |
| 776 |
> |
substantially longer time simulation. Implicit solvent Langevin |
| 777 |
> |
dynamics simulation of met-enkephalin not only outperforms explicit |
| 778 |
> |
solvent simulation on computation efficiency, but also agrees very |
| 779 |
> |
well with explicit solvent simulation on dynamics |
| 780 |
> |
properties\cite{Shen2002}. Recently, applying Langevin dynamics with |
| 781 |
> |
UNRES model, Liow and his coworkers suggest that protein folding |
| 782 |
> |
pathways can be possibly exploited within a reasonable amount of |
| 783 |
> |
time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics |
| 784 |
> |
also enhances the sampling of the system and increases the |
| 785 |
> |
probability of crossing energy barrier\cite{Banerjee2004, Cui2003}. |
| 786 |
> |
Combining Langevin dynamics with Kramers's theory, Klimov and |
| 787 |
> |
Thirumalai identified the free-energy barrier by studying the |
| 788 |
> |
viscosity dependence of the protein folding rates\cite{Klimov1997}. |
| 789 |
> |
In order to account for solvent induced interactions missing from |
| 790 |
> |
implicit solvent model, Kaya incorporated desolvation free energy |
| 791 |
> |
barrier into implicit coarse-grained solvent model in protein |
| 792 |
> |
folding/unfolding study and discovered a higher free energy barrier |
| 793 |
> |
between the native and denatured states. Because of its stability |
| 794 |
> |
against noise, Langevin dynamics is very suitable for studying |
| 795 |
> |
remagnetization processes in various |
| 796 |
> |
systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For |
| 797 |
> |
instance, the oscillation power spectrum of nanoparticles from |
| 798 |
> |
Langevin dynamics simulation has the same peak frequencies for |
| 799 |
> |
different wave vectors,which recovers the property of magnetic |
| 800 |
> |
excitations in small finite structures\cite{Berkov2005a}. In an |
| 801 |
> |
attempt to reduce the computational cost of simulation, multiple |
| 802 |
> |
time stepping (MTS) methods have been introduced and have been of |
| 803 |
> |
great interest to macromolecule and protein |
| 804 |
> |
community\cite{Tuckerman1992}. Relying on the observation that |
| 805 |
> |
forces between distant atoms generally demonstrate slower |
| 806 |
> |
fluctuations than forces between close atoms, MTS method are |
| 807 |
> |
generally implemented by evaluating the slowly fluctuating forces |
| 808 |
> |
less frequently than the fast ones. Unfortunately, nonlinear |
| 809 |
> |
instability resulting from increasing timestep in MTS simulation |
| 810 |
> |
have became a critical obstruction preventing the long time |
| 811 |
> |
simulation. Due to the coupling to the heat bath, Langevin dynamics |
| 812 |
> |
has been shown to be able to damp out the resonance artifact more |
| 813 |
> |
efficiently\cite{Sandu1999}. |
| 814 |
|
|
| 815 |
< |
%\section{\label{methodSection:moleculeScale}Molecular-Scale Modeling} |
| 815 |
> |
%review rigid body dynamics |
| 816 |
> |
Rigid bodies are frequently involved in the modeling of different |
| 817 |
> |
areas, from engineering, physics, to chemistry. For example, |
| 818 |
> |
missiles and vehicle are usually modeled by rigid bodies. The |
| 819 |
> |
movement of the objects in 3D gaming engine or other physics |
| 820 |
> |
simulator is governed by the rigid body dynamics. In molecular |
| 821 |
> |
simulation, rigid body is used to simplify the model in |
| 822 |
> |
protein-protein docking study\cite{Gray2003}. |
| 823 |
> |
|
| 824 |
> |
It is very important to develop stable and efficient methods to |
| 825 |
> |
integrate the equations of motion of orientational degrees of |
| 826 |
> |
freedom. Euler angles are the nature choice to describe the |
| 827 |
> |
rotational degrees of freedom. However, due to its singularity, the |
| 828 |
> |
numerical integration of corresponding equations of motion is very |
| 829 |
> |
inefficient and inaccurate. Although an alternative integrator using |
| 830 |
> |
different sets of Euler angles can overcome this |
| 831 |
> |
difficulty\cite{Ryckaert1977, Andersen1983}, the computational |
| 832 |
> |
penalty and the lost of angular momentum conservation still remain. |
| 833 |
> |
In 1977, a singularity free representation utilizing quaternions was |
| 834 |
> |
developed by Evans\cite{Evans1977}. Unfortunately, this approach |
| 835 |
> |
suffer from the nonseparable Hamiltonian resulted from quaternion |
| 836 |
> |
representation, which prevents the symplectic algorithm to be |
| 837 |
> |
utilized. Another different approach is to apply holonomic |
| 838 |
> |
constraints to the atoms belonging to the rigid |
| 839 |
> |
body\cite{Barojas1973}. Each atom moves independently under the |
| 840 |
> |
normal forces deriving from potential energy and constraint forces |
| 841 |
> |
which are used to guarantee the rigidness. However, due to their |
| 842 |
> |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
| 843 |
> |
when the number of constraint increases. |
| 844 |
> |
|
| 845 |
> |
The break through in geometric literature suggests that, in order to |
| 846 |
> |
develop a long-term integration scheme, one should preserve the |
| 847 |
> |
geometric structure of the flow. Matubayasi and Nakahara developed a |
| 848 |
> |
time-reversible integrator for rigid bodies in quaternion |
| 849 |
> |
representation. Although it is not symplectic, this integrator still |
| 850 |
> |
demonstrates a better long-time energy conservation than traditional |
| 851 |
> |
methods because of the time-reversible nature. Extending |
| 852 |
> |
Trotter-Suzuki to general system with a flat phase space, Miller and |
| 853 |
> |
his colleagues devised an novel symplectic, time-reversible and |
| 854 |
> |
volume-preserving integrator in quaternion representation. However, |
| 855 |
> |
all of the integrators in quaternion representation suffer from the |
| 856 |
> |
computational penalty of constructing a rotation matrix from |
| 857 |
> |
quaternions to evolve coordinates and velocities at every time step. |
| 858 |
> |
An alternative integration scheme utilizing rotation matrix directly |
| 859 |
> |
is RSHAKE\cite{Kol1997}, in which a conjugate momentum to rotation |
| 860 |
> |
matrix is introduced to re-formulate the Hamiltonian's equation and |
| 861 |
> |
the Hamiltonian is evolved in a constraint manifold by iteratively |
| 862 |
> |
satisfying the orthogonality constraint. However, RSHAKE is |
| 863 |
> |
inefficient because of the iterative procedure. An extremely |
| 864 |
> |
efficient integration scheme in rotation matrix representation, |
| 865 |
> |
which also preserves the same structural properties of the |
| 866 |
> |
Hamiltonian flow as Miller's integrator, is proposed by Dullweber, |
| 867 |
> |
Leimkuhler and McLachlan (DLM)\cite{Dullweber1997}. |
| 868 |
> |
|
| 869 |
> |
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
| 870 |
> |
Combining Langevin or Brownian dynamics with rigid body dynamics, |
| 871 |
> |
one can study the slow processes in biomolecular systems. Modeling |
| 872 |
> |
the DNA as a chain of rigid spheres beads, which subject to harmonic |
| 873 |
> |
potentials as well as excluded volume potentials, Mielke and his |
| 874 |
> |
coworkers discover rapid superhelical stress generations from the |
| 875 |
> |
stochastic simulation of twin supercoiling DNA with response to |
| 876 |
> |
induced torques\cite{Mielke2004}. Membrane fusion is another key |
| 877 |
> |
biological process which controls a variety of physiological |
| 878 |
> |
functions, such as release of neurotransmitters \textit{etc}. A |
| 879 |
> |
typical fusion event happens on the time scale of millisecond, which |
| 880 |
> |
is impracticable to study using all atomistic model with newtonian |
| 881 |
> |
mechanics. With the help of coarse-grained rigid body model and |
| 882 |
> |
stochastic dynamics, the fusion pathways were exploited by many |
| 883 |
> |
researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the |
| 884 |
> |
difficulty of numerical integration of anisotropy rotation, most of |
| 885 |
> |
the rigid body models are simply modeled by sphere, cylinder, |
| 886 |
> |
ellipsoid or other regular shapes in stochastic simulations. In an |
| 887 |
> |
effort to account for the diffusion anisotropy of the arbitrary |
| 888 |
> |
particles, Fernandes and de la Torre improved the original Brownian |
| 889 |
> |
dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
| 890 |
> |
incorporating a generalized $6\times6$ diffusion tensor and |
| 891 |
> |
introducing a simple rotation evolution scheme consisting of three |
| 892 |
> |
consecutive rotations\cite{Fernandes2002}. Unfortunately, unexpected |
| 893 |
> |
error and bias are introduced into the system due to the arbitrary |
| 894 |
> |
order of applying the noncommuting rotation |
| 895 |
> |
operators\cite{Beard2003}. Based on the observation the momentum |
| 896 |
> |
relaxation time is much less than the time step, one may ignore the |
| 897 |
> |
inertia in Brownian dynamics. However, assumption of the zero |
| 898 |
> |
average acceleration is not always true for cooperative motion which |
| 899 |
> |
is common in protein motion. An inertial Brownian dynamics (IBD) was |
| 900 |
> |
proposed to address this issue by adding an inertial correction |
| 901 |
> |
term\cite{Beard2001}. As a complement to IBD which has a lower bound |
| 902 |
> |
in time step because of the inertial relaxation time, long-time-step |
| 903 |
> |
inertial dynamics (LTID) can be used to investigate the inertial |
| 904 |
> |
behavior of the polymer segments in low friction |
| 905 |
> |
regime\cite{Beard2001}. LTID can also deal with the rotational |
| 906 |
> |
dynamics for nonskew bodies without translation-rotation coupling by |
| 907 |
> |
separating the translation and rotation motion and taking advantage |
| 908 |
> |
of the analytical solution of hydrodynamics properties. However, |
| 909 |
> |
typical nonskew bodies like cylinder and ellipsoid are inadequate to |
| 910 |
> |
represent most complex macromolecule assemblies. These intricate |
| 911 |
> |
molecules have been represented by a set of beads and their |
| 912 |
> |
hydrodynamics properties can be calculated using variant |
| 913 |
> |
hydrodynamic interaction tensors. |
| 914 |
> |
|
| 915 |
> |
The goal of the present work is to develop a Langevin dynamics |
| 916 |
> |
algorithm for arbitrary rigid particles by integrating the accurate |
| 917 |
> |
estimation of friction tensor from hydrodynamics theory into the |
| 918 |
> |
sophisticated rigid body dynamics. |
| 919 |
> |
|
| 920 |
> |
|
| 921 |
> |
\subsection{Friction Tensor} |
| 922 |
> |
|
| 923 |
> |
For an arbitrary rigid body moves in a fluid, it may experience |
| 924 |
> |
friction force $f_r$ or friction torque $\tau _r$ along the opposite |
| 925 |
> |
direction of the velocity $v$ or angular velocity $\omega$ at |
| 926 |
> |
arbitrary origin $P$, |
| 927 |
> |
\begin{equation} |
| 928 |
> |
\left( \begin{array}{l} |
| 929 |
> |
f_r \\ |
| 930 |
> |
\tau _r \\ |
| 931 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 932 |
> |
{\Xi _{P,t} } & {\Xi _{P,c}^T } \\ |
| 933 |
> |
{\Xi _{P,c} } & {\Xi _{P,r} } \\ |
| 934 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 935 |
> |
\nu \\ |
| 936 |
> |
\omega \\ |
| 937 |
> |
\end{array} \right) |
| 938 |
> |
\end{equation} |
| 939 |
> |
where $\Xi _{P,t}t$ is the translation friction tensor, $\Xi _{P,r}$ |
| 940 |
> |
is the rotational friction tensor and $\Xi _{P,c}$ is the |
| 941 |
> |
translation-rotation coupling tensor. The procedure of calculating |
| 942 |
> |
friction tensor using hydrodynamic tensor and comparison between |
| 943 |
> |
bead model and shell model were elaborated by Carrasco \textit{et |
| 944 |
> |
al}\cite{Carrasco1999}. An important property of the friction tensor |
| 945 |
> |
is that the translational friction tensor is independent of origin |
| 946 |
> |
while the rotational and coupling are sensitive to the choice of the |
| 947 |
> |
origin \cite{Brenner1967}, which can be described by |
| 948 |
> |
\begin{equation} |
| 949 |
> |
\begin{array}{c} |
| 950 |
> |
\Xi _{P,t} = \Xi _{O,t} = \Xi _t \\ |
| 951 |
> |
\Xi _{P,c} = \Xi _{O,c} - r_{OP} \times \Xi _t \\ |
| 952 |
> |
\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 \\ |
| 953 |
> |
\end{array} |
| 954 |
> |
\end{equation} |
| 955 |
> |
Where $O$ is another origin and $r_{OP}$ is the vector joining $O$ |
| 956 |
> |
and $P$. It is also worthy of mention that both of translational and |
| 957 |
> |
rotational frictional tensors are always symmetric. In contrast, |
| 958 |
> |
coupling tensor is only symmetric at center of reaction: |
| 959 |
> |
\begin{equation} |
| 960 |
> |
\Xi _{R,c} = \Xi _{R,c}^T |
| 961 |
> |
\end{equation} |
| 962 |
> |
The proper location for applying friction force is the center of |
| 963 |
> |
reaction, at which the trace of rotational resistance tensor reaches |
| 964 |
> |
minimum. |
| 965 |
> |
|
| 966 |
> |
\subsection{Rigid body dynamics} |
| 967 |
> |
|
| 968 |
> |
The Hamiltonian of rigid body can be separated in terms of potential |
| 969 |
> |
energy $V(r,A)$ and kinetic energy $T(p,\pi)$, |
| 970 |
> |
\[ |
| 971 |
> |
H = V(r,A) + T(v,\pi ) |
| 972 |
> |
\] |
| 973 |
> |
A second-order symplectic method is now obtained by the composition |
| 974 |
> |
of the flow maps, |
| 975 |
> |
\[ |
| 976 |
> |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 977 |
> |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 978 |
> |
\] |
| 979 |
> |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 980 |
> |
sub-flows which corresponding to force and torque respectively, |
| 981 |
> |
\[ |
| 982 |
> |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 983 |
> |
_{\Delta t/2,\tau }. |
| 984 |
> |
\] |
| 985 |
> |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 986 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 987 |
> |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
| 988 |
> |
|
| 989 |
> |
Furthermore, kinetic potential can be separated to translational |
| 990 |
> |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 991 |
> |
\begin{equation} |
| 992 |
> |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 993 |
> |
\end{equation} |
| 994 |
> |
where $ T^t (p) = \frac{1}{2}v^T m v $ and $T^r (\pi )$ is defined |
| 995 |
> |
by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 996 |
> |
corresponding flow maps are given by |
| 997 |
> |
\[ |
| 998 |
> |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 999 |
> |
_{\Delta t,T^r }. |
| 1000 |
> |
\] |
| 1001 |
> |
The free rigid body is an example of Lie-Poisson system with |
| 1002 |
> |
Hamiltonian function |
| 1003 |
> |
\begin{equation} |
| 1004 |
> |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1005 |
> |
\label{introEquation:rotationalKineticRB} |
| 1006 |
> |
\end{equation} |
| 1007 |
> |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
| 1008 |
> |
Lie-Poisson structure matrix, |
| 1009 |
> |
\begin{equation} |
| 1010 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 1011 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1012 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1013 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1014 |
> |
\end{array}} \right) |
| 1015 |
> |
\end{equation} |
| 1016 |
> |
Thus, the dynamics of free rigid body is governed by |
| 1017 |
> |
\begin{equation} |
| 1018 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1019 |
> |
\end{equation} |
| 1020 |
> |
One may notice that each $T_i^r$ in Equation |
| 1021 |
> |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1022 |
> |
instance, the equations of motion due to $T_1^r$ are given by |
| 1023 |
> |
\begin{equation} |
| 1024 |
> |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}A = AR_1 |
| 1025 |
> |
\label{introEqaution:RBMotionSingleTerm} |
| 1026 |
> |
\end{equation} |
| 1027 |
> |
where |
| 1028 |
> |
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1029 |
> |
0 & 0 & 0 \\ |
| 1030 |
> |
0 & 0 & {\pi _1 } \\ |
| 1031 |
> |
0 & { - \pi _1 } & 0 \\ |
| 1032 |
> |
\end{array}} \right). |
| 1033 |
> |
\] |
| 1034 |
> |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1035 |
> |
\[ |
| 1036 |
> |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),A(\Delta t) = |
| 1037 |
> |
A(0)e^{\Delta tR_1 } |
| 1038 |
> |
\] |
| 1039 |
> |
with |
| 1040 |
> |
\[ |
| 1041 |
> |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
| 1042 |
> |
0 & 0 & 0 \\ |
| 1043 |
> |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
| 1044 |
> |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1045 |
> |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1046 |
> |
\] |
| 1047 |
> |
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1048 |
> |
tR_1 }$, we can use Cayley transformation, |
| 1049 |
> |
\[ |
| 1050 |
> |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1051 |
> |
) |
| 1052 |
> |
\] |
| 1053 |
> |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1054 |
> |
manner. |
| 1055 |
> |
|
| 1056 |
> |
In order to construct a second-order symplectic method, we split the |
| 1057 |
> |
angular kinetic Hamiltonian function into five terms |
| 1058 |
> |
\[ |
| 1059 |
> |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1060 |
> |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1061 |
> |
(\pi _1 ) |
| 1062 |
> |
\]. |
| 1063 |
> |
Concatenating flows corresponding to these five terms, we can obtain |
| 1064 |
> |
the flow map for free rigid body, |
| 1065 |
> |
\[ |
| 1066 |
> |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1067 |
> |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1068 |
> |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1069 |
> |
_1 }. |
| 1070 |
> |
\] |
| 1071 |
> |
|
| 1072 |
> |
The equations of motion corresponding to potential energy and |
| 1073 |
> |
kinetic energy are listed in the below table, |
| 1074 |
> |
\begin{center} |
| 1075 |
> |
\begin{tabular}{|l|l|} |
| 1076 |
> |
\hline |
| 1077 |
> |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 1078 |
> |
Potential & Kinetic \\ |
| 1079 |
> |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
| 1080 |
> |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
| 1081 |
> |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
| 1082 |
> |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
| 1083 |
> |
\hline |
| 1084 |
> |
\end{tabular} |
| 1085 |
> |
\end{center} |
| 1086 |
> |
|
| 1087 |
> |
Finally, we obtain the overall symplectic flow maps for free moving |
| 1088 |
> |
rigid body |
| 1089 |
> |
\begin{align*} |
| 1090 |
> |
\varphi _{\Delta t} = &\varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \circ \\ |
| 1091 |
> |
&\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 \\ |
| 1092 |
> |
&\varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1093 |
> |
\label{introEquation:overallRBFlowMaps} |
| 1094 |
> |
\end{align*} |
| 1095 |
> |
|
| 1096 |
> |
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
| 1097 |
> |
|
| 1098 |
> |
Consider a Langevin equation of motions in generalized coordinates |
| 1099 |
> |
\begin{equation} |
| 1100 |
> |
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
| 1101 |
> |
\label{LDGeneralizedForm} |
| 1102 |
> |
\end{equation} |
| 1103 |
> |
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
| 1104 |
> |
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
| 1105 |
> |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
| 1106 |
> |
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
| 1107 |
> |
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
| 1108 |
> |
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
| 1109 |
> |
system in Newtownian mechanics typically refers to lab-fixed frame, |
| 1110 |
> |
it is also convenient to handle the rotation of rigid body in |
| 1111 |
> |
body-fixed frame. Thus the friction and random forces are calculated |
| 1112 |
> |
in body-fixed frame and converted back to lab-fixed frame by: |
| 1113 |
> |
\[ |
| 1114 |
> |
\begin{array}{l} |
| 1115 |
> |
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
| 1116 |
> |
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
| 1117 |
> |
\end{array}. |
| 1118 |
> |
\] |
| 1119 |
> |
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
| 1120 |
> |
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
| 1121 |
> |
angular velocity $\omega _i$, |
| 1122 |
> |
\begin{equation} |
| 1123 |
> |
F_{r,i}^b (t) = \left( \begin{array}{l} |
| 1124 |
> |
f_{r,i}^b (t) \\ |
| 1125 |
> |
\tau _{r,i}^b (t) \\ |
| 1126 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1127 |
> |
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
| 1128 |
> |
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
| 1129 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 1130 |
> |
v_{R,i}^b (t) \\ |
| 1131 |
> |
\omega _i (t) \\ |
| 1132 |
> |
\end{array} \right), |
| 1133 |
> |
\end{equation} |
| 1134 |
> |
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
| 1135 |
> |
with zero mean and variance |
| 1136 |
> |
\begin{equation} |
| 1137 |
> |
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
| 1138 |
> |
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
| 1139 |
> |
2k_B T\Xi _R \delta (t - t'). |
| 1140 |
> |
\end{equation} |
| 1141 |
> |
The equation of motion for $v_i$ can be written as |
| 1142 |
> |
\begin{equation} |
| 1143 |
> |
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
| 1144 |
> |
f_{r,i}^l (t) |
| 1145 |
> |
\end{equation} |
| 1146 |
> |
Since the frictional force is applied at the center of resistance |
| 1147 |
> |
which generally does not coincide with the center of mass, an extra |
| 1148 |
> |
torque is exerted at the center of mass. Thus, the net body-fixed |
| 1149 |
> |
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
| 1150 |
> |
given by |
| 1151 |
> |
\begin{equation} |
| 1152 |
> |
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
| 1153 |
> |
\end{equation} |
| 1154 |
> |
where $r_{MR}$ is the vector from the center of mass to the center |
| 1155 |
> |
of the resistance. Instead of integrating angular velocity in |
| 1156 |
> |
lab-fixed frame, we consider the equation of motion of angular |
| 1157 |
> |
momentum in body-fixed frame |
| 1158 |
> |
\begin{equation} |
| 1159 |
> |
\dot \pi _i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b |
| 1160 |
> |
(t) + \tau _{r,i}^b(t) |
| 1161 |
> |
\end{equation} |
| 1162 |
> |
|
| 1163 |
> |
Embedding the friction terms into force and torque, one can |
| 1164 |
> |
integrate the langevin equations of motion for rigid body of |
| 1165 |
> |
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
| 1166 |
> |
$h= \delta t$: |
| 1167 |
> |
|
| 1168 |
> |
{\tt part one:} |
| 1169 |
> |
\begin{align*} |
| 1170 |
> |
v_i (t + h/2) &\leftarrow v_i (t) + \frac{{hf_{t,i}^l (t)}}{{2m_i }} \\ |
| 1171 |
> |
\pi _i (t + h/2) &\leftarrow \pi _i (t) + \frac{{h\tau _{t,i}^b (t)}}{2} \\ |
| 1172 |
> |
r_i (t + h) &\leftarrow r_i (t) + hv_i (t + h/2) \\ |
| 1173 |
> |
A_i (t + h) &\leftarrow rotate\left( {h\pi _i (t + h/2)I^{ - 1} } \right) \\ |
| 1174 |
> |
\end{align*} |
| 1175 |
> |
In this context, the $\mathrm{rotate}$ function is the reversible |
| 1176 |
> |
product of five consecutive body-fixed rotations, |
| 1177 |
> |
\begin{equation} |
| 1178 |
> |
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
| 1179 |
> |
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
| 1180 |
> |
/ 2) \cdot \mathsf{G}_x(a_x /2), |
| 1181 |
> |
\end{equation} |
| 1182 |
> |
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
| 1183 |
> |
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
| 1184 |
> |
angular momentum ($\pi$) by an angle $\theta$ around body-fixed axis |
| 1185 |
> |
$\alpha$, |
| 1186 |
> |
\begin{equation} |
| 1187 |
> |
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 1188 |
> |
\begin{array}{lcl} |
| 1189 |
> |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 1190 |
> |
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
| 1191 |
> |
j}(0). |
| 1192 |
> |
\end{array} |
| 1193 |
> |
\right. |
| 1194 |
> |
\end{equation} |
| 1195 |
> |
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
| 1196 |
> |
rotation matrix. For example, in the small-angle limit, the |
| 1197 |
> |
rotation matrix around the body-fixed x-axis can be approximated as |
| 1198 |
> |
\begin{equation} |
| 1199 |
> |
\mathsf{R}_x(\theta) \approx \left( |
| 1200 |
> |
\begin{array}{ccc} |
| 1201 |
> |
1 & 0 & 0 \\ |
| 1202 |
> |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
| 1203 |
> |
\theta^2 / 4} \\ |
| 1204 |
> |
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
| 1205 |
> |
\theta^2 / 4} |
| 1206 |
> |
\end{array} |
| 1207 |
> |
\right). |
| 1208 |
> |
\end{equation} |
| 1209 |
> |
All other rotations follow in a straightforward manner. |
| 1210 |
> |
|
| 1211 |
> |
After the first part of the propagation, the friction and random |
| 1212 |
> |
forces are generated at the center of resistance in body-fixed frame |
| 1213 |
> |
and converted back into lab-fixed frame |
| 1214 |
> |
\[ |
| 1215 |
> |
f_{t,i}^l (t + h) = - \left( {\frac{{\partial V}}{{\partial r_i }}} |
| 1216 |
> |
\right)_{r_i (t + h)} + A_i^T (t + h)[F_{f,i}^b (t + h) + F_{r,i}^b |
| 1217 |
> |
(t + h)], |
| 1218 |
> |
\] |
| 1219 |
> |
while the system torque in lab-fixed frame is transformed into |
| 1220 |
> |
body-fixed frame, |
| 1221 |
> |
\[ |
| 1222 |
> |
\tau _{t,i}^b (t + h) = A\tau _{s,i}^l (t) + \tau _{n,i}^b (t) + |
| 1223 |
> |
\tau _{r,i}^b (t). |
| 1224 |
> |
\] |
| 1225 |
> |
Once the forces and torques have been obtained at the new time step, |
| 1226 |
> |
the velocities can be advanced to the same time value. |
| 1227 |
> |
|
| 1228 |
> |
{\tt part two:} |
| 1229 |
> |
\begin{align*} |
| 1230 |
> |
v_i (t) &\leftarrow v_i (t + h/2) + \frac{{hf_{t,i}^l (t + h)}}{{2m_i }} \\ |
| 1231 |
> |
\pi _i (t) &\leftarrow \pi _i (t + h/2) + \frac{{h\tau _{t,i}^b (t + h)}}{2} \\ |
| 1232 |
> |
\end{align*} |
| 1233 |
> |
|
| 1234 |
> |
\subsection{Results and discussion} |
