570 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
571 |
|
\end{equation} |
572 |
|
The most obvious change being that matrix $J$ now depends on $x$. |
573 |
– |
The free rigid body is an example of Poisson system (actually a |
574 |
– |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
575 |
– |
energy. |
576 |
– |
\begin{equation} |
577 |
– |
J(\pi ) = \left( {\begin{array}{*{20}c} |
578 |
– |
0 & {\pi _3 } & { - \pi _2 } \\ |
579 |
– |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
580 |
– |
{\pi _2 } & { - \pi _1 } & 0 \\ |
581 |
– |
\end{array}} \right) |
582 |
– |
\end{equation} |
573 |
|
|
584 |
– |
\begin{equation} |
585 |
– |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
586 |
– |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
587 |
– |
\end{equation} |
588 |
– |
|
574 |
|
\subsection{\label{introSection:exactFlow}Exact Flow} |
575 |
|
|
576 |
|
Let $x(t)$ be the exact solution of the ODE system, |
822 |
|
% |
823 |
|
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
824 |
|
q(\Delta t)} \right]. % |
825 |
< |
\label{introEquation:positionVerlet1} |
825 |
> |
\label{introEquation:positionVerlet2} |
826 |
|
\end{align} |
827 |
|
|
828 |
|
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
831 |
|
error of splitting method in terms of commutator of the |
832 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
833 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
834 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
834 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
835 |
|
\begin{equation} |
836 |
|
\exp (hX + hY) = \exp (hZ) |
837 |
|
\end{equation} |
846 |
|
\] |
847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
848 |
|
can obtain |
849 |
< |
\begin{eqnarray*} |
849 |
> |
\begin{equation} |
850 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
851 |
|
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
852 |
|
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
853 |
|
\ldots ) |
854 |
< |
\end{eqnarray*} |
854 |
> |
\end{equation} |
855 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
856 |
|
error of Spring splitting is proportional to $h^3$. The same |
857 |
|
procedure can be applied to general splitting, of the form |
883 |
|
|
884 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
885 |
|
|
886 |
< |
As a special discipline of molecular modeling, Molecular dynamics |
887 |
< |
has proven to be a powerful tool for studying the functions of |
888 |
< |
biological systems, providing structural, thermodynamic and |
889 |
< |
dynamical information. |
890 |
< |
|
891 |
< |
\subsection{\label{introSec:mdInit}Initialization} |
892 |
< |
|
893 |
< |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
886 |
> |
As one of the principal tools of molecular modeling, Molecular |
887 |
> |
dynamics has proven to be a powerful tool for studying the functions |
888 |
> |
of biological systems, providing structural, thermodynamic and |
889 |
> |
dynamical information. The basic idea of molecular dynamics is that |
890 |
> |
macroscopic properties are related to microscopic behavior and |
891 |
> |
microscopic behavior can be calculated from the trajectories in |
892 |
> |
simulations. For instance, instantaneous temperature of an |
893 |
> |
Hamiltonian system of $N$ particle can be measured by |
894 |
> |
\[ |
895 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
896 |
> |
\] |
897 |
> |
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
898 |
> |
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
899 |
> |
the boltzman constant. |
900 |
|
|
901 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
901 |
> |
A typical molecular dynamics run consists of three essential steps: |
902 |
> |
\begin{enumerate} |
903 |
> |
\item Initialization |
904 |
> |
\begin{enumerate} |
905 |
> |
\item Preliminary preparation |
906 |
> |
\item Minimization |
907 |
> |
\item Heating |
908 |
> |
\item Equilibration |
909 |
> |
\end{enumerate} |
910 |
> |
\item Production |
911 |
> |
\item Analysis |
912 |
> |
\end{enumerate} |
913 |
> |
These three individual steps will be covered in the following |
914 |
> |
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
915 |
> |
initialization of a simulation. Sec.~\ref{introSec:production} will |
916 |
> |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
917 |
> |
provides the theoretical tools for trajectory analysis. |
918 |
> |
|
919 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
920 |
> |
|
921 |
> |
\subsubsection{Preliminary preparation} |
922 |
> |
|
923 |
> |
When selecting the starting structure of a molecule for molecular |
924 |
> |
simulation, one may retrieve its Cartesian coordinates from public |
925 |
> |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
926 |
> |
thousands of crystal structures of molecules are discovered every |
927 |
> |
year, many more remain unknown due to the difficulties of |
928 |
> |
purification and crystallization. Even for the molecule with known |
929 |
> |
structure, some important information is missing. For example, the |
930 |
> |
missing hydrogen atom which acts as donor in hydrogen bonding must |
931 |
> |
be added. Moreover, in order to include electrostatic interaction, |
932 |
> |
one may need to specify the partial charges for individual atoms. |
933 |
> |
Under some circumstances, we may even need to prepare the system in |
934 |
> |
a special setup. For instance, when studying transport phenomenon in |
935 |
> |
membrane system, we may prepare the lipids in bilayer structure |
936 |
> |
instead of placing lipids randomly in solvent, since we are not |
937 |
> |
interested in self-aggregation and it takes a long time to happen. |
938 |
> |
|
939 |
> |
\subsubsection{Minimization} |
940 |
|
|
941 |
+ |
It is quite possible that some of molecules in the system from |
942 |
+ |
preliminary preparation may be overlapped with each other. This |
943 |
+ |
close proximity leads to high potential energy which consequently |
944 |
+ |
jeopardizes any molecular dynamics simulations. To remove these |
945 |
+ |
steric overlaps, one typically performs energy minimization to find |
946 |
+ |
a more reasonable conformation. Several energy minimization methods |
947 |
+ |
have been developed to exploit the energy surface and to locate the |
948 |
+ |
local minimum. While converging slowly near the minimum, steepest |
949 |
+ |
descent method is extremely robust when systems are far from |
950 |
+ |
harmonic. Thus, it is often used to refine structure from |
951 |
+ |
crystallographic data. Relied on the gradient or hessian, advanced |
952 |
+ |
methods like conjugate gradient and Newton-Raphson converge rapidly |
953 |
+ |
to a local minimum, while become unstable if the energy surface is |
954 |
+ |
far from quadratic. Another factor must be taken into account, when |
955 |
+ |
choosing energy minimization method, is the size of the system. |
956 |
+ |
Steepest descent and conjugate gradient can deal with models of any |
957 |
+ |
size. Because of the limit of computation power to calculate hessian |
958 |
+ |
matrix and insufficient storage capacity to store them, most |
959 |
+ |
Newton-Raphson methods can not be used with very large models. |
960 |
+ |
|
961 |
+ |
\subsubsection{Heating} |
962 |
+ |
|
963 |
+ |
Typically, Heating is performed by assigning random velocities |
964 |
+ |
according to a Gaussian distribution for a temperature. Beginning at |
965 |
+ |
a lower temperature and gradually increasing the temperature by |
966 |
+ |
assigning greater random velocities, we end up with setting the |
967 |
+ |
temperature of the system to a final temperature at which the |
968 |
+ |
simulation will be conducted. In heating phase, we should also keep |
969 |
+ |
the system from drifting or rotating as a whole. Equivalently, the |
970 |
+ |
net linear momentum and angular momentum of the system should be |
971 |
+ |
shifted to zero. |
972 |
+ |
|
973 |
+ |
\subsubsection{Equilibration} |
974 |
+ |
|
975 |
+ |
The purpose of equilibration is to allow the system to evolve |
976 |
+ |
spontaneously for a period of time and reach equilibrium. The |
977 |
+ |
procedure is continued until various statistical properties, such as |
978 |
+ |
temperature, pressure, energy, volume and other structural |
979 |
+ |
properties \textit{etc}, become independent of time. Strictly |
980 |
+ |
speaking, minimization and heating are not necessary, provided the |
981 |
+ |
equilibration process is long enough. However, these steps can serve |
982 |
+ |
as a means to arrive at an equilibrated structure in an effective |
983 |
+ |
way. |
984 |
+ |
|
985 |
+ |
\subsection{\label{introSection:production}Production} |
986 |
+ |
|
987 |
+ |
Production run is the most important steps of the simulation, in |
988 |
+ |
which the equilibrated structure is used as a starting point and the |
989 |
+ |
motions of the molecules are collected for later analysis. In order |
990 |
+ |
to capture the macroscopic properties of the system, the molecular |
991 |
+ |
dynamics simulation must be performed in correct and efficient way. |
992 |
+ |
|
993 |
+ |
The most expensive part of a molecular dynamics simulation is the |
994 |
+ |
calculation of non-bonded forces, such as van der Waals force and |
995 |
+ |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
996 |
+ |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
997 |
+ |
which making large simulations prohibitive in the absence of any |
998 |
+ |
computation saving techniques. |
999 |
+ |
|
1000 |
+ |
A natural approach to avoid system size issue is to represent the |
1001 |
+ |
bulk behavior by a finite number of the particles. However, this |
1002 |
+ |
approach will suffer from the surface effect. To offset this, |
1003 |
+ |
\textit{Periodic boundary condition} is developed to simulate bulk |
1004 |
+ |
properties with a relatively small number of particles. In this |
1005 |
+ |
method, the simulation box is replicated throughout space to form an |
1006 |
+ |
infinite lattice. During the simulation, when a particle moves in |
1007 |
+ |
the primary cell, its image in other cells move in exactly the same |
1008 |
+ |
direction with exactly the same orientation. Thus, as a particle |
1009 |
+ |
leaves the primary cell, one of its images will enter through the |
1010 |
+ |
opposite face. |
1011 |
+ |
%\begin{figure} |
1012 |
+ |
%\centering |
1013 |
+ |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1014 |
+ |
%\caption[An illustration of periodic boundary conditions]{A 2-D |
1015 |
+ |
%illustration of periodic boundary conditions. As one particle leaves |
1016 |
+ |
%the right of the simulation box, an image of it enters the left.} |
1017 |
+ |
%\label{introFig:pbc} |
1018 |
+ |
%\end{figure} |
1019 |
+ |
|
1020 |
+ |
%cutoff and minimum image convention |
1021 |
+ |
Another important technique to improve the efficiency of force |
1022 |
+ |
evaluation is to apply cutoff where particles farther than a |
1023 |
+ |
predetermined distance, are not included in the calculation |
1024 |
+ |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1025 |
+ |
discontinuity in the potential energy curve |
1026 |
+ |
(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the |
1027 |
+ |
potential to ensure the potential curve go smoothly to zero at the |
1028 |
+ |
cutoff radius. Cutoff strategy works pretty well for Lennard-Jones |
1029 |
+ |
interaction because of its short range nature. However, simply |
1030 |
+ |
truncating the electrostatic interaction with the use of cutoff has |
1031 |
+ |
been shown to lead to severe artifacts in simulations. Ewald |
1032 |
+ |
summation, in which the slowly conditionally convergent Coulomb |
1033 |
+ |
potential is transformed into direct and reciprocal sums with rapid |
1034 |
+ |
and absolute convergence, has proved to minimize the periodicity |
1035 |
+ |
artifacts in liquid simulations. Taking the advantages of the fast |
1036 |
+ |
Fourier transform (FFT) for calculating discrete Fourier transforms, |
1037 |
+ |
the particle mesh-based methods are accelerated from $O(N^{3/2})$ to |
1038 |
+ |
$O(N logN)$. An alternative approach is \emph{fast multipole |
1039 |
+ |
method}, which treats Coulombic interaction exactly at short range, |
1040 |
+ |
and approximate the potential at long range through multipolar |
1041 |
+ |
expansion. In spite of their wide acceptances at the molecular |
1042 |
+ |
simulation community, these two methods are hard to be implemented |
1043 |
+ |
correctly and efficiently. Instead, we use a damped and |
1044 |
+ |
charge-neutralized Coulomb potential method developed by Wolf and |
1045 |
+ |
his coworkers. The shifted Coulomb potential for particle $i$ and |
1046 |
+ |
particle $j$ at distance $r_{rj}$ is given by: |
1047 |
+ |
\begin{equation} |
1048 |
+ |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1049 |
+ |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1050 |
+ |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
1051 |
+ |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
1052 |
+ |
\end{equation} |
1053 |
+ |
where $\alpha$ is the convergence parameter. Due to the lack of |
1054 |
+ |
inherent periodicity and rapid convergence,this method is extremely |
1055 |
+ |
efficient and easy to implement. |
1056 |
+ |
%\begin{figure} |
1057 |
+ |
%\centering |
1058 |
+ |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1059 |
+ |
%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} |
1060 |
+ |
%\label{introFigure:shiftedCoulomb} |
1061 |
+ |
%\end{figure} |
1062 |
+ |
|
1063 |
+ |
%multiple time step |
1064 |
+ |
|
1065 |
+ |
\subsection{\label{introSection:Analysis} Analysis} |
1066 |
+ |
|
1067 |
+ |
Recently, advanced visualization technique are widely applied to |
1068 |
+ |
monitor the motions of molecules. Although the dynamics of the |
1069 |
+ |
system can be described qualitatively from animation, quantitative |
1070 |
+ |
trajectory analysis are more appreciable. According to the |
1071 |
+ |
principles of Statistical Mechanics, |
1072 |
+ |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1073 |
+ |
thermodynamics properties, analyze fluctuations of structural |
1074 |
+ |
parameters, and investigate time-dependent processes of the molecule |
1075 |
+ |
from the trajectories. |
1076 |
+ |
|
1077 |
+ |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
1078 |
+ |
|
1079 |
+ |
Thermodynamics properties, which can be expressed in terms of some |
1080 |
+ |
function of the coordinates and momenta of all particles in the |
1081 |
+ |
system, can be directly computed from molecular dynamics. The usual |
1082 |
+ |
way to measure the pressure is based on virial theorem of Clausius |
1083 |
+ |
which states that the virial is equal to $-3Nk_BT$. For a system |
1084 |
+ |
with forces between particles, the total virial, $W$, contains the |
1085 |
+ |
contribution from external pressure and interaction between the |
1086 |
+ |
particles: |
1087 |
+ |
\[ |
1088 |
+ |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
1089 |
+ |
f_{ij} } } \right\rangle |
1090 |
+ |
\] |
1091 |
+ |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
1092 |
+ |
distance $r_{ij}$. Thus, the expression for the pressure is given |
1093 |
+ |
by: |
1094 |
+ |
\begin{equation} |
1095 |
+ |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
1096 |
+ |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1097 |
+ |
\end{equation} |
1098 |
+ |
|
1099 |
+ |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
1100 |
+ |
|
1101 |
+ |
Structural Properties of a simple fluid can be described by a set of |
1102 |
+ |
distribution functions. Among these functions,\emph{pair |
1103 |
+ |
distribution function}, also known as \emph{radial distribution |
1104 |
+ |
function}, is of most fundamental importance to liquid-state theory. |
1105 |
+ |
Pair distribution function can be gathered by Fourier transforming |
1106 |
+ |
raw data from a series of neutron diffraction experiments and |
1107 |
+ |
integrating over the surface factor \cite{Powles73}. The experiment |
1108 |
+ |
result can serve as a criterion to justify the correctness of the |
1109 |
+ |
theory. Moreover, various equilibrium thermodynamic and structural |
1110 |
+ |
properties can also be expressed in terms of radial distribution |
1111 |
+ |
function \cite{allen87:csl}. |
1112 |
+ |
|
1113 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
1114 |
+ |
particle $i$ will be located at a distance $r$ from a another |
1115 |
+ |
particle $j$ in the system |
1116 |
+ |
\[ |
1117 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1118 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1119 |
+ |
\] |
1120 |
+ |
Note that the delta function can be replaced by a histogram in |
1121 |
+ |
computer simulation. Figure |
1122 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1123 |
+ |
distribution function for the liquid argon system. The occurrence of |
1124 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1125 |
+ |
to find particles at certain radial values than at others. This is a |
1126 |
+ |
result of the attractive interaction at such distances. Because of |
1127 |
+ |
the strong repulsive forces at short distance, the probability of |
1128 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
1129 |
+ |
other is essentially zero. |
1130 |
+ |
|
1131 |
+ |
%\begin{figure} |
1132 |
+ |
%\centering |
1133 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
1134 |
+ |
%\caption[Pair distribution function for the liquid argon |
1135 |
+ |
%]{Pair distribution function for the liquid argon} |
1136 |
+ |
%\label{introFigure:pairDistributionFunction} |
1137 |
+ |
%\end{figure} |
1138 |
+ |
|
1139 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
1140 |
+ |
Properties} |
1141 |
+ |
|
1142 |
+ |
Time-dependent properties are usually calculated using \emph{time |
1143 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
1144 |
+ |
at two different time |
1145 |
+ |
\begin{equation} |
1146 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1147 |
+ |
\label{introEquation:timeCorrelationFunction} |
1148 |
+ |
\end{equation} |
1149 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
1150 |
+ |
function is called \emph{auto correlation function}. One example of |
1151 |
+ |
auto correlation function is velocity auto-correlation function |
1152 |
+ |
which is directly related to transport properties of molecular |
1153 |
+ |
liquids: |
1154 |
+ |
\[ |
1155 |
+ |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1156 |
+ |
\right\rangle } dt |
1157 |
+ |
\] |
1158 |
+ |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1159 |
+ |
function which is averaging over time origins and over all the |
1160 |
+ |
atoms, dipole autocorrelation are calculated for the entire system. |
1161 |
+ |
The dipole autocorrelation function is given by: |
1162 |
+ |
\[ |
1163 |
+ |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1164 |
+ |
\right\rangle |
1165 |
+ |
\] |
1166 |
+ |
Here $u_{tot}$ is the net dipole of the entire system and is given |
1167 |
+ |
by |
1168 |
+ |
\[ |
1169 |
+ |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1170 |
+ |
\] |
1171 |
+ |
In principle, many time correlation functions can be related with |
1172 |
+ |
Fourier transforms of the infrared, Raman, and inelastic neutron |
1173 |
+ |
scattering spectra of molecular liquids. In practice, one can |
1174 |
+ |
extract the IR spectrum from the intensity of dipole fluctuation at |
1175 |
+ |
each frequency using the following relationship: |
1176 |
+ |
\[ |
1177 |
+ |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1178 |
+ |
i2\pi vt} dt} |
1179 |
+ |
\] |
1180 |
+ |
|
1181 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1182 |
|
|
1183 |
|
Rigid bodies are frequently involved in the modeling of different |
1211 |
|
The break through in geometric literature suggests that, in order to |
1212 |
|
develop a long-term integration scheme, one should preserve the |
1213 |
|
symplectic structure of the flow. Introducing conjugate momentum to |
1214 |
< |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
1214 |
> |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
1215 |
|
symplectic integrator, RSHAKE, was proposed to evolve the |
1216 |
|
Hamiltonian system in a constraint manifold by iteratively |
1217 |
< |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
1217 |
> |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
1218 |
|
method using quaternion representation was developed by Omelyan. |
1219 |
|
However, both of these methods are iterative and inefficient. In |
1220 |
|
this section, we will present a symplectic Lie-Poisson integrator |
1221 |
< |
for rigid body developed by Dullweber and his coworkers\cite{}. |
1221 |
> |
for rigid body developed by Dullweber and his |
1222 |
> |
coworkers\cite{Dullweber1997} in depth. |
1223 |
|
|
954 |
– |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
955 |
– |
|
1224 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
1225 |
< |
|
1225 |
> |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
1226 |
> |
function |
1227 |
|
\begin{equation} |
1228 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
1229 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
1238 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1239 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
1240 |
|
\begin{equation} |
1241 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
1241 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1242 |
|
\end{equation} |
1243 |
|
which is used to ensure rotation matrix's orthogonality. |
1244 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1245 |
|
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
1246 |
|
\begin{equation} |
1247 |
< |
Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0 . \\ |
1247 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1248 |
|
\label{introEquation:RBFirstOrderConstraint} |
1249 |
|
\end{equation} |
1250 |
|
|
1256 |
|
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1257 |
|
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1258 |
|
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1259 |
< |
\frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
1259 |
> |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
1260 |
|
\end{array} |
1261 |
|
\] |
1262 |
|
|
1263 |
< |
|
1263 |
> |
In general, there are two ways to satisfy the holonomic constraints. |
1264 |
> |
We can use constraint force provided by lagrange multiplier on the |
1265 |
> |
normal manifold to keep the motion on constraint space. Or we can |
1266 |
> |
simply evolve the system in constraint manifold. The two method are |
1267 |
> |
proved to be equivalent. The holonomic constraint and equations of |
1268 |
> |
motions define a constraint manifold for rigid body |
1269 |
|
\[ |
1270 |
< |
M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} |
1271 |
< |
\right\} . |
1270 |
> |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1271 |
> |
\right\}. |
1272 |
|
\] |
1273 |
|
|
1274 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
1275 |
< |
|
1276 |
< |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations} |
1277 |
< |
|
1004 |
< |
|
1005 |
< |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1006 |
< |
|
1007 |
< |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
1008 |
< |
|
1009 |
< |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
1010 |
< |
|
1011 |
< |
\begin{equation} |
1012 |
< |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1013 |
< |
\label{introEquation:bathGLE} |
1014 |
< |
\end{equation} |
1015 |
< |
where $H_B$ is harmonic bath Hamiltonian, |
1274 |
> |
Unfortunately, this constraint manifold is not the cotangent bundle |
1275 |
> |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
1276 |
> |
transformation, the cotangent space and the phase space are |
1277 |
> |
diffeomorphic. Introducing |
1278 |
|
\[ |
1279 |
< |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1018 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
1279 |
> |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1280 |
|
\] |
1281 |
< |
and $\Delta U$ is bilinear system-bath coupling, |
1281 |
> |
the mechanical system subject to a holonomic constraint manifold $M$ |
1282 |
> |
can be re-formulated as a Hamiltonian system on the cotangent space |
1283 |
|
\[ |
1284 |
< |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1284 |
> |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
1285 |
> |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
1286 |
|
\] |
1287 |
< |
Completing the square, |
1287 |
> |
|
1288 |
> |
For a body fixed vector $X_i$ with respect to the center of mass of |
1289 |
> |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
1290 |
> |
given as |
1291 |
> |
\begin{equation} |
1292 |
> |
X_i^{lab} = Q X_i + q. |
1293 |
> |
\end{equation} |
1294 |
> |
Therefore, potential energy $V(q,Q)$ is defined by |
1295 |
|
\[ |
1296 |
< |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
1027 |
< |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1028 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1029 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
1030 |
< |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1296 |
> |
V(q,Q) = V(Q X_0 + q). |
1297 |
|
\] |
1298 |
< |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
1298 |
> |
Hence, the force and torque are given by |
1299 |
|
\[ |
1300 |
< |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1035 |
< |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1036 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1037 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1300 |
> |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, |
1301 |
|
\] |
1302 |
< |
where |
1302 |
> |
and |
1303 |
|
\[ |
1304 |
< |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1042 |
< |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1304 |
> |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
1305 |
|
\] |
1306 |
< |
Since the first two terms of the new Hamiltonian depend only on the |
1045 |
< |
system coordinates, we can get the equations of motion for |
1046 |
< |
Generalized Langevin Dynamics by Hamilton's equations |
1047 |
< |
\ref{introEquation:motionHamiltonianCoordinate, |
1048 |
< |
introEquation:motionHamiltonianMomentum}, |
1049 |
< |
\begin{align} |
1050 |
< |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
1051 |
< |
&= m\ddot x |
1052 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
1053 |
< |
\label{introEquation:Lp5} |
1054 |
< |
\end{align} |
1055 |
< |
, and |
1056 |
< |
\begin{align} |
1057 |
< |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
1058 |
< |
&= m\ddot x_\alpha |
1059 |
< |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
1060 |
< |
\end{align} |
1306 |
> |
respectively. |
1307 |
|
|
1308 |
< |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
1309 |
< |
|
1308 |
> |
As a common choice to describe the rotation dynamics of the rigid |
1309 |
> |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
1310 |
> |
rewrite the equations of motion, |
1311 |
> |
\begin{equation} |
1312 |
> |
\begin{array}{l} |
1313 |
> |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1314 |
> |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
1315 |
> |
\end{array} |
1316 |
> |
\label{introEqaution:RBMotionPI} |
1317 |
> |
\end{equation} |
1318 |
> |
, as well as holonomic constraints, |
1319 |
|
\[ |
1320 |
< |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
1320 |
> |
\begin{array}{l} |
1321 |
> |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
1322 |
> |
Q^T Q = 1 \\ |
1323 |
> |
\end{array} |
1324 |
|
\] |
1325 |
|
|
1326 |
+ |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1327 |
+ |
so(3)^ \star$, the hat-map isomorphism, |
1328 |
+ |
\begin{equation} |
1329 |
+ |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1330 |
+ |
{\begin{array}{*{20}c} |
1331 |
+ |
0 & { - v_3 } & {v_2 } \\ |
1332 |
+ |
{v_3 } & 0 & { - v_1 } \\ |
1333 |
+ |
{ - v_2 } & {v_1 } & 0 \\ |
1334 |
+ |
\end{array}} \right), |
1335 |
+ |
\label{introEquation:hatmapIsomorphism} |
1336 |
+ |
\end{equation} |
1337 |
+ |
will let us associate the matrix products with traditional vector |
1338 |
+ |
operations |
1339 |
|
\[ |
1340 |
< |
L(x + y) = L(x) + L(y) |
1340 |
> |
\hat vu = v \times u |
1341 |
|
\] |
1342 |
|
|
1343 |
+ |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1344 |
+ |
matrix, |
1345 |
+ |
\begin{equation} |
1346 |
+ |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
1347 |
+ |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
1348 |
+ |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1349 |
+ |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
1350 |
+ |
\end{equation} |
1351 |
+ |
Since $\Lambda$ is symmetric, the last term of Equation |
1352 |
+ |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1353 |
+ |
multiplier $\Lambda$ is absent from the equations of motion. This |
1354 |
+ |
unique property eliminate the requirement of iterations which can |
1355 |
+ |
not be avoided in other methods\cite{}. |
1356 |
+ |
|
1357 |
+ |
Applying hat-map isomorphism, we obtain the equation of motion for |
1358 |
+ |
angular momentum on body frame |
1359 |
+ |
\begin{equation} |
1360 |
+ |
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
1361 |
+ |
F_i (r,Q)} \right) \times X_i }. |
1362 |
+ |
\label{introEquation:bodyAngularMotion} |
1363 |
+ |
\end{equation} |
1364 |
+ |
In the same manner, the equation of motion for rotation matrix is |
1365 |
+ |
given by |
1366 |
|
\[ |
1367 |
< |
L(ax) = aL(x) |
1367 |
> |
\dot Q = Qskew(I^{ - 1} \pi ) |
1368 |
|
\] |
1369 |
|
|
1370 |
+ |
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
1371 |
+ |
Lie-Poisson Integrator for Free Rigid Body} |
1372 |
+ |
|
1373 |
+ |
If there is not external forces exerted on the rigid body, the only |
1374 |
+ |
contribution to the rotational is from the kinetic potential (the |
1375 |
+ |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
1376 |
+ |
rigid body is an example of Lie-Poisson system with Hamiltonian |
1377 |
+ |
function |
1378 |
+ |
\begin{equation} |
1379 |
+ |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1380 |
+ |
\label{introEquation:rotationalKineticRB} |
1381 |
+ |
\end{equation} |
1382 |
+ |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
1383 |
+ |
Lie-Poisson structure matrix, |
1384 |
+ |
\begin{equation} |
1385 |
+ |
J(\pi ) = \left( {\begin{array}{*{20}c} |
1386 |
+ |
0 & {\pi _3 } & { - \pi _2 } \\ |
1387 |
+ |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
1388 |
+ |
{\pi _2 } & { - \pi _1 } & 0 \\ |
1389 |
+ |
\end{array}} \right) |
1390 |
+ |
\end{equation} |
1391 |
+ |
Thus, the dynamics of free rigid body is governed by |
1392 |
+ |
\begin{equation} |
1393 |
+ |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
1394 |
+ |
\end{equation} |
1395 |
+ |
|
1396 |
+ |
One may notice that each $T_i^r$ in Equation |
1397 |
+ |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1398 |
+ |
instance, the equations of motion due to $T_1^r$ are given by |
1399 |
+ |
\begin{equation} |
1400 |
+ |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
1401 |
+ |
\label{introEqaution:RBMotionSingleTerm} |
1402 |
+ |
\end{equation} |
1403 |
+ |
where |
1404 |
+ |
\[ R_1 = \left( {\begin{array}{*{20}c} |
1405 |
+ |
0 & 0 & 0 \\ |
1406 |
+ |
0 & 0 & {\pi _1 } \\ |
1407 |
+ |
0 & { - \pi _1 } & 0 \\ |
1408 |
+ |
\end{array}} \right). |
1409 |
+ |
\] |
1410 |
+ |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1411 |
|
\[ |
1412 |
< |
L(\dot x) = pL(x) - px(0) |
1412 |
> |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
1413 |
> |
Q(0)e^{\Delta tR_1 } |
1414 |
|
\] |
1415 |
< |
|
1415 |
> |
with |
1416 |
|
\[ |
1417 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
1417 |
> |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
1418 |
> |
0 & 0 & 0 \\ |
1419 |
> |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
1420 |
> |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
1421 |
> |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1422 |
|
\] |
1423 |
< |
|
1423 |
> |
To reduce the cost of computing expensive functions in $e^{\Delta |
1424 |
> |
tR_1 }$, we can use Cayley transformation, |
1425 |
|
\[ |
1426 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
1426 |
> |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1427 |
> |
) |
1428 |
|
\] |
1429 |
+ |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1430 |
+ |
manner. |
1431 |
|
|
1432 |
< |
Some relatively important transformation, |
1432 |
> |
In order to construct a second-order symplectic method, we split the |
1433 |
> |
angular kinetic Hamiltonian function can into five terms |
1434 |
|
\[ |
1435 |
< |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
1435 |
> |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1436 |
> |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1437 |
> |
(\pi _1 ) |
1438 |
> |
\]. |
1439 |
> |
Concatenating flows corresponding to these five terms, we can obtain |
1440 |
> |
an symplectic integrator, |
1441 |
> |
\[ |
1442 |
> |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
1443 |
> |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1444 |
> |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1445 |
> |
_1 }. |
1446 |
|
\] |
1447 |
|
|
1448 |
+ |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
1449 |
+ |
$F(\pi )$ and $G(\pi )$ is defined by |
1450 |
|
\[ |
1451 |
< |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
1451 |
> |
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
1452 |
> |
) |
1453 |
|
\] |
1454 |
< |
|
1454 |
> |
If the Poisson bracket of a function $F$ with an arbitrary smooth |
1455 |
> |
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
1456 |
> |
conserved quantity in Poisson system. We can easily verify that the |
1457 |
> |
norm of the angular momentum, $\parallel \pi |
1458 |
> |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
1459 |
> |
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
1460 |
> |
then by the chain rule |
1461 |
|
\[ |
1462 |
< |
L(1) = \frac{1}{p} |
1462 |
> |
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
1463 |
> |
}}{2})\pi |
1464 |
|
\] |
1465 |
+ |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
1466 |
+ |
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
1467 |
+ |
Lie-Poisson integrator is found to be extremely efficient and stable |
1468 |
+ |
which can be explained by the fact the small angle approximation is |
1469 |
+ |
used and the norm of the angular momentum is conserved. |
1470 |
|
|
1471 |
< |
First, the bath coordinates, |
1471 |
> |
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
1472 |
> |
Splitting for Rigid Body} |
1473 |
> |
|
1474 |
> |
The Hamiltonian of rigid body can be separated in terms of kinetic |
1475 |
> |
energy and potential energy, |
1476 |
|
\[ |
1477 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
1104 |
< |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
1105 |
< |
}}L(x) |
1477 |
> |
H = T(p,\pi ) + V(q,Q) |
1478 |
|
\] |
1479 |
< |
\[ |
1480 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
1481 |
< |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
1479 |
> |
The equations of motion corresponding to potential energy and |
1480 |
> |
kinetic energy are listed in the below table, |
1481 |
> |
\begin{center} |
1482 |
> |
\begin{tabular}{|l|l|} |
1483 |
> |
\hline |
1484 |
> |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
1485 |
> |
Potential & Kinetic \\ |
1486 |
> |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
1487 |
> |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
1488 |
> |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
1489 |
> |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
1490 |
> |
\hline |
1491 |
> |
\end{tabular} |
1492 |
> |
\end{center} |
1493 |
> |
A second-order symplectic method is now obtained by the composition |
1494 |
> |
of the flow maps, |
1495 |
> |
\[ |
1496 |
> |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1497 |
> |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1498 |
|
\] |
1499 |
< |
Then, the system coordinates, |
1500 |
< |
\begin{align} |
1501 |
< |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1502 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
1503 |
< |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
1504 |
< |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
1505 |
< |
}}\omega _\alpha ^2 L(x)} \right\}} |
1506 |
< |
% |
1507 |
< |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1120 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
1121 |
< |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
1122 |
< |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
1123 |
< |
\end{align} |
1124 |
< |
Then, the inverse transform, |
1499 |
> |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
1500 |
> |
sub-flows which corresponding to force and torque respectively, |
1501 |
> |
\[ |
1502 |
> |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1503 |
> |
_{\Delta t/2,\tau }. |
1504 |
> |
\] |
1505 |
> |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
1506 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
1507 |
> |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
1508 |
|
|
1509 |
+ |
Furthermore, kinetic potential can be separated to translational |
1510 |
+ |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
1511 |
+ |
\begin{equation} |
1512 |
+ |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
1513 |
+ |
\end{equation} |
1514 |
+ |
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
1515 |
+ |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
1516 |
+ |
corresponding flow maps are given by |
1517 |
+ |
\[ |
1518 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
1519 |
+ |
_{\Delta t,T^r }. |
1520 |
+ |
\] |
1521 |
+ |
Finally, we obtain the overall symplectic flow maps for free moving |
1522 |
+ |
rigid body |
1523 |
+ |
\begin{equation} |
1524 |
+ |
\begin{array}{c} |
1525 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
1526 |
+ |
\circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \\ |
1527 |
+ |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1528 |
+ |
\end{array} |
1529 |
+ |
\label{introEquation:overallRBFlowMaps} |
1530 |
+ |
\end{equation} |
1531 |
+ |
|
1532 |
+ |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1533 |
+ |
As an alternative to newtonian dynamics, Langevin dynamics, which |
1534 |
+ |
mimics a simple heat bath with stochastic and dissipative forces, |
1535 |
+ |
has been applied in a variety of studies. This section will review |
1536 |
+ |
the theory of Langevin dynamics simulation. A brief derivation of |
1537 |
+ |
generalized Langevin equation will be given first. Follow that, we |
1538 |
+ |
will discuss the physical meaning of the terms appearing in the |
1539 |
+ |
equation as well as the calculation of friction tensor from |
1540 |
+ |
hydrodynamics theory. |
1541 |
+ |
|
1542 |
+ |
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
1543 |
+ |
|
1544 |
+ |
Harmonic bath model, in which an effective set of harmonic |
1545 |
+ |
oscillators are used to mimic the effect of a linearly responding |
1546 |
+ |
environment, has been widely used in quantum chemistry and |
1547 |
+ |
statistical mechanics. One of the successful applications of |
1548 |
+ |
Harmonic bath model is the derivation of Deriving Generalized |
1549 |
+ |
Langevin Dynamics. Lets consider a system, in which the degree of |
1550 |
+ |
freedom $x$ is assumed to couple to the bath linearly, giving a |
1551 |
+ |
Hamiltonian of the form |
1552 |
+ |
\begin{equation} |
1553 |
+ |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1554 |
+ |
\label{introEquation:bathGLE}. |
1555 |
+ |
\end{equation} |
1556 |
+ |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
1557 |
+ |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
1558 |
+ |
\[ |
1559 |
+ |
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1560 |
+ |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
1561 |
+ |
\right\}} |
1562 |
+ |
\] |
1563 |
+ |
where the index $\alpha$ runs over all the bath degrees of freedom, |
1564 |
+ |
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
1565 |
+ |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
1566 |
+ |
coupling, |
1567 |
+ |
\[ |
1568 |
+ |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1569 |
+ |
\] |
1570 |
+ |
where $g_\alpha$ are the coupling constants between the bath and the |
1571 |
+ |
coordinate $x$. Introducing |
1572 |
+ |
\[ |
1573 |
+ |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1574 |
+ |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1575 |
+ |
\] and combining the last two terms in Equation |
1576 |
+ |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
1577 |
+ |
Hamiltonian as |
1578 |
+ |
\[ |
1579 |
+ |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1580 |
+ |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1581 |
+ |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1582 |
+ |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1583 |
+ |
\] |
1584 |
+ |
Since the first two terms of the new Hamiltonian depend only on the |
1585 |
+ |
system coordinates, we can get the equations of motion for |
1586 |
+ |
Generalized Langevin Dynamics by Hamilton's equations |
1587 |
+ |
\ref{introEquation:motionHamiltonianCoordinate, |
1588 |
+ |
introEquation:motionHamiltonianMomentum}, |
1589 |
+ |
\begin{equation} |
1590 |
+ |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
1591 |
+ |
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
1592 |
+ |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)}, |
1593 |
+ |
\label{introEquation:coorMotionGLE} |
1594 |
+ |
\end{equation} |
1595 |
+ |
and |
1596 |
+ |
\begin{equation} |
1597 |
+ |
m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - |
1598 |
+ |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
1599 |
+ |
\label{introEquation:bathMotionGLE} |
1600 |
+ |
\end{equation} |
1601 |
+ |
|
1602 |
+ |
In order to derive an equation for $x$, the dynamics of the bath |
1603 |
+ |
variables $x_\alpha$ must be solved exactly first. As an integral |
1604 |
+ |
transform which is particularly useful in solving linear ordinary |
1605 |
+ |
differential equations, Laplace transform is the appropriate tool to |
1606 |
+ |
solve this problem. The basic idea is to transform the difficult |
1607 |
+ |
differential equations into simple algebra problems which can be |
1608 |
+ |
solved easily. Then applying inverse Laplace transform, also known |
1609 |
+ |
as the Bromwich integral, we can retrieve the solutions of the |
1610 |
+ |
original problems. |
1611 |
+ |
|
1612 |
+ |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
1613 |
+ |
transform of f(t) is a new function defined as |
1614 |
+ |
\[ |
1615 |
+ |
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1616 |
+ |
\] |
1617 |
+ |
where $p$ is real and $L$ is called the Laplace Transform |
1618 |
+ |
Operator. Below are some important properties of Laplace transform |
1619 |
+ |
\begin{equation} |
1620 |
+ |
\begin{array}{c} |
1621 |
+ |
L(x + y) = L(x) + L(y) \\ |
1622 |
+ |
L(ax) = aL(x) \\ |
1623 |
+ |
L(\dot x) = pL(x) - px(0) \\ |
1624 |
+ |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
1625 |
+ |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
1626 |
+ |
\end{array} |
1627 |
+ |
\end{equation} |
1628 |
+ |
|
1629 |
+ |
Applying Laplace transform to the bath coordinates, we obtain |
1630 |
+ |
\[ |
1631 |
+ |
\begin{array}{c} |
1632 |
+ |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1633 |
+ |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1634 |
+ |
\end{array} |
1635 |
+ |
\] |
1636 |
+ |
By the same way, the system coordinates become |
1637 |
+ |
\[ |
1638 |
+ |
\begin{array}{c} |
1639 |
+ |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1640 |
+ |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1641 |
+ |
\end{array} |
1642 |
+ |
\] |
1643 |
+ |
|
1644 |
+ |
With the help of some relatively important inverse Laplace |
1645 |
+ |
transformations: |
1646 |
+ |
\[ |
1647 |
+ |
\begin{array}{c} |
1648 |
+ |
L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ |
1649 |
+ |
L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ |
1650 |
+ |
L(1) = \frac{1}{p} \\ |
1651 |
+ |
\end{array} |
1652 |
+ |
\] |
1653 |
+ |
, we obtain |
1654 |
|
\begin{align} |
1655 |
|
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
1656 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1670 |
|
(\omega _\alpha t)} \right\}} |
1671 |
|
\end{align} |
1672 |
|
|
1673 |
+ |
Introducing a \emph{dynamic friction kernel} |
1674 |
|
\begin{equation} |
1675 |
+ |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1676 |
+ |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
1677 |
+ |
\label{introEquation:dynamicFrictionKernelDefinition} |
1678 |
+ |
\end{equation} |
1679 |
+ |
and \emph{a random force} |
1680 |
+ |
\begin{equation} |
1681 |
+ |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
1682 |
+ |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
1683 |
+ |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
1684 |
+ |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), |
1685 |
+ |
\label{introEquation:randomForceDefinition} |
1686 |
+ |
\end{equation} |
1687 |
+ |
the equation of motion can be rewritten as |
1688 |
+ |
\begin{equation} |
1689 |
|
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
1690 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
1691 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
1692 |
|
\end{equation} |
1693 |
< |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
1694 |
< |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
1693 |
> |
which is known as the \emph{generalized Langevin equation}. |
1694 |
> |
|
1695 |
> |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel} |
1696 |
> |
|
1697 |
> |
One may notice that $R(t)$ depends only on initial conditions, which |
1698 |
> |
implies it is completely deterministic within the context of a |
1699 |
> |
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
1700 |
> |
uncorrelated to $x$ and $\dot x$, |
1701 |
|
\[ |
1702 |
< |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1703 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
1702 |
> |
\begin{array}{l} |
1703 |
> |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
1704 |
> |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
1705 |
> |
\end{array} |
1706 |
|
\] |
1707 |
< |
For an infinite harmonic bath, we can use the spectral density and |
1708 |
< |
an integral over frequencies. |
1707 |
> |
This property is what we expect from a truly random process. As long |
1708 |
> |
as the model, which is gaussian distribution in general, chosen for |
1709 |
> |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
1710 |
> |
still remains. |
1711 |
|
|
1712 |
+ |
%dynamic friction kernel |
1713 |
+ |
The convolution integral |
1714 |
|
\[ |
1715 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
1161 |
< |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
1162 |
< |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
1163 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
1715 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } |
1716 |
|
\] |
1717 |
< |
The random forces depend only on initial conditions. |
1717 |
> |
depends on the entire history of the evolution of $x$, which implies |
1718 |
> |
that the bath retains memory of previous motions. In other words, |
1719 |
> |
the bath requires a finite time to respond to change in the motion |
1720 |
> |
of the system. For a sluggish bath which responds slowly to changes |
1721 |
> |
in the system coordinate, we may regard $\xi(t)$ as a constant |
1722 |
> |
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes |
1723 |
> |
\[ |
1724 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
1725 |
> |
\] |
1726 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1727 |
> |
\[ |
1728 |
> |
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
1729 |
> |
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
1730 |
> |
\] |
1731 |
> |
which can be used to describe dynamic caging effect. The other |
1732 |
> |
extreme is the bath that responds infinitely quickly to motions in |
1733 |
> |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
1734 |
> |
time: |
1735 |
> |
\[ |
1736 |
> |
\xi (t) = 2\xi _0 \delta (t) |
1737 |
> |
\] |
1738 |
> |
Hence, the convolution integral becomes |
1739 |
> |
\[ |
1740 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
1741 |
> |
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
1742 |
> |
\] |
1743 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1744 |
> |
\begin{equation} |
1745 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
1746 |
> |
x(t) + R(t) \label{introEquation:LangevinEquation} |
1747 |
> |
\end{equation} |
1748 |
> |
which is known as the Langevin equation. The static friction |
1749 |
> |
coefficient $\xi _0$ can either be calculated from spectral density |
1750 |
> |
or be determined by Stokes' law for regular shaped particles.A |
1751 |
> |
briefly review on calculating friction tensor for arbitrary shaped |
1752 |
> |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
1753 |
|
|
1754 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
1755 |
< |
So we can define a new set of coordinates, |
1755 |
> |
|
1756 |
> |
Defining a new set of coordinates, |
1757 |
|
\[ |
1758 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1759 |
|
^2 }}x(0) |
1760 |
< |
\] |
1761 |
< |
This makes |
1760 |
> |
\], |
1761 |
> |
we can rewrite $R(T)$ as |
1762 |
|
\[ |
1763 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
1763 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1764 |
|
\] |
1765 |
|
And since the $q$ coordinates are harmonic oscillators, |
1766 |
|
\[ |
1767 |
< |
\begin{array}{l} |
1767 |
> |
\begin{array}{c} |
1768 |
> |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1769 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1770 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1771 |
+ |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1772 |
+ |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1773 |
+ |
= kT\xi (t) \\ |
1774 |
|
\end{array} |
1775 |
+ |
\] |
1776 |
+ |
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1777 |
+ |
\begin{equation} |
1778 |
+ |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1779 |
+ |
\label{introEquation:secondFluctuationDissipation}. |
1780 |
+ |
\end{equation} |
1781 |
+ |
In effect, it acts as a constraint on the possible ways in which one |
1782 |
+ |
can model the random force and friction kernel. |
1783 |
+ |
|
1784 |
+ |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1785 |
+ |
Theoretically, the friction kernel can be determined using velocity |
1786 |
+ |
autocorrelation function. However, this approach become impractical |
1787 |
+ |
when the system become more and more complicate. Instead, various |
1788 |
+ |
approaches based on hydrodynamics have been developed to calculate |
1789 |
+ |
the friction coefficients. The friction effect is isotropic in |
1790 |
+ |
Equation, \zeta can be taken as a scalar. In general, friction |
1791 |
+ |
tensor \Xi is a $6\times 6$ matrix given by |
1792 |
+ |
\[ |
1793 |
+ |
\Xi = \left( {\begin{array}{*{20}c} |
1794 |
+ |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
1795 |
+ |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
1796 |
+ |
\end{array}} \right). |
1797 |
|
\] |
1798 |
+ |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
1799 |
+ |
tensor and rotational resistance (friction) tensor respectively, |
1800 |
+ |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
1801 |
+ |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
1802 |
+ |
particle moves in a fluid, it may experience friction force or |
1803 |
+ |
torque along the opposite direction of the velocity or angular |
1804 |
+ |
velocity, |
1805 |
+ |
\[ |
1806 |
+ |
\left( \begin{array}{l} |
1807 |
+ |
F_R \\ |
1808 |
+ |
\tau _R \\ |
1809 |
+ |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
1810 |
+ |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
1811 |
+ |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
1812 |
+ |
\end{array}} \right)\left( \begin{array}{l} |
1813 |
+ |
v \\ |
1814 |
+ |
w \\ |
1815 |
+ |
\end{array} \right) |
1816 |
+ |
\] |
1817 |
+ |
where $F_r$ is the friction force and $\tau _R$ is the friction |
1818 |
+ |
toque. |
1819 |
|
|
1820 |
< |
\begin{align} |
1186 |
< |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
1187 |
< |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
1188 |
< |
(t)q_\beta (0)} \right\rangle } } |
1189 |
< |
% |
1190 |
< |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
1191 |
< |
\right\rangle \cos (\omega _\alpha t)} |
1192 |
< |
% |
1193 |
< |
&= kT\xi (t) |
1194 |
< |
\end{align} |
1820 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
1821 |
|
|
1822 |
< |
\begin{equation} |
1823 |
< |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1824 |
< |
\label{introEquation:secondFluctuationDissipation} |
1825 |
< |
\end{equation} |
1822 |
> |
For a spherical particle, the translational and rotational friction |
1823 |
> |
constant can be calculated from Stoke's law, |
1824 |
> |
\[ |
1825 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
1826 |
> |
{6\pi \eta R} & 0 & 0 \\ |
1827 |
> |
0 & {6\pi \eta R} & 0 \\ |
1828 |
> |
0 & 0 & {6\pi \eta R} \\ |
1829 |
> |
\end{array}} \right) |
1830 |
> |
\] |
1831 |
> |
and |
1832 |
> |
\[ |
1833 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
1834 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
1835 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
1836 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
1837 |
> |
\end{array}} \right) |
1838 |
> |
\] |
1839 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
1840 |
> |
hydrodynamics radius. |
1841 |
|
|
1842 |
< |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
1842 |
> |
Other non-spherical shape, such as cylinder and ellipsoid |
1843 |
> |
\textit{etc}, are widely used as reference for developing new |
1844 |
> |
hydrodynamics theory, because their properties can be calculated |
1845 |
> |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
1846 |
> |
also called a triaxial ellipsoid, which is given in Cartesian |
1847 |
> |
coordinates by |
1848 |
> |
\[ |
1849 |
> |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
1850 |
> |
}} = 1 |
1851 |
> |
\] |
1852 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
1853 |
> |
due to the complexity of the elliptic integral, only the ellipsoid |
1854 |
> |
with the restriction of two axes having to be equal, \textit{i.e.} |
1855 |
> |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
1856 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
1857 |
> |
\[ |
1858 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
1859 |
> |
} }}{b}, |
1860 |
> |
\] |
1861 |
> |
and oblate, |
1862 |
> |
\[ |
1863 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
1864 |
> |
}}{a} |
1865 |
> |
\], |
1866 |
> |
one can write down the translational and rotational resistance |
1867 |
> |
tensors |
1868 |
> |
\[ |
1869 |
> |
\begin{array}{l} |
1870 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
1871 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
1872 |
> |
\end{array}, |
1873 |
> |
\] |
1874 |
> |
and |
1875 |
> |
\[ |
1876 |
> |
\begin{array}{l} |
1877 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
1878 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
1879 |
> |
\end{array}. |
1880 |
> |
\] |
1881 |
|
|
1882 |
< |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1204 |
< |
\subsection{\label{introSection:analyticalApproach}Analytical |
1205 |
< |
Approach} |
1882 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
1883 |
|
|
1884 |
< |
\subsection{\label{introSection:approximationApproach}Approximation |
1885 |
< |
Approach} |
1884 |
> |
Unlike spherical and other regular shaped molecules, there is not |
1885 |
> |
analytical solution for friction tensor of any arbitrary shaped |
1886 |
> |
rigid molecules. The ellipsoid of revolution model and general |
1887 |
> |
triaxial ellipsoid model have been used to approximate the |
1888 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
1889 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
1890 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
1891 |
> |
unique\cite{Wegener79} as well as the intrinsic coupling between |
1892 |
> |
translational and rotational motion of rigid body\cite{}, general |
1893 |
> |
ellipsoid is not always suitable for modeling arbitrarily shaped |
1894 |
> |
rigid molecule. A number of studies have been devoted to determine |
1895 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
1896 |
> |
advanced method\cite{} where the molecule of interest was modeled by |
1897 |
> |
combinations of spheres(beads)\cite{} and the hydrodynamics |
1898 |
> |
properties of the molecule can be calculated using the hydrodynamic |
1899 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
1900 |
> |
immersed in a continuous medium. Due to hydrodynamics interaction, |
1901 |
> |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
1902 |
> |
unperturbed velocity $v_i$, |
1903 |
> |
\[ |
1904 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
1905 |
> |
\] |
1906 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
1907 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
1908 |
> |
proportional to its ``net'' velocity |
1909 |
> |
\begin{equation} |
1910 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
1911 |
> |
\label{introEquation:tensorExpression} |
1912 |
> |
\end{equation} |
1913 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
1914 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
1915 |
> |
\ref{introEquation:tensorExpression} |
1916 |
> |
\begin{equation} |
1917 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
1918 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
1919 |
> |
\label{introEquation:oseenTensor} |
1920 |
> |
\end{equation} |
1921 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
1922 |
> |
A second order expression for element of different size was |
1923 |
> |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
1924 |
> |
la Torre and Bloomfield, |
1925 |
> |
\begin{equation} |
1926 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
1927 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
1928 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
1929 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
1930 |
> |
\label{introEquation:RPTensorNonOverlapped} |
1931 |
> |
\end{equation} |
1932 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
1933 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
1934 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
1935 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
1936 |
> |
\begin{equation} |
1937 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
1938 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
1939 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
1940 |
> |
\label{introEquation:RPTensorOverlapped} |
1941 |
> |
\end{equation} |
1942 |
|
|
1943 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
1944 |
< |
Body} |
1943 |
> |
To calculate the resistance tensor at an arbitrary origin $O$, we |
1944 |
> |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
1945 |
> |
$B_{ij}$ blocks |
1946 |
> |
\begin{equation} |
1947 |
> |
B = \left( {\begin{array}{*{20}c} |
1948 |
> |
{B_{11} } & \ldots & {B_{1N} } \\ |
1949 |
> |
\vdots & \ddots & \vdots \\ |
1950 |
> |
{B_{N1} } & \cdots & {B_{NN} } \\ |
1951 |
> |
\end{array}} \right), |
1952 |
> |
\end{equation} |
1953 |
> |
where $B_{ij}$ is given by |
1954 |
> |
\[ |
1955 |
> |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
1956 |
> |
)T_{ij} |
1957 |
> |
\] |
1958 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
1959 |
> |
$B$, we obtain |
1960 |
|
|
1961 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
1961 |
> |
\[ |
1962 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
1963 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
1964 |
> |
\vdots & \ddots & \vdots \\ |
1965 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
1966 |
> |
\end{array}} \right) |
1967 |
> |
\] |
1968 |
> |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
1969 |
> |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
1970 |
> |
\[ |
1971 |
> |
U_i = \left( {\begin{array}{*{20}c} |
1972 |
> |
0 & { - z_i } & {y_i } \\ |
1973 |
> |
{z_i } & 0 & { - x_i } \\ |
1974 |
> |
{ - y_i } & {x_i } & 0 \\ |
1975 |
> |
\end{array}} \right) |
1976 |
> |
\] |
1977 |
> |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
1978 |
> |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
1979 |
> |
arbitrary origin $O$ can be written as |
1980 |
> |
\begin{equation} |
1981 |
> |
\begin{array}{l} |
1982 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
1983 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
1984 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
1985 |
> |
\end{array} |
1986 |
> |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
1987 |
> |
\end{equation} |
1988 |
> |
|
1989 |
> |
The resistance tensor depends on the origin to which they refer. The |
1990 |
> |
proper location for applying friction force is the center of |
1991 |
> |
resistance (reaction), at which the trace of rotational resistance |
1992 |
> |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
1993 |
> |
resistance is defined as an unique point of the rigid body at which |
1994 |
> |
the translation-rotation coupling tensor are symmetric, |
1995 |
> |
\begin{equation} |
1996 |
> |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
1997 |
> |
\label{introEquation:definitionCR} |
1998 |
> |
\end{equation} |
1999 |
> |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
2000 |
> |
we can easily find out that the translational resistance tensor is |
2001 |
> |
origin independent, while the rotational resistance tensor and |
2002 |
> |
translation-rotation coupling resistance tensor depend on the |
2003 |
> |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
2004 |
> |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
2005 |
> |
obtain the resistance tensor at $P$ by |
2006 |
> |
\begin{equation} |
2007 |
> |
\begin{array}{l} |
2008 |
> |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
2009 |
> |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
2010 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
2011 |
> |
\end{array} |
2012 |
> |
\label{introEquation:resistanceTensorTransformation} |
2013 |
> |
\end{equation} |
2014 |
> |
where |
2015 |
> |
\[ |
2016 |
> |
U_{OP} = \left( {\begin{array}{*{20}c} |
2017 |
> |
0 & { - z_{OP} } & {y_{OP} } \\ |
2018 |
> |
{z_i } & 0 & { - x_{OP} } \\ |
2019 |
> |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
2020 |
> |
\end{array}} \right) |
2021 |
> |
\] |
2022 |
> |
Using Equations \ref{introEquation:definitionCR} and |
2023 |
> |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
2024 |
> |
the position of center of resistance, |
2025 |
> |
\[ |
2026 |
> |
\left( \begin{array}{l} |
2027 |
> |
x_{OR} \\ |
2028 |
> |
y_{OR} \\ |
2029 |
> |
z_{OR} \\ |
2030 |
> |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
2031 |
> |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
2032 |
> |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
2033 |
> |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
2034 |
> |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
2035 |
> |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
2036 |
> |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
2037 |
> |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
2038 |
> |
\end{array} \right). |
2039 |
> |
\] |
2040 |
> |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
2041 |
> |
joining center of resistance $R$ and origin $O$. |