| 315 |
|
isolated and conserve energy, Microcanonical ensemble(NVE) has a |
| 316 |
|
partition function like, |
| 317 |
|
\begin{equation} |
| 318 |
< |
\Omega (N,V,E) = e^{\beta TS} |
| 319 |
< |
\label{introEqaution:NVEPartition}. |
| 318 |
> |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 319 |
|
\end{equation} |
| 320 |
|
A canonical ensemble(NVT)is an ensemble of systems, each of which |
| 321 |
|
can share its energy with a large heat reservoir. The distribution |
| 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 $x$. |
| 574 |
– |
The free rigid body is an example of Poisson system (actually a |
| 575 |
– |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
| 576 |
– |
energy. |
| 577 |
– |
\begin{equation} |
| 578 |
– |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 579 |
– |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 580 |
– |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 581 |
– |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 582 |
– |
\end{array}} \right) |
| 583 |
– |
\end{equation} |
| 573 |
|
|
| 585 |
– |
\begin{equation} |
| 586 |
– |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
| 587 |
– |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
| 588 |
– |
\end{equation} |
| 589 |
– |
|
| 574 |
|
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 575 |
|
|
| 576 |
|
Let $x(t)$ be the exact solution of the ODE system, |
| 619 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 620 |
|
a \emph{symplectic} flow if it satisfies, |
| 621 |
|
\begin{equation} |
| 622 |
< |
'\varphi^T J '\varphi = J. |
| 622 |
> |
{\varphi '}^T J \varphi ' = J. |
| 623 |
|
\end{equation} |
| 624 |
|
According to Liouville's theorem, the symplectic volume is invariant |
| 625 |
|
under a Hamiltonian flow, which is the basis for classical |
| 627 |
|
field on a symplectic manifold can be shown to be a |
| 628 |
|
symplectomorphism. As to the Poisson system, |
| 629 |
|
\begin{equation} |
| 630 |
< |
'\varphi ^T J '\varphi = J \circ \varphi |
| 630 |
> |
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 631 |
|
\end{equation} |
| 632 |
|
is the property must be preserved by the integrator. |
| 633 |
|
|
| 645 |
|
In other words, the flow of this vector field is reversible if and |
| 646 |
|
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 647 |
|
|
| 648 |
< |
When designing any numerical methods, one should always try to |
| 648 |
> |
A \emph{first integral}, or conserved quantity of a general |
| 649 |
> |
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 650 |
> |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 651 |
> |
\[ |
| 652 |
> |
\frac{{dG(x(t))}}{{dt}} = 0. |
| 653 |
> |
\] |
| 654 |
> |
Using chain rule, one may obtain, |
| 655 |
> |
\[ |
| 656 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
| 657 |
> |
\] |
| 658 |
> |
which is the condition for conserving \emph{first integral}. For a |
| 659 |
> |
canonical Hamiltonian system, the time evolution of an arbitrary |
| 660 |
> |
smooth function $G$ is given by, |
| 661 |
> |
\begin{equation} |
| 662 |
> |
\begin{array}{c} |
| 663 |
> |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 664 |
> |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 665 |
> |
\end{array} |
| 666 |
> |
\label{introEquation:firstIntegral1} |
| 667 |
> |
\end{equation} |
| 668 |
> |
Using poisson bracket notion, Equation |
| 669 |
> |
\ref{introEquation:firstIntegral1} can be rewritten as |
| 670 |
> |
\[ |
| 671 |
> |
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 672 |
> |
\] |
| 673 |
> |
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 674 |
> |
integral} of a Hamiltonian system is |
| 675 |
> |
\[ |
| 676 |
> |
\left\{ {G,H} \right\} = 0. |
| 677 |
> |
\] |
| 678 |
> |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 679 |
> |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 680 |
> |
0$. |
| 681 |
> |
|
| 682 |
> |
|
| 683 |
> |
When designing any numerical methods, one should always try to |
| 684 |
|
preserve the structural properties of the original ODE and its flow. |
| 685 |
|
|
| 686 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 704 |
|
ordinary implicit Runge-Kutta methods are not suitable for |
| 705 |
|
Hamiltonian system. Recently, various high-order explicit |
| 706 |
|
Runge--Kutta methods have been developed to overcome this |
| 707 |
< |
instability \cite{}. However, due to computational penalty involved |
| 708 |
< |
in implementing the Runge-Kutta methods, they do not attract too |
| 709 |
< |
much attention from Molecular Dynamics community. Instead, splitting |
| 710 |
< |
have been widely accepted since they exploit natural decompositions |
| 711 |
< |
of the system\cite{Tuckerman92}. |
| 707 |
> |
instability. However, due to computational penalty involved in |
| 708 |
> |
implementing the Runge-Kutta methods, they do not attract too much |
| 709 |
> |
attention from Molecular Dynamics community. Instead, splitting have |
| 710 |
> |
been widely accepted since they exploit natural decompositions of |
| 711 |
> |
the system\cite{Tuckerman92}. |
| 712 |
|
|
| 713 |
|
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
| 714 |
|
|
| 755 |
|
splitting gives a second-order decomposition, |
| 756 |
|
\begin{equation} |
| 757 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 758 |
< |
_{1,h/2} , |
| 740 |
< |
\label{introEqaution:secondOrderSplitting} |
| 758 |
> |
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 759 |
|
\end{equation} |
| 760 |
|
which has a local error proportional to $h^3$. Sprang splitting's |
| 761 |
|
popularity in molecular simulation community attribute to its |
| 762 |
|
symmetric property, |
| 763 |
|
\begin{equation} |
| 764 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 765 |
< |
\lable{introEquation:timeReversible} |
| 765 |
> |
\label{introEquation:timeReversible} |
| 766 |
|
\end{equation} |
| 767 |
|
|
| 768 |
|
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
| 820 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
| 821 |
|
\label{introEquation:positionVerlet1} \\% |
| 822 |
|
% |
| 823 |
< |
q(\Delta t) = q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 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} |
| 846 |
|
\] |
| 847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
|
can obtain |
| 849 |
< |
\begin{eqnarray} |
| 849 |
> |
\begin{eqnarray*} |
| 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 |
< |
h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 + \ldots ) |
| 853 |
< |
\end{eqnarray} |
| 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*} |
| 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. |
| 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 |
> |
\[ |
| 895 |
> |
T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
| 896 |
> |
\] |
| 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{introSec:mdInit}Initialization} |
| 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, including the force evaluation |
| 917 |
> |
and the numerical integration schemes of the equations of motion . |
| 918 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 919 |
> |
trajectory analysis. |
| 920 |
|
|
| 921 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 921 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 922 |
|
|
| 923 |
< |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 923 |
> |
\subsubsection{Preliminary preparation} |
| 924 |
|
|
| 925 |
< |
A rigid body is a body in which the distance between any two given |
| 926 |
< |
points of a rigid body remains constant regardless of external |
| 927 |
< |
forces exerted on it. A rigid body therefore conserves its shape |
| 928 |
< |
during its motion. |
| 925 |
> |
When selecting the starting structure of a molecule for molecular |
| 926 |
> |
simulation, one may retrieve its Cartesian coordinates from public |
| 927 |
> |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
| 928 |
> |
thousands of crystal structures of molecules are discovered every |
| 929 |
> |
year, many more remain unknown due to the difficulties of |
| 930 |
> |
purification and crystallization. Even for the molecule with known |
| 931 |
> |
structure, some important information is missing. For example, the |
| 932 |
> |
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 933 |
> |
be added. Moreover, in order to include electrostatic interaction, |
| 934 |
> |
one may need to specify the partial charges for individual atoms. |
| 935 |
> |
Under some circumstances, we may even need to prepare the system in |
| 936 |
> |
a special setup. For instance, when studying transport phenomenon in |
| 937 |
> |
membrane system, we may prepare the lipids in bilayer structure |
| 938 |
> |
instead of placing lipids randomly in solvent, since we are not |
| 939 |
> |
interested in self-aggregation and it takes a long time to happen. |
| 940 |
|
|
| 941 |
< |
Applications of dynamics of rigid bodies. |
| 941 |
> |
\subsubsection{Minimization} |
| 942 |
|
|
| 943 |
< |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 944 |
< |
|
| 945 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 946 |
< |
|
| 947 |
< |
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
| 948 |
< |
|
| 949 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 950 |
< |
|
| 951 |
< |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 952 |
< |
|
| 953 |
< |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 954 |
< |
|
| 955 |
< |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 943 |
> |
It is quite possible that some of molecules in the system from |
| 944 |
> |
preliminary preparation may be overlapped with each other. This |
| 945 |
> |
close proximity leads to high potential energy which consequently |
| 946 |
> |
jeopardizes any molecular dynamics simulations. To remove these |
| 947 |
> |
steric overlaps, one typically performs energy minimization to find |
| 948 |
> |
a more reasonable conformation. Several energy minimization methods |
| 949 |
> |
have been developed to exploit the energy surface and to locate the |
| 950 |
> |
local minimum. While converging slowly near the minimum, steepest |
| 951 |
> |
descent method is extremely robust when systems are far from |
| 952 |
> |
harmonic. Thus, it is often used to refine structure from |
| 953 |
> |
crystallographic data. Relied on the gradient or hessian, advanced |
| 954 |
> |
methods like conjugate gradient and Newton-Raphson converge rapidly |
| 955 |
> |
to a local minimum, while become unstable if the energy surface is |
| 956 |
> |
far from quadratic. Another factor must be taken into account, when |
| 957 |
> |
choosing energy minimization method, is the size of the system. |
| 958 |
> |
Steepest descent and conjugate gradient can deal with models of any |
| 959 |
> |
size. Because of the limit of computation power to calculate hessian |
| 960 |
> |
matrix and insufficient storage capacity to store them, most |
| 961 |
> |
Newton-Raphson methods can not be used with very large models. |
| 962 |
|
|
| 963 |
+ |
\subsubsection{Heating} |
| 964 |
+ |
|
| 965 |
+ |
Typically, Heating is performed by assigning random velocities |
| 966 |
+ |
according to a Gaussian distribution for a temperature. Beginning at |
| 967 |
+ |
a lower temperature and gradually increasing the temperature by |
| 968 |
+ |
assigning greater random velocities, we end up with setting the |
| 969 |
+ |
temperature of the system to a final temperature at which the |
| 970 |
+ |
simulation will be conducted. In heating phase, we should also keep |
| 971 |
+ |
the system from drifting or rotating as a whole. Equivalently, the |
| 972 |
+ |
net linear momentum and angular momentum of the system should be |
| 973 |
+ |
shifted to zero. |
| 974 |
+ |
|
| 975 |
+ |
\subsubsection{Equilibration} |
| 976 |
+ |
|
| 977 |
+ |
The purpose of equilibration is to allow the system to evolve |
| 978 |
+ |
spontaneously for a period of time and reach equilibrium. The |
| 979 |
+ |
procedure is continued until various statistical properties, such as |
| 980 |
+ |
temperature, pressure, energy, volume and other structural |
| 981 |
+ |
properties \textit{etc}, become independent of time. Strictly |
| 982 |
+ |
speaking, minimization and heating are not necessary, provided the |
| 983 |
+ |
equilibration process is long enough. However, these steps can serve |
| 984 |
+ |
as a means to arrive at an equilibrated structure in an effective |
| 985 |
+ |
way. |
| 986 |
+ |
|
| 987 |
+ |
\subsection{\label{introSection:production}Production} |
| 988 |
+ |
|
| 989 |
+ |
\subsubsection{\label{introSec:forceCalculation}The Force Calculation} |
| 990 |
+ |
|
| 991 |
+ |
\subsubsection{\label{introSection:integrationSchemes} Integration |
| 992 |
+ |
Schemes} |
| 993 |
+ |
|
| 994 |
+ |
\subsection{\label{introSection:Analysis} Analysis} |
| 995 |
+ |
|
| 996 |
+ |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 997 |
+ |
|
| 998 |
+ |
Rigid bodies are frequently involved in the modeling of different |
| 999 |
+ |
areas, from engineering, physics, to chemistry. For example, |
| 1000 |
+ |
missiles and vehicle are usually modeled by rigid bodies. The |
| 1001 |
+ |
movement of the objects in 3D gaming engine or other physics |
| 1002 |
+ |
simulator is governed by the rigid body dynamics. In molecular |
| 1003 |
+ |
simulation, rigid body is used to simplify the model in |
| 1004 |
+ |
protein-protein docking study{\cite{Gray03}}. |
| 1005 |
+ |
|
| 1006 |
+ |
It is very important to develop stable and efficient methods to |
| 1007 |
+ |
integrate the equations of motion of orientational degrees of |
| 1008 |
+ |
freedom. Euler angles are the nature choice to describe the |
| 1009 |
+ |
rotational degrees of freedom. However, due to its singularity, the |
| 1010 |
+ |
numerical integration of corresponding equations of motion is very |
| 1011 |
+ |
inefficient and inaccurate. Although an alternative integrator using |
| 1012 |
+ |
different sets of Euler angles can overcome this difficulty\cite{}, |
| 1013 |
+ |
the computational penalty and the lost of angular momentum |
| 1014 |
+ |
conservation still remain. A singularity free representation |
| 1015 |
+ |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
| 1016 |
+ |
this approach suffer from the nonseparable Hamiltonian resulted from |
| 1017 |
+ |
quaternion representation, which prevents the symplectic algorithm |
| 1018 |
+ |
to be utilized. Another different approach is to apply holonomic |
| 1019 |
+ |
constraints to the atoms belonging to the rigid body. Each atom |
| 1020 |
+ |
moves independently under the normal forces deriving from potential |
| 1021 |
+ |
energy and constraint forces which are used to guarantee the |
| 1022 |
+ |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
| 1023 |
+ |
algorithm converge very slowly when the number of constraint |
| 1024 |
+ |
increases. |
| 1025 |
+ |
|
| 1026 |
+ |
The break through in geometric literature suggests that, in order to |
| 1027 |
+ |
develop a long-term integration scheme, one should preserve the |
| 1028 |
+ |
symplectic structure of the flow. Introducing conjugate momentum to |
| 1029 |
+ |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
| 1030 |
+ |
symplectic integrator, RSHAKE, was proposed to evolve the |
| 1031 |
+ |
Hamiltonian system in a constraint manifold by iteratively |
| 1032 |
+ |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
| 1033 |
+ |
method using quaternion representation was developed by Omelyan. |
| 1034 |
+ |
However, both of these methods are iterative and inefficient. In |
| 1035 |
+ |
this section, we will present a symplectic Lie-Poisson integrator |
| 1036 |
+ |
for rigid body developed by Dullweber and his |
| 1037 |
+ |
coworkers\cite{Dullweber1997} in depth. |
| 1038 |
+ |
|
| 1039 |
+ |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 1040 |
+ |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
| 1041 |
+ |
function |
| 1042 |
|
\begin{equation} |
| 1043 |
< |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1044 |
< |
\label{introEquation:bathGLE} |
| 1043 |
> |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1044 |
> |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1045 |
> |
\label{introEquation:RBHamiltonian} |
| 1046 |
|
\end{equation} |
| 1047 |
< |
where $H_B$ is harmonic bath Hamiltonian, |
| 1047 |
> |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 1048 |
> |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 1049 |
> |
$J$, a diagonal matrix, is defined by |
| 1050 |
|
\[ |
| 1051 |
< |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 906 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
| 1051 |
> |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1052 |
|
\] |
| 1053 |
< |
and $\Delta U$ is bilinear system-bath coupling, |
| 1053 |
> |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1054 |
> |
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 1055 |
> |
\begin{equation} |
| 1056 |
> |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 1057 |
> |
\end{equation} |
| 1058 |
> |
which is used to ensure rotation matrix's orthogonality. |
| 1059 |
> |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 1060 |
> |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1061 |
> |
\begin{equation} |
| 1062 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1063 |
> |
\label{introEquation:RBFirstOrderConstraint} |
| 1064 |
> |
\end{equation} |
| 1065 |
> |
|
| 1066 |
> |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1067 |
> |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1068 |
> |
the equations of motion, |
| 1069 |
|
\[ |
| 1070 |
< |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1070 |
> |
\begin{array}{c} |
| 1071 |
> |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1072 |
> |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1073 |
> |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1074 |
> |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 1075 |
> |
\end{array} |
| 1076 |
|
\] |
| 1077 |
< |
Completing the square, |
| 1077 |
> |
|
| 1078 |
> |
In general, there are two ways to satisfy the holonomic constraints. |
| 1079 |
> |
We can use constraint force provided by lagrange multiplier on the |
| 1080 |
> |
normal manifold to keep the motion on constraint space. Or we can |
| 1081 |
> |
simply evolve the system in constraint manifold. The two method are |
| 1082 |
> |
proved to be equivalent. The holonomic constraint and equations of |
| 1083 |
> |
motions define a constraint manifold for rigid body |
| 1084 |
|
\[ |
| 1085 |
< |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
| 1086 |
< |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 916 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 917 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
| 918 |
< |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1085 |
> |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1086 |
> |
\right\}. |
| 1087 |
|
\] |
| 1088 |
< |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
| 1089 |
< |
\[ |
| 1090 |
< |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1091 |
< |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1092 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1093 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1088 |
> |
|
| 1089 |
> |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1090 |
> |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 1091 |
> |
transformation, the cotangent space and the phase space are |
| 1092 |
> |
diffeomorphic. Introducing |
| 1093 |
> |
\[ |
| 1094 |
> |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1095 |
|
\] |
| 1096 |
< |
where |
| 1096 |
> |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 1097 |
> |
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1098 |
|
\[ |
| 1099 |
< |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1100 |
< |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1099 |
> |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1100 |
> |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1101 |
|
\] |
| 932 |
– |
Since the first two terms of the new Hamiltonian depend only on the |
| 933 |
– |
system coordinates, we can get the equations of motion for |
| 934 |
– |
Generalized Langevin Dynamics by Hamilton's equations |
| 935 |
– |
\ref{introEquation:motionHamiltonianCoordinate, |
| 936 |
– |
introEquation:motionHamiltonianMomentum}, |
| 937 |
– |
\begin{align} |
| 938 |
– |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 939 |
– |
&= m\ddot x |
| 940 |
– |
&= - \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)} |
| 941 |
– |
\label{introEquation:Lp5} |
| 942 |
– |
\end{align} |
| 943 |
– |
, and |
| 944 |
– |
\begin{align} |
| 945 |
– |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
| 946 |
– |
&= m\ddot x_\alpha |
| 947 |
– |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
| 948 |
– |
\end{align} |
| 1102 |
|
|
| 1103 |
< |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
| 1104 |
< |
|
| 1103 |
> |
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1104 |
> |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1105 |
> |
given as |
| 1106 |
> |
\begin{equation} |
| 1107 |
> |
X_i^{lab} = Q X_i + q. |
| 1108 |
> |
\end{equation} |
| 1109 |
> |
Therefore, potential energy $V(q,Q)$ is defined by |
| 1110 |
|
\[ |
| 1111 |
< |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
| 1111 |
> |
V(q,Q) = V(Q X_0 + q). |
| 1112 |
|
\] |
| 1113 |
< |
|
| 1113 |
> |
Hence, the force and torque are given by |
| 1114 |
|
\[ |
| 1115 |
< |
L(x + y) = L(x) + L(y) |
| 1115 |
> |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, |
| 1116 |
|
\] |
| 1117 |
< |
|
| 1117 |
> |
and |
| 1118 |
|
\[ |
| 1119 |
< |
L(ax) = aL(x) |
| 1119 |
> |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1120 |
|
\] |
| 1121 |
+ |
respectively. |
| 1122 |
|
|
| 1123 |
+ |
As a common choice to describe the rotation dynamics of the rigid |
| 1124 |
+ |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1125 |
+ |
rewrite the equations of motion, |
| 1126 |
+ |
\begin{equation} |
| 1127 |
+ |
\begin{array}{l} |
| 1128 |
+ |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1129 |
+ |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1130 |
+ |
\end{array} |
| 1131 |
+ |
\label{introEqaution:RBMotionPI} |
| 1132 |
+ |
\end{equation} |
| 1133 |
+ |
, as well as holonomic constraints, |
| 1134 |
|
\[ |
| 1135 |
< |
L(\dot x) = pL(x) - px(0) |
| 1135 |
> |
\begin{array}{l} |
| 1136 |
> |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1137 |
> |
Q^T Q = 1 \\ |
| 1138 |
> |
\end{array} |
| 1139 |
|
\] |
| 1140 |
|
|
| 1141 |
+ |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1142 |
+ |
so(3)^ \star$, the hat-map isomorphism, |
| 1143 |
+ |
\begin{equation} |
| 1144 |
+ |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1145 |
+ |
{\begin{array}{*{20}c} |
| 1146 |
+ |
0 & { - v_3 } & {v_2 } \\ |
| 1147 |
+ |
{v_3 } & 0 & { - v_1 } \\ |
| 1148 |
+ |
{ - v_2 } & {v_1 } & 0 \\ |
| 1149 |
+ |
\end{array}} \right), |
| 1150 |
+ |
\label{introEquation:hatmapIsomorphism} |
| 1151 |
+ |
\end{equation} |
| 1152 |
+ |
will let us associate the matrix products with traditional vector |
| 1153 |
+ |
operations |
| 1154 |
|
\[ |
| 1155 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
| 1155 |
> |
\hat vu = v \times u |
| 1156 |
|
\] |
| 1157 |
|
|
| 1158 |
+ |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1159 |
+ |
matrix, |
| 1160 |
+ |
\begin{equation} |
| 1161 |
+ |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1162 |
+ |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1163 |
+ |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1164 |
+ |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1165 |
+ |
\end{equation} |
| 1166 |
+ |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1167 |
+ |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1168 |
+ |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1169 |
+ |
unique property eliminate the requirement of iterations which can |
| 1170 |
+ |
not be avoided in other methods\cite{}. |
| 1171 |
+ |
|
| 1172 |
+ |
Applying hat-map isomorphism, we obtain the equation of motion for |
| 1173 |
+ |
angular momentum on body frame |
| 1174 |
+ |
\begin{equation} |
| 1175 |
+ |
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1176 |
+ |
F_i (r,Q)} \right) \times X_i }. |
| 1177 |
+ |
\label{introEquation:bodyAngularMotion} |
| 1178 |
+ |
\end{equation} |
| 1179 |
+ |
In the same manner, the equation of motion for rotation matrix is |
| 1180 |
+ |
given by |
| 1181 |
|
\[ |
| 1182 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
| 1182 |
> |
\dot Q = Qskew(I^{ - 1} \pi ) |
| 1183 |
|
\] |
| 1184 |
|
|
| 1185 |
< |
Some relatively important transformation, |
| 1185 |
> |
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1186 |
> |
Lie-Poisson Integrator for Free Rigid Body} |
| 1187 |
> |
|
| 1188 |
> |
If there is not external forces exerted on the rigid body, the only |
| 1189 |
> |
contribution to the rotational is from the kinetic potential (the |
| 1190 |
> |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
| 1191 |
> |
rigid body is an example of Lie-Poisson system with Hamiltonian |
| 1192 |
> |
function |
| 1193 |
> |
\begin{equation} |
| 1194 |
> |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1195 |
> |
\label{introEquation:rotationalKineticRB} |
| 1196 |
> |
\end{equation} |
| 1197 |
> |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
| 1198 |
> |
Lie-Poisson structure matrix, |
| 1199 |
> |
\begin{equation} |
| 1200 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 1201 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1202 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1203 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1204 |
> |
\end{array}} \right) |
| 1205 |
> |
\end{equation} |
| 1206 |
> |
Thus, the dynamics of free rigid body is governed by |
| 1207 |
> |
\begin{equation} |
| 1208 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1209 |
> |
\end{equation} |
| 1210 |
> |
|
| 1211 |
> |
One may notice that each $T_i^r$ in Equation |
| 1212 |
> |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1213 |
> |
instance, the equations of motion due to $T_1^r$ are given by |
| 1214 |
> |
\begin{equation} |
| 1215 |
> |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
| 1216 |
> |
\label{introEqaution:RBMotionSingleTerm} |
| 1217 |
> |
\end{equation} |
| 1218 |
> |
where |
| 1219 |
> |
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1220 |
> |
0 & 0 & 0 \\ |
| 1221 |
> |
0 & 0 & {\pi _1 } \\ |
| 1222 |
> |
0 & { - \pi _1 } & 0 \\ |
| 1223 |
> |
\end{array}} \right). |
| 1224 |
> |
\] |
| 1225 |
> |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1226 |
|
\[ |
| 1227 |
< |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
| 1227 |
> |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
| 1228 |
> |
Q(0)e^{\Delta tR_1 } |
| 1229 |
|
\] |
| 1230 |
+ |
with |
| 1231 |
+ |
\[ |
| 1232 |
+ |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
| 1233 |
+ |
0 & 0 & 0 \\ |
| 1234 |
+ |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
| 1235 |
+ |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1236 |
+ |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1237 |
+ |
\] |
| 1238 |
+ |
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1239 |
+ |
tR_1 }$, we can use Cayley transformation, |
| 1240 |
+ |
\[ |
| 1241 |
+ |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1242 |
+ |
) |
| 1243 |
+ |
\] |
| 1244 |
+ |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1245 |
+ |
manner. |
| 1246 |
|
|
| 1247 |
+ |
In order to construct a second-order symplectic method, we split the |
| 1248 |
+ |
angular kinetic Hamiltonian function can into five terms |
| 1249 |
|
\[ |
| 1250 |
< |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
| 1250 |
> |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1251 |
> |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1252 |
> |
(\pi _1 ) |
| 1253 |
> |
\]. |
| 1254 |
> |
Concatenating flows corresponding to these five terms, we can obtain |
| 1255 |
> |
an symplectic integrator, |
| 1256 |
> |
\[ |
| 1257 |
> |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1258 |
> |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1259 |
> |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1260 |
> |
_1 }. |
| 1261 |
|
\] |
| 1262 |
|
|
| 1263 |
+ |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1264 |
+ |
$F(\pi )$ and $G(\pi )$ is defined by |
| 1265 |
|
\[ |
| 1266 |
< |
L(1) = \frac{1}{p} |
| 1266 |
> |
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1267 |
> |
) |
| 1268 |
|
\] |
| 1269 |
+ |
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1270 |
+ |
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1271 |
+ |
conserved quantity in Poisson system. We can easily verify that the |
| 1272 |
+ |
norm of the angular momentum, $\parallel \pi |
| 1273 |
+ |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1274 |
+ |
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1275 |
+ |
then by the chain rule |
| 1276 |
+ |
\[ |
| 1277 |
+ |
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1278 |
+ |
}}{2})\pi |
| 1279 |
+ |
\] |
| 1280 |
+ |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1281 |
+ |
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1282 |
+ |
Lie-Poisson integrator is found to be extremely efficient and stable |
| 1283 |
+ |
which can be explained by the fact the small angle approximation is |
| 1284 |
+ |
used and the norm of the angular momentum is conserved. |
| 1285 |
|
|
| 1286 |
< |
First, the bath coordinates, |
| 1286 |
> |
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
| 1287 |
> |
Splitting for Rigid Body} |
| 1288 |
> |
|
| 1289 |
> |
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1290 |
> |
energy and potential energy, |
| 1291 |
|
\[ |
| 1292 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
| 992 |
< |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
| 993 |
< |
}}L(x) |
| 1292 |
> |
H = T(p,\pi ) + V(q,Q) |
| 1293 |
|
\] |
| 1294 |
+ |
The equations of motion corresponding to potential energy and |
| 1295 |
+ |
kinetic energy are listed in the below table, |
| 1296 |
+ |
\begin{center} |
| 1297 |
+ |
\begin{tabular}{|l|l|} |
| 1298 |
+ |
\hline |
| 1299 |
+ |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 1300 |
+ |
Potential & Kinetic \\ |
| 1301 |
+ |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
| 1302 |
+ |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
| 1303 |
+ |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
| 1304 |
+ |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
| 1305 |
+ |
\hline |
| 1306 |
+ |
\end{tabular} |
| 1307 |
+ |
\end{center} |
| 1308 |
+ |
A second-order symplectic method is now obtained by the composition |
| 1309 |
+ |
of the flow maps, |
| 1310 |
|
\[ |
| 1311 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
| 1312 |
< |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
| 1311 |
> |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1312 |
> |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1313 |
|
\] |
| 1314 |
< |
Then, the system coordinates, |
| 1315 |
< |
\begin{align} |
| 1316 |
< |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1317 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
| 1318 |
< |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
| 1319 |
< |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
| 1320 |
< |
}}\omega _\alpha ^2 L(x)} \right\}} |
| 1321 |
< |
% |
| 1322 |
< |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1008 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
| 1009 |
< |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
| 1010 |
< |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
| 1011 |
< |
\end{align} |
| 1012 |
< |
Then, the inverse transform, |
| 1314 |
> |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 1315 |
> |
sub-flows which corresponding to force and torque respectively, |
| 1316 |
> |
\[ |
| 1317 |
> |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1318 |
> |
_{\Delta t/2,\tau }. |
| 1319 |
> |
\] |
| 1320 |
> |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1321 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 1322 |
> |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
| 1323 |
|
|
| 1324 |
+ |
Furthermore, kinetic potential can be separated to translational |
| 1325 |
+ |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1326 |
+ |
\begin{equation} |
| 1327 |
+ |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1328 |
+ |
\end{equation} |
| 1329 |
+ |
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1330 |
+ |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1331 |
+ |
corresponding flow maps are given by |
| 1332 |
+ |
\[ |
| 1333 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1334 |
+ |
_{\Delta t,T^r }. |
| 1335 |
+ |
\] |
| 1336 |
+ |
Finally, we obtain the overall symplectic flow maps for free moving |
| 1337 |
+ |
rigid body |
| 1338 |
+ |
\begin{equation} |
| 1339 |
+ |
\begin{array}{c} |
| 1340 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1341 |
+ |
\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 } \\ |
| 1342 |
+ |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1343 |
+ |
\end{array} |
| 1344 |
+ |
\label{introEquation:overallRBFlowMaps} |
| 1345 |
+ |
\end{equation} |
| 1346 |
+ |
|
| 1347 |
+ |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1348 |
+ |
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1349 |
+ |
mimics a simple heat bath with stochastic and dissipative forces, |
| 1350 |
+ |
has been applied in a variety of studies. This section will review |
| 1351 |
+ |
the theory of Langevin dynamics simulation. A brief derivation of |
| 1352 |
+ |
generalized Langevin equation will be given first. Follow that, we |
| 1353 |
+ |
will discuss the physical meaning of the terms appearing in the |
| 1354 |
+ |
equation as well as the calculation of friction tensor from |
| 1355 |
+ |
hydrodynamics theory. |
| 1356 |
+ |
|
| 1357 |
+ |
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1358 |
+ |
|
| 1359 |
+ |
Harmonic bath model, in which an effective set of harmonic |
| 1360 |
+ |
oscillators are used to mimic the effect of a linearly responding |
| 1361 |
+ |
environment, has been widely used in quantum chemistry and |
| 1362 |
+ |
statistical mechanics. One of the successful applications of |
| 1363 |
+ |
Harmonic bath model is the derivation of Deriving Generalized |
| 1364 |
+ |
Langevin Dynamics. Lets consider a system, in which the degree of |
| 1365 |
+ |
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1366 |
+ |
Hamiltonian of the form |
| 1367 |
+ |
\begin{equation} |
| 1368 |
+ |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1369 |
+ |
\label{introEquation:bathGLE}. |
| 1370 |
+ |
\end{equation} |
| 1371 |
+ |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
| 1372 |
+ |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
| 1373 |
+ |
\[ |
| 1374 |
+ |
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1375 |
+ |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1376 |
+ |
\right\}} |
| 1377 |
+ |
\] |
| 1378 |
+ |
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1379 |
+ |
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
| 1380 |
+ |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
| 1381 |
+ |
coupling, |
| 1382 |
+ |
\[ |
| 1383 |
+ |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1384 |
+ |
\] |
| 1385 |
+ |
where $g_\alpha$ are the coupling constants between the bath and the |
| 1386 |
+ |
coordinate $x$. Introducing |
| 1387 |
+ |
\[ |
| 1388 |
+ |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1389 |
+ |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1390 |
+ |
\] and combining the last two terms in Equation |
| 1391 |
+ |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
| 1392 |
+ |
Hamiltonian as |
| 1393 |
+ |
\[ |
| 1394 |
+ |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1395 |
+ |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1396 |
+ |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1397 |
+ |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1398 |
+ |
\] |
| 1399 |
+ |
Since the first two terms of the new Hamiltonian depend only on the |
| 1400 |
+ |
system coordinates, we can get the equations of motion for |
| 1401 |
+ |
Generalized Langevin Dynamics by Hamilton's equations |
| 1402 |
+ |
\ref{introEquation:motionHamiltonianCoordinate, |
| 1403 |
+ |
introEquation:motionHamiltonianMomentum}, |
| 1404 |
+ |
\begin{equation} |
| 1405 |
+ |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1406 |
+ |
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
| 1407 |
+ |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)}, |
| 1408 |
+ |
\label{introEquation:coorMotionGLE} |
| 1409 |
+ |
\end{equation} |
| 1410 |
+ |
and |
| 1411 |
+ |
\begin{equation} |
| 1412 |
+ |
m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - |
| 1413 |
+ |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1414 |
+ |
\label{introEquation:bathMotionGLE} |
| 1415 |
+ |
\end{equation} |
| 1416 |
+ |
|
| 1417 |
+ |
In order to derive an equation for $x$, the dynamics of the bath |
| 1418 |
+ |
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1419 |
+ |
transform which is particularly useful in solving linear ordinary |
| 1420 |
+ |
differential equations, Laplace transform is the appropriate tool to |
| 1421 |
+ |
solve this problem. The basic idea is to transform the difficult |
| 1422 |
+ |
differential equations into simple algebra problems which can be |
| 1423 |
+ |
solved easily. Then applying inverse Laplace transform, also known |
| 1424 |
+ |
as the Bromwich integral, we can retrieve the solutions of the |
| 1425 |
+ |
original problems. |
| 1426 |
+ |
|
| 1427 |
+ |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1428 |
+ |
transform of f(t) is a new function defined as |
| 1429 |
+ |
\[ |
| 1430 |
+ |
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1431 |
+ |
\] |
| 1432 |
+ |
where $p$ is real and $L$ is called the Laplace Transform |
| 1433 |
+ |
Operator. Below are some important properties of Laplace transform |
| 1434 |
+ |
\begin{equation} |
| 1435 |
+ |
\begin{array}{c} |
| 1436 |
+ |
L(x + y) = L(x) + L(y) \\ |
| 1437 |
+ |
L(ax) = aL(x) \\ |
| 1438 |
+ |
L(\dot x) = pL(x) - px(0) \\ |
| 1439 |
+ |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
| 1440 |
+ |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
| 1441 |
+ |
\end{array} |
| 1442 |
+ |
\end{equation} |
| 1443 |
+ |
|
| 1444 |
+ |
Applying Laplace transform to the bath coordinates, we obtain |
| 1445 |
+ |
\[ |
| 1446 |
+ |
\begin{array}{c} |
| 1447 |
+ |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
| 1448 |
+ |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1449 |
+ |
\end{array} |
| 1450 |
+ |
\] |
| 1451 |
+ |
By the same way, the system coordinates become |
| 1452 |
+ |
\[ |
| 1453 |
+ |
\begin{array}{c} |
| 1454 |
+ |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1455 |
+ |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1456 |
+ |
\end{array} |
| 1457 |
+ |
\] |
| 1458 |
+ |
|
| 1459 |
+ |
With the help of some relatively important inverse Laplace |
| 1460 |
+ |
transformations: |
| 1461 |
+ |
\[ |
| 1462 |
+ |
\begin{array}{c} |
| 1463 |
+ |
L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ |
| 1464 |
+ |
L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ |
| 1465 |
+ |
L(1) = \frac{1}{p} \\ |
| 1466 |
+ |
\end{array} |
| 1467 |
+ |
\] |
| 1468 |
+ |
, we obtain |
| 1469 |
|
\begin{align} |
| 1470 |
|
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
| 1471 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1485 |
|
(\omega _\alpha t)} \right\}} |
| 1486 |
|
\end{align} |
| 1487 |
|
|
| 1488 |
+ |
Introducing a \emph{dynamic friction kernel} |
| 1489 |
|
\begin{equation} |
| 1490 |
+ |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1491 |
+ |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 1492 |
+ |
\label{introEquation:dynamicFrictionKernelDefinition} |
| 1493 |
+ |
\end{equation} |
| 1494 |
+ |
and \emph{a random force} |
| 1495 |
+ |
\begin{equation} |
| 1496 |
+ |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 1497 |
+ |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 1498 |
+ |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 1499 |
+ |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), |
| 1500 |
+ |
\label{introEquation:randomForceDefinition} |
| 1501 |
+ |
\end{equation} |
| 1502 |
+ |
the equation of motion can be rewritten as |
| 1503 |
+ |
\begin{equation} |
| 1504 |
|
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
| 1505 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1506 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1507 |
|
\end{equation} |
| 1508 |
< |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
| 1509 |
< |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
| 1508 |
> |
which is known as the \emph{generalized Langevin equation}. |
| 1509 |
> |
|
| 1510 |
> |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel} |
| 1511 |
> |
|
| 1512 |
> |
One may notice that $R(t)$ depends only on initial conditions, which |
| 1513 |
> |
implies it is completely deterministic within the context of a |
| 1514 |
> |
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1515 |
> |
uncorrelated to $x$ and $\dot x$, |
| 1516 |
|
\[ |
| 1517 |
< |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1518 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 1517 |
> |
\begin{array}{l} |
| 1518 |
> |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
| 1519 |
> |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
| 1520 |
> |
\end{array} |
| 1521 |
|
\] |
| 1522 |
< |
For an infinite harmonic bath, we can use the spectral density and |
| 1523 |
< |
an integral over frequencies. |
| 1522 |
> |
This property is what we expect from a truly random process. As long |
| 1523 |
> |
as the model, which is gaussian distribution in general, chosen for |
| 1524 |
> |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
| 1525 |
> |
still remains. |
| 1526 |
|
|
| 1527 |
+ |
%dynamic friction kernel |
| 1528 |
+ |
The convolution integral |
| 1529 |
|
\[ |
| 1530 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 1049 |
< |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 1050 |
< |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 1051 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
| 1530 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } |
| 1531 |
|
\] |
| 1532 |
< |
The random forces depend only on initial conditions. |
| 1532 |
> |
depends on the entire history of the evolution of $x$, which implies |
| 1533 |
> |
that the bath retains memory of previous motions. In other words, |
| 1534 |
> |
the bath requires a finite time to respond to change in the motion |
| 1535 |
> |
of the system. For a sluggish bath which responds slowly to changes |
| 1536 |
> |
in the system coordinate, we may regard $\xi(t)$ as a constant |
| 1537 |
> |
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes |
| 1538 |
> |
\[ |
| 1539 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
| 1540 |
> |
\] |
| 1541 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1542 |
> |
\[ |
| 1543 |
> |
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1544 |
> |
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1545 |
> |
\] |
| 1546 |
> |
which can be used to describe dynamic caging effect. The other |
| 1547 |
> |
extreme is the bath that responds infinitely quickly to motions in |
| 1548 |
> |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
| 1549 |
> |
time: |
| 1550 |
> |
\[ |
| 1551 |
> |
\xi (t) = 2\xi _0 \delta (t) |
| 1552 |
> |
\] |
| 1553 |
> |
Hence, the convolution integral becomes |
| 1554 |
> |
\[ |
| 1555 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1556 |
> |
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1557 |
> |
\] |
| 1558 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1559 |
> |
\begin{equation} |
| 1560 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1561 |
> |
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1562 |
> |
\end{equation} |
| 1563 |
> |
which is known as the Langevin equation. The static friction |
| 1564 |
> |
coefficient $\xi _0$ can either be calculated from spectral density |
| 1565 |
> |
or be determined by Stokes' law for regular shaped particles.A |
| 1566 |
> |
briefly review on calculating friction tensor for arbitrary shaped |
| 1567 |
> |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1568 |
|
|
| 1569 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 1570 |
< |
So we can define a new set of coordinates, |
| 1570 |
> |
|
| 1571 |
> |
Defining a new set of coordinates, |
| 1572 |
|
\[ |
| 1573 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1574 |
|
^2 }}x(0) |
| 1575 |
< |
\] |
| 1576 |
< |
This makes |
| 1575 |
> |
\], |
| 1576 |
> |
we can rewrite $R(T)$ as |
| 1577 |
|
\[ |
| 1578 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
| 1578 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1579 |
|
\] |
| 1580 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1581 |
|
\[ |
| 1582 |
< |
\begin{array}{l} |
| 1582 |
> |
\begin{array}{c} |
| 1583 |
> |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1584 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1585 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1586 |
+ |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
| 1587 |
+ |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1588 |
+ |
= kT\xi (t) \\ |
| 1589 |
|
\end{array} |
| 1590 |
|
\] |
| 1591 |
< |
|
| 1073 |
< |
\begin{align} |
| 1074 |
< |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
| 1075 |
< |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
| 1076 |
< |
(t)q_\beta (0)} \right\rangle } } |
| 1077 |
< |
% |
| 1078 |
< |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
| 1079 |
< |
\right\rangle \cos (\omega _\alpha t)} |
| 1080 |
< |
% |
| 1081 |
< |
&= kT\xi (t) |
| 1082 |
< |
\end{align} |
| 1083 |
< |
|
| 1591 |
> |
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1592 |
|
\begin{equation} |
| 1593 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 1594 |
< |
\label{introEquation:secondFluctuationDissipation} |
| 1594 |
> |
\label{introEquation:secondFluctuationDissipation}. |
| 1595 |
|
\end{equation} |
| 1596 |
+ |
In effect, it acts as a constraint on the possible ways in which one |
| 1597 |
+ |
can model the random force and friction kernel. |
| 1598 |
|
|
| 1089 |
– |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
| 1090 |
– |
|
| 1599 |
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1600 |
< |
\subsection{\label{introSection:analyticalApproach}Analytical |
| 1601 |
< |
Approach} |
| 1600 |
> |
Theoretically, the friction kernel can be determined using velocity |
| 1601 |
> |
autocorrelation function. However, this approach become impractical |
| 1602 |
> |
when the system become more and more complicate. Instead, various |
| 1603 |
> |
approaches based on hydrodynamics have been developed to calculate |
| 1604 |
> |
the friction coefficients. The friction effect is isotropic in |
| 1605 |
> |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1606 |
> |
tensor \Xi is a $6\times 6$ matrix given by |
| 1607 |
> |
\[ |
| 1608 |
> |
\Xi = \left( {\begin{array}{*{20}c} |
| 1609 |
> |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1610 |
> |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 1611 |
> |
\end{array}} \right). |
| 1612 |
> |
\] |
| 1613 |
> |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 1614 |
> |
tensor and rotational resistance (friction) tensor respectively, |
| 1615 |
> |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 1616 |
> |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 1617 |
> |
particle moves in a fluid, it may experience friction force or |
| 1618 |
> |
torque along the opposite direction of the velocity or angular |
| 1619 |
> |
velocity, |
| 1620 |
> |
\[ |
| 1621 |
> |
\left( \begin{array}{l} |
| 1622 |
> |
F_R \\ |
| 1623 |
> |
\tau _R \\ |
| 1624 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1625 |
> |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1626 |
> |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1627 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 1628 |
> |
v \\ |
| 1629 |
> |
w \\ |
| 1630 |
> |
\end{array} \right) |
| 1631 |
> |
\] |
| 1632 |
> |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1633 |
> |
toque. |
| 1634 |
|
|
| 1635 |
< |
\subsection{\label{introSection:approximationApproach}Approximation |
| 1096 |
< |
Approach} |
| 1635 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
| 1636 |
|
|
| 1637 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1638 |
< |
Body} |
| 1637 |
> |
For a spherical particle, the translational and rotational friction |
| 1638 |
> |
constant can be calculated from Stoke's law, |
| 1639 |
> |
\[ |
| 1640 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1641 |
> |
{6\pi \eta R} & 0 & 0 \\ |
| 1642 |
> |
0 & {6\pi \eta R} & 0 \\ |
| 1643 |
> |
0 & 0 & {6\pi \eta R} \\ |
| 1644 |
> |
\end{array}} \right) |
| 1645 |
> |
\] |
| 1646 |
> |
and |
| 1647 |
> |
\[ |
| 1648 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1649 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1650 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1651 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1652 |
> |
\end{array}} \right) |
| 1653 |
> |
\] |
| 1654 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1655 |
> |
hydrodynamics radius. |
| 1656 |
> |
|
| 1657 |
> |
Other non-spherical shape, such as cylinder and ellipsoid |
| 1658 |
> |
\textit{etc}, are widely used as reference for developing new |
| 1659 |
> |
hydrodynamics theory, because their properties can be calculated |
| 1660 |
> |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 1661 |
> |
also called a triaxial ellipsoid, which is given in Cartesian |
| 1662 |
> |
coordinates by |
| 1663 |
> |
\[ |
| 1664 |
> |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 1665 |
> |
}} = 1 |
| 1666 |
> |
\] |
| 1667 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 1668 |
> |
due to the complexity of the elliptic integral, only the ellipsoid |
| 1669 |
> |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 1670 |
> |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 1671 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 1672 |
> |
\[ |
| 1673 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1674 |
> |
} }}{b}, |
| 1675 |
> |
\] |
| 1676 |
> |
and oblate, |
| 1677 |
> |
\[ |
| 1678 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1679 |
> |
}}{a} |
| 1680 |
> |
\], |
| 1681 |
> |
one can write down the translational and rotational resistance |
| 1682 |
> |
tensors |
| 1683 |
> |
\[ |
| 1684 |
> |
\begin{array}{l} |
| 1685 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1686 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1687 |
> |
\end{array}, |
| 1688 |
> |
\] |
| 1689 |
> |
and |
| 1690 |
> |
\[ |
| 1691 |
> |
\begin{array}{l} |
| 1692 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1693 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1694 |
> |
\end{array}. |
| 1695 |
> |
\] |
| 1696 |
> |
|
| 1697 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
| 1698 |
> |
|
| 1699 |
> |
Unlike spherical and other regular shaped molecules, there is not |
| 1700 |
> |
analytical solution for friction tensor of any arbitrary shaped |
| 1701 |
> |
rigid molecules. The ellipsoid of revolution model and general |
| 1702 |
> |
triaxial ellipsoid model have been used to approximate the |
| 1703 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1704 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1705 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
| 1706 |
> |
unique\cite{Wegener79} as well as the intrinsic coupling between |
| 1707 |
> |
translational and rotational motion of rigid body\cite{}, general |
| 1708 |
> |
ellipsoid is not always suitable for modeling arbitrarily shaped |
| 1709 |
> |
rigid molecule. A number of studies have been devoted to determine |
| 1710 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
| 1711 |
> |
advanced method\cite{} where the molecule of interest was modeled by |
| 1712 |
> |
combinations of spheres(beads)\cite{} and the hydrodynamics |
| 1713 |
> |
properties of the molecule can be calculated using the hydrodynamic |
| 1714 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 1715 |
> |
immersed in a continuous medium. Due to hydrodynamics interaction, |
| 1716 |
> |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 1717 |
> |
unperturbed velocity $v_i$, |
| 1718 |
> |
\[ |
| 1719 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1720 |
> |
\] |
| 1721 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1722 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1723 |
> |
proportional to its ``net'' velocity |
| 1724 |
> |
\begin{equation} |
| 1725 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1726 |
> |
\label{introEquation:tensorExpression} |
| 1727 |
> |
\end{equation} |
| 1728 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1729 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1730 |
> |
\ref{introEquation:tensorExpression} |
| 1731 |
> |
\begin{equation} |
| 1732 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1733 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1734 |
> |
\label{introEquation:oseenTensor} |
| 1735 |
> |
\end{equation} |
| 1736 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1737 |
> |
A second order expression for element of different size was |
| 1738 |
> |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
| 1739 |
> |
la Torre and Bloomfield, |
| 1740 |
> |
\begin{equation} |
| 1741 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1742 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1743 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1744 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1745 |
> |
\label{introEquation:RPTensorNonOverlapped} |
| 1746 |
> |
\end{equation} |
| 1747 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1748 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1749 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1750 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
| 1751 |
> |
\begin{equation} |
| 1752 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1753 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1754 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1755 |
> |
\label{introEquation:RPTensorOverlapped} |
| 1756 |
> |
\end{equation} |
| 1757 |
> |
|
| 1758 |
> |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1759 |
> |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1760 |
> |
$B_{ij}$ blocks |
| 1761 |
> |
\begin{equation} |
| 1762 |
> |
B = \left( {\begin{array}{*{20}c} |
| 1763 |
> |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1764 |
> |
\vdots & \ddots & \vdots \\ |
| 1765 |
> |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1766 |
> |
\end{array}} \right), |
| 1767 |
> |
\end{equation} |
| 1768 |
> |
where $B_{ij}$ is given by |
| 1769 |
> |
\[ |
| 1770 |
> |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1771 |
> |
)T_{ij} |
| 1772 |
> |
\] |
| 1773 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 1774 |
> |
$B$, we obtain |
| 1775 |
> |
|
| 1776 |
> |
\[ |
| 1777 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1778 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1779 |
> |
\vdots & \ddots & \vdots \\ |
| 1780 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1781 |
> |
\end{array}} \right) |
| 1782 |
> |
\] |
| 1783 |
> |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1784 |
> |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1785 |
> |
\[ |
| 1786 |
> |
U_i = \left( {\begin{array}{*{20}c} |
| 1787 |
> |
0 & { - z_i } & {y_i } \\ |
| 1788 |
> |
{z_i } & 0 & { - x_i } \\ |
| 1789 |
> |
{ - y_i } & {x_i } & 0 \\ |
| 1790 |
> |
\end{array}} \right) |
| 1791 |
> |
\] |
| 1792 |
> |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1793 |
> |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1794 |
> |
arbitrary origin $O$ can be written as |
| 1795 |
> |
\begin{equation} |
| 1796 |
> |
\begin{array}{l} |
| 1797 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1798 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1799 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1800 |
> |
\end{array} |
| 1801 |
> |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1802 |
> |
\end{equation} |
| 1803 |
> |
|
| 1804 |
> |
The resistance tensor depends on the origin to which they refer. The |
| 1805 |
> |
proper location for applying friction force is the center of |
| 1806 |
> |
resistance (reaction), at which the trace of rotational resistance |
| 1807 |
> |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1808 |
> |
resistance is defined as an unique point of the rigid body at which |
| 1809 |
> |
the translation-rotation coupling tensor are symmetric, |
| 1810 |
> |
\begin{equation} |
| 1811 |
> |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1812 |
> |
\label{introEquation:definitionCR} |
| 1813 |
> |
\end{equation} |
| 1814 |
> |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1815 |
> |
we can easily find out that the translational resistance tensor is |
| 1816 |
> |
origin independent, while the rotational resistance tensor and |
| 1817 |
> |
translation-rotation coupling resistance tensor depend on the |
| 1818 |
> |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1819 |
> |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1820 |
> |
obtain the resistance tensor at $P$ by |
| 1821 |
> |
\begin{equation} |
| 1822 |
> |
\begin{array}{l} |
| 1823 |
> |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 1824 |
> |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 1825 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
| 1826 |
> |
\end{array} |
| 1827 |
> |
\label{introEquation:resistanceTensorTransformation} |
| 1828 |
> |
\end{equation} |
| 1829 |
> |
where |
| 1830 |
> |
\[ |
| 1831 |
> |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 1832 |
> |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 1833 |
> |
{z_i } & 0 & { - x_{OP} } \\ |
| 1834 |
> |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 1835 |
> |
\end{array}} \right) |
| 1836 |
> |
\] |
| 1837 |
> |
Using Equations \ref{introEquation:definitionCR} and |
| 1838 |
> |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 1839 |
> |
the position of center of resistance, |
| 1840 |
> |
\[ |
| 1841 |
> |
\left( \begin{array}{l} |
| 1842 |
> |
x_{OR} \\ |
| 1843 |
> |
y_{OR} \\ |
| 1844 |
> |
z_{OR} \\ |
| 1845 |
> |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
| 1846 |
> |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 1847 |
> |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 1848 |
> |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 1849 |
> |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
| 1850 |
> |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 1851 |
> |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 1852 |
> |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 1853 |
> |
\end{array} \right). |
| 1854 |
> |
\] |
| 1855 |
> |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1856 |
> |
joining center of resistance $R$ and origin $O$. |
| 1857 |
> |
|
| 1858 |
> |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
