| 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. |
| 139 |
|
average 7\% increase in computation time using the DLM method in |
| 140 |
|
place of quaternions. This cost is more than justified when |
| 141 |
|
comparing the energy conservation of the two methods as illustrated |
| 142 |
< |
in Fig.~\ref{timestep}. |
| 142 |
> |
in Fig.~\ref{methodFig:timestep}. |
| 143 |
|
|
| 144 |
|
\begin{figure} |
| 145 |
|
\centering |
| 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. Details on the constraint |
| 478 |
– |
algorithms are given in section \ref{oopseSec:rattle}. |
| 479 |
– |
|
| 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( |
| 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} |
| 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) \\ |
| 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 |
|
|
| 595 |
< |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
| 595 |
> |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 596 |
|
|
| 611 |
– |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 612 |
– |
|
| 597 |
|
A comprehensive understanding of structure¨Cfunction relations of |
| 598 |
|
biological membrane system ultimately relies on structure and |
| 599 |
|
dynamics of lipid bilayer, which are strongly affected by the |
| 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 |
| 742 |
|
\begin{equation} |
| 743 |
|
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 744 |
|
-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 |
– |
%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) |
| 745 |
|
\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} |
