| 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 |
| 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 |
– |
|
| 813 |
|
It is very important to develop stable and efficient methods to |
| 814 |
|
integrate the equations of motion of orientational degrees of |
| 815 |
|
freedom. Euler angles are the nature choice to describe the |
| 833 |
|
|
| 834 |
|
The break through in geometric literature suggests that, in order to |
| 835 |
|
develop a long-term integration scheme, one should preserve the |
| 836 |
< |
geometric structure of the flow. Matubayasi and Nakahara developed a |
| 836 |
> |
geometric structure of the flow. Matubayasi developed a |
| 837 |
|
time-reversible integrator for rigid bodies in quaternion |
| 838 |
|
representation. Although it is not symplectic, this integrator still |
| 839 |
|
demonstrates a better long-time energy conservation than traditional |
| 906 |
|
estimation of friction tensor from hydrodynamics theory into the |
| 907 |
|
sophisticated rigid body dynamics. |
| 908 |
|
|
| 909 |
< |
|
| 910 |
< |
\subsection{Friction Tensor} |
| 911 |
< |
|
| 912 |
< |
For an arbitrary rigid body moves in a fluid, it may experience |
| 913 |
< |
friction force $f_r$ or friction torque $\tau _r$ along the opposite |
| 914 |
< |
direction of the velocity $v$ or angular velocity $\omega$ at |
| 915 |
< |
arbitrary origin $P$, |
| 916 |
< |
\begin{equation} |
| 917 |
< |
\left( \begin{array}{l} |
| 918 |
< |
f_r \\ |
| 919 |
< |
\tau _r \\ |
| 920 |
< |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 921 |
< |
{\Xi _{P,t} } & {\Xi _{P,c}^T } \\ |
| 922 |
< |
{\Xi _{P,c} } & {\Xi _{P,r} } \\ |
| 923 |
< |
\end{array}} \right)\left( \begin{array}{l} |
| 924 |
< |
\nu \\ |
| 925 |
< |
\omega \\ |
| 909 |
> |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 910 |
> |
Theoretically, the friction kernel can be determined using velocity |
| 911 |
> |
autocorrelation function. However, this approach become impractical |
| 912 |
> |
when the system become more and more complicate. Instead, various |
| 913 |
> |
approaches based on hydrodynamics have been developed to calculate |
| 914 |
> |
the friction coefficients. The friction effect is isotropic in |
| 915 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 916 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 917 |
> |
\[ |
| 918 |
> |
\Xi = \left( {\begin{array}{*{20}c} |
| 919 |
> |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 920 |
> |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 921 |
> |
\end{array}} \right). |
| 922 |
> |
\] |
| 923 |
> |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 924 |
> |
tensor and rotational resistance (friction) tensor respectively, |
| 925 |
> |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 926 |
> |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 927 |
> |
particle moves in a fluid, it may experience friction force or |
| 928 |
> |
torque along the opposite direction of the velocity or angular |
| 929 |
> |
velocity, |
| 930 |
> |
\[ |
| 931 |
> |
\left( \begin{array}{l} |
| 932 |
> |
F_R \\ |
| 933 |
> |
\tau _R \\ |
| 934 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 935 |
> |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 936 |
> |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 937 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 938 |
> |
v \\ |
| 939 |
> |
w \\ |
| 940 |
|
\end{array} \right) |
| 941 |
< |
\end{equation} |
| 942 |
< |
where $\Xi _{P,t}t$ is the translation friction tensor, $\Xi _{P,r}$ |
| 943 |
< |
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. |
| 941 |
> |
\] |
| 942 |
> |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 943 |
> |
toque. |
| 944 |
|
|
| 945 |
< |
\subsection{Rigid body dynamics} |
| 945 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
| 946 |
|
|
| 947 |
< |
The Hamiltonian of rigid body can be separated in terms of potential |
| 948 |
< |
energy $V(r,A)$ and kinetic energy $T(p,\pi)$, |
| 947 |
> |
For a spherical particle, the translational and rotational friction |
| 948 |
> |
constant can be calculated from Stoke's law, |
| 949 |
|
\[ |
| 950 |
< |
H = V(r,A) + T(v,\pi ) |
| 951 |
< |
\] |
| 952 |
< |
A second-order symplectic method is now obtained by the composition |
| 953 |
< |
of the flow maps, |
| 950 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 951 |
> |
{6\pi \eta R} & 0 & 0 \\ |
| 952 |
> |
0 & {6\pi \eta R} & 0 \\ |
| 953 |
> |
0 & 0 & {6\pi \eta R} \\ |
| 954 |
> |
\end{array}} \right) |
| 955 |
> |
\] |
| 956 |
> |
and |
| 957 |
|
\[ |
| 958 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 959 |
< |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 958 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 959 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 960 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
| 961 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
| 962 |
> |
\end{array}} \right) |
| 963 |
|
\] |
| 964 |
< |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 965 |
< |
sub-flows which corresponding to force and torque respectively, |
| 964 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 965 |
> |
hydrodynamics radius. |
| 966 |
> |
|
| 967 |
> |
Other non-spherical shape, such as cylinder and ellipsoid |
| 968 |
> |
\textit{etc}, are widely used as reference for developing new |
| 969 |
> |
hydrodynamics theory, because their properties can be calculated |
| 970 |
> |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 971 |
> |
also called a triaxial ellipsoid, which is given in Cartesian |
| 972 |
> |
coordinates by\cite{Perrin1934, Perrin1936} |
| 973 |
|
\[ |
| 974 |
< |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 975 |
< |
_{\Delta t/2,\tau }. |
| 974 |
> |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 975 |
> |
}} = 1 |
| 976 |
|
\] |
| 977 |
< |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 978 |
< |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 979 |
< |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
| 977 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 978 |
> |
due to the complexity of the elliptic integral, only the ellipsoid |
| 979 |
> |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 980 |
> |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 981 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 982 |
> |
\[ |
| 983 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 984 |
> |
} }}{b}, |
| 985 |
> |
\] |
| 986 |
> |
and oblate, |
| 987 |
> |
\[ |
| 988 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 989 |
> |
}}{a} |
| 990 |
> |
\], |
| 991 |
> |
one can write down the translational and rotational resistance |
| 992 |
> |
tensors |
| 993 |
> |
\[ |
| 994 |
> |
\begin{array}{l} |
| 995 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 996 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 997 |
> |
\end{array}, |
| 998 |
> |
\] |
| 999 |
> |
and |
| 1000 |
> |
\[ |
| 1001 |
> |
\begin{array}{l} |
| 1002 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1003 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1004 |
> |
\end{array}. |
| 1005 |
> |
\] |
| 1006 |
|
|
| 1007 |
< |
Furthermore, kinetic potential can be separated to translational |
| 1008 |
< |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1009 |
< |
\begin{equation} |
| 1010 |
< |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1011 |
< |
\end{equation} |
| 1012 |
< |
where $ T^t (p) = \frac{1}{2}v^T m v $ and $T^r (\pi )$ is defined |
| 1013 |
< |
by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1014 |
< |
corresponding flow maps are given by |
| 1007 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
| 1008 |
> |
|
| 1009 |
> |
Unlike spherical and other regular shaped molecules, there is not |
| 1010 |
> |
analytical solution for friction tensor of any arbitrary shaped |
| 1011 |
> |
rigid molecules. The ellipsoid of revolution model and general |
| 1012 |
> |
triaxial ellipsoid model have been used to approximate the |
| 1013 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1014 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1015 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
| 1016 |
> |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 1017 |
> |
translational and rotational motion of rigid body, general ellipsoid |
| 1018 |
> |
is not always suitable for modeling arbitrarily shaped rigid |
| 1019 |
> |
molecule. A number of studies have been devoted to determine the |
| 1020 |
> |
friction tensor for irregularly shaped rigid bodies using more |
| 1021 |
> |
advanced method where the molecule of interest was modeled by |
| 1022 |
> |
combinations of spheres(beads)\cite{Carrasco1999} and the |
| 1023 |
> |
hydrodynamics properties of the molecule can be calculated using the |
| 1024 |
> |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
| 1025 |
> |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
| 1026 |
> |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
| 1027 |
> |
than its unperturbed velocity $v_i$, |
| 1028 |
|
\[ |
| 1029 |
< |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 997 |
< |
_{\Delta t,T^r }. |
| 1029 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1030 |
|
\] |
| 1031 |
< |
The free rigid body is an example of Lie-Poisson system with |
| 1032 |
< |
Hamiltonian function |
| 1031 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1032 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1033 |
> |
proportional to its ``net'' velocity |
| 1034 |
|
\begin{equation} |
| 1035 |
< |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1036 |
< |
\label{introEquation:rotationalKineticRB} |
| 1035 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1036 |
> |
\label{introEquation:tensorExpression} |
| 1037 |
|
\end{equation} |
| 1038 |
< |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
| 1039 |
< |
Lie-Poisson structure matrix, |
| 1038 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1039 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1040 |
> |
\ref{introEquation:tensorExpression} |
| 1041 |
|
\begin{equation} |
| 1042 |
< |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 1043 |
< |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1010 |
< |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1011 |
< |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1012 |
< |
\end{array}} \right) |
| 1042 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1043 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} |
| 1044 |
|
\end{equation} |
| 1045 |
< |
Thus, the dynamics of free rigid body is governed by |
| 1045 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1046 |
> |
A second order expression for element of different size was |
| 1047 |
> |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
| 1048 |
> |
Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
| 1049 |
|
\begin{equation} |
| 1050 |
< |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1050 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1051 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1052 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1053 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1054 |
> |
\label{introEquation:RPTensorNonOverlapped} |
| 1055 |
|
\end{equation} |
| 1056 |
< |
One may notice that each $T_i^r$ in Equation |
| 1057 |
< |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1058 |
< |
instance, the equations of motion due to $T_1^r$ are given by |
| 1056 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1057 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1058 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1059 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
| 1060 |
|
\begin{equation} |
| 1061 |
< |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}A = AR_1 |
| 1062 |
< |
\label{introEqaution:RBMotionSingleTerm} |
| 1061 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1062 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1063 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1064 |
> |
\label{introEquation:RPTensorOverlapped} |
| 1065 |
|
\end{equation} |
| 1066 |
< |
where |
| 1067 |
< |
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1068 |
< |
0 & 0 & 0 \\ |
| 1069 |
< |
0 & 0 & {\pi _1 } \\ |
| 1070 |
< |
0 & { - \pi _1 } & 0 \\ |
| 1071 |
< |
\end{array}} \right). |
| 1072 |
< |
\] |
| 1073 |
< |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1066 |
> |
|
| 1067 |
> |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1068 |
> |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1069 |
> |
$B_{ij}$ blocks |
| 1070 |
> |
\begin{equation} |
| 1071 |
> |
B = \left( {\begin{array}{*{20}c} |
| 1072 |
> |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1073 |
> |
\vdots & \ddots & \vdots \\ |
| 1074 |
> |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1075 |
> |
\end{array}} \right), |
| 1076 |
> |
\end{equation} |
| 1077 |
> |
where $B_{ij}$ is given by |
| 1078 |
|
\[ |
| 1079 |
< |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),A(\Delta t) = |
| 1080 |
< |
A(0)e^{\Delta tR_1 } |
| 1079 |
> |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1080 |
> |
)T_{ij} |
| 1081 |
|
\] |
| 1082 |
< |
with |
| 1082 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 1083 |
> |
$B$, we obtain |
| 1084 |
> |
|
| 1085 |
|
\[ |
| 1086 |
< |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
| 1087 |
< |
0 & 0 & 0 \\ |
| 1088 |
< |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
| 1089 |
< |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1090 |
< |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1086 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1087 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1088 |
> |
\vdots & \ddots & \vdots \\ |
| 1089 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1090 |
> |
\end{array}} \right) |
| 1091 |
|
\] |
| 1092 |
< |
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1093 |
< |
tR_1 }$, we can use Cayley transformation, |
| 1092 |
> |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1093 |
> |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1094 |
|
\[ |
| 1095 |
< |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1096 |
< |
) |
| 1095 |
> |
U_i = \left( {\begin{array}{*{20}c} |
| 1096 |
> |
0 & { - z_i } & {y_i } \\ |
| 1097 |
> |
{z_i } & 0 & { - x_i } \\ |
| 1098 |
> |
{ - y_i } & {x_i } & 0 \\ |
| 1099 |
> |
\end{array}} \right) |
| 1100 |
|
\] |
| 1101 |
< |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1102 |
< |
manner. |
| 1101 |
> |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1102 |
> |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1103 |
> |
arbitrary origin $O$ can be written as |
| 1104 |
> |
\begin{equation} |
| 1105 |
> |
\begin{array}{l} |
| 1106 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1107 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1108 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1109 |
> |
\end{array} |
| 1110 |
> |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1111 |
> |
\end{equation} |
| 1112 |
|
|
| 1113 |
< |
In order to construct a second-order symplectic method, we split the |
| 1114 |
< |
angular kinetic Hamiltonian function into five terms |
| 1113 |
> |
The resistance tensor depends on the origin to which they refer. The |
| 1114 |
> |
proper location for applying friction force is the center of |
| 1115 |
> |
resistance (reaction), at which the trace of rotational resistance |
| 1116 |
> |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1117 |
> |
resistance is defined as an unique point of the rigid body at which |
| 1118 |
> |
the translation-rotation coupling tensor are symmetric, |
| 1119 |
> |
\begin{equation} |
| 1120 |
> |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1121 |
> |
\label{introEquation:definitionCR} |
| 1122 |
> |
\end{equation} |
| 1123 |
> |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1124 |
> |
we can easily find out that the translational resistance tensor is |
| 1125 |
> |
origin independent, while the rotational resistance tensor and |
| 1126 |
> |
translation-rotation coupling resistance tensor depend on the |
| 1127 |
> |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1128 |
> |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1129 |
> |
obtain the resistance tensor at $P$ by |
| 1130 |
> |
\begin{equation} |
| 1131 |
> |
\begin{array}{l} |
| 1132 |
> |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 1133 |
> |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 1134 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
| 1135 |
> |
\end{array} |
| 1136 |
> |
\label{introEquation:resistanceTensorTransformation} |
| 1137 |
> |
\end{equation} |
| 1138 |
> |
where |
| 1139 |
|
\[ |
| 1140 |
< |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1141 |
< |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1142 |
< |
(\pi _1 ) |
| 1143 |
< |
\]. |
| 1144 |
< |
Concatenating flows corresponding to these five terms, we can obtain |
| 1062 |
< |
the flow map for free rigid body, |
| 1063 |
< |
\[ |
| 1064 |
< |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1065 |
< |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1066 |
< |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1067 |
< |
_1 }. |
| 1140 |
> |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 1141 |
> |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 1142 |
> |
{z_i } & 0 & { - x_{OP} } \\ |
| 1143 |
> |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 1144 |
> |
\end{array}} \right) |
| 1145 |
|
\] |
| 1146 |
+ |
Using Equations \ref{introEquation:definitionCR} and |
| 1147 |
+ |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 1148 |
+ |
the position of center of resistance, |
| 1149 |
+ |
\begin{eqnarray*} |
| 1150 |
+ |
\left( \begin{array}{l} |
| 1151 |
+ |
x_{OR} \\ |
| 1152 |
+ |
y_{OR} \\ |
| 1153 |
+ |
z_{OR} \\ |
| 1154 |
+ |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
| 1155 |
+ |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 1156 |
+ |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 1157 |
+ |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 1158 |
+ |
\end{array}} \right)^{ - 1} \\ |
| 1159 |
+ |
& & \left( \begin{array}{l} |
| 1160 |
+ |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 1161 |
+ |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 1162 |
+ |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 1163 |
+ |
\end{array} \right) \\ |
| 1164 |
+ |
\end{eqnarray*} |
| 1165 |
|
|
| 1166 |
< |
The equations of motion corresponding to potential energy and |
| 1167 |
< |
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} |
| 1166 |
> |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1167 |
> |
joining center of resistance $R$ and origin $O$. |
| 1168 |
|
|
| 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 |
– |
|
| 1169 |
|
\subsection{Langevin dynamics for rigid particles of arbitrary shape} |
| 1170 |
|
|
| 1171 |
|
Consider a Langevin equation of motions in generalized coordinates |
