| 27 |
|
\end{equation} |
| 28 |
|
A point mass interacting with other bodies moves with the |
| 29 |
|
acceleration along the direction of the force acting on it. Let |
| 30 |
< |
$F_ij$ be the force that particle $i$ exerts on particle $j$, and |
| 31 |
< |
$F_ji$ be the force that particle $j$ exerts on particle $i$. |
| 30 |
> |
$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and |
| 31 |
> |
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
| 32 |
|
Newton¡¯s third law states that |
| 33 |
|
\begin{equation} |
| 34 |
< |
F_ij = -F_ji |
| 34 |
> |
F_{ij} = -F_{ji} |
| 35 |
|
\label{introEquation:newtonThirdLaw} |
| 36 |
|
\end{equation} |
| 37 |
|
|
| 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 |
| 393 |
|
\begin{equation} |
| 394 |
|
\rho \propto e^{ - \beta H} |
| 395 |
|
\label{introEquation:densityAndHamiltonian} |
| 396 |
+ |
\end{equation} |
| 397 |
+ |
|
| 398 |
+ |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
| 399 |
+ |
Lets consider a region in the phase space, |
| 400 |
+ |
\begin{equation} |
| 401 |
+ |
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
| 402 |
+ |
\end{equation} |
| 403 |
+ |
If this region is small enough, the density $\rho$ can be regarded |
| 404 |
+ |
as uniform over the whole phase space. Thus, the number of phase |
| 405 |
+ |
points inside this region is given by, |
| 406 |
+ |
\begin{equation} |
| 407 |
+ |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
| 408 |
+ |
dp_1 } ..dp_f. |
| 409 |
+ |
\end{equation} |
| 410 |
+ |
|
| 411 |
+ |
\begin{equation} |
| 412 |
+ |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
| 413 |
+ |
\frac{d}{{dt}}(\delta v) = 0. |
| 414 |
+ |
\end{equation} |
| 415 |
+ |
With the help of stationary assumption |
| 416 |
+ |
(\ref{introEquation:stationary}), we obtain the principle of the |
| 417 |
+ |
\emph{conservation of extension in phase space}, |
| 418 |
+ |
\begin{equation} |
| 419 |
+ |
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
| 420 |
+ |
...dq_f dp_1 } ..dp_f = 0. |
| 421 |
+ |
\label{introEquation:volumePreserving} |
| 422 |
|
\end{equation} |
| 423 |
|
|
| 424 |
+ |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
| 425 |
+ |
|
| 426 |
|
Liouville's theorem can be expresses in a variety of different forms |
| 427 |
|
which are convenient within different contexts. For any two function |
| 428 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
| 458 |
|
\label{introEquation:liouvilleTheoremInOperator} |
| 459 |
|
\end{equation} |
| 460 |
|
|
| 434 |
– |
|
| 461 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
| 462 |
|
|
| 463 |
|
Various thermodynamic properties can be calculated from Molecular |
| 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 $x$. |
| 547 |
– |
The free rigid body is an example of Poisson system (actually a |
| 548 |
– |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
| 549 |
– |
energy. |
| 550 |
– |
\begin{equation} |
| 551 |
– |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 552 |
– |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 553 |
– |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 554 |
– |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 555 |
– |
\end{array}} \right) |
| 556 |
– |
\end{equation} |
| 573 |
|
|
| 574 |
< |
\begin{equation} |
| 559 |
< |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
| 560 |
< |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
| 561 |
< |
\end{equation} |
| 574 |
> |
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 575 |
|
|
| 563 |
– |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 576 |
|
Let $x(t)$ be the exact solution of the ODE system, |
| 577 |
|
\begin{equation} |
| 578 |
|
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
| 582 |
|
x(t+\tau) =\varphi_\tau(x(t)) |
| 583 |
|
\] |
| 584 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 585 |
< |
space to itself. In most cases, it is not easy to find the exact |
| 574 |
< |
flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$, |
| 575 |
< |
which is usually called integrator. The order of an integrator |
| 576 |
< |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
| 577 |
< |
order $p$, |
| 585 |
> |
space to itself. The flow has the continuous group property, |
| 586 |
|
\begin{equation} |
| 587 |
+ |
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
| 588 |
+ |
+ \tau _2 } . |
| 589 |
+ |
\end{equation} |
| 590 |
+ |
In particular, |
| 591 |
+ |
\begin{equation} |
| 592 |
+ |
\varphi _\tau \circ \varphi _{ - \tau } = I |
| 593 |
+ |
\end{equation} |
| 594 |
+ |
Therefore, the exact flow is self-adjoint, |
| 595 |
+ |
\begin{equation} |
| 596 |
+ |
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 597 |
+ |
\end{equation} |
| 598 |
+ |
The exact flow can also be written in terms of the of an operator, |
| 599 |
+ |
\begin{equation} |
| 600 |
+ |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 601 |
+ |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 602 |
+ |
\label{introEquation:exponentialOperator} |
| 603 |
+ |
\end{equation} |
| 604 |
+ |
|
| 605 |
+ |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
| 606 |
+ |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
| 607 |
+ |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
| 608 |
+ |
the Taylor series of $\psi_\tau$ agree to order $p$, |
| 609 |
+ |
\begin{equation} |
| 610 |
|
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 611 |
|
\end{equation} |
| 612 |
|
|
| 613 |
+ |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 614 |
+ |
|
| 615 |
|
The hidden geometric properties of ODE and its flow play important |
| 616 |
< |
roles in numerical studies. Let $\varphi$ be the flow of Hamiltonian |
| 617 |
< |
vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies, |
| 616 |
> |
roles in numerical studies. Many of them can be found in systems |
| 617 |
> |
which occur naturally in applications. |
| 618 |
> |
|
| 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. It is possible |
| 633 |
< |
to construct a \emph{volume-preserving} flow for a source free($ |
| 634 |
< |
\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi = |
| 635 |
< |
1$. Changing the variables $y = h(x)$ in a |
| 636 |
< |
ODE\ref{introEquation:ODE} will result in a new system, |
| 632 |
> |
is the property must be preserved by the integrator. |
| 633 |
> |
|
| 634 |
> |
It is possible to construct a \emph{volume-preserving} flow for a |
| 635 |
> |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
| 636 |
> |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
| 637 |
> |
be volume-preserving. |
| 638 |
> |
|
| 639 |
> |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
| 640 |
> |
will result in a new system, |
| 641 |
|
\[ |
| 642 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 643 |
|
\] |
| 644 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 645 |
|
In other words, the flow of this vector field is reversible if and |
| 646 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. When |
| 647 |
< |
designing any numerical methods, one should always try to preserve |
| 648 |
< |
the structural properties of the original ODE and its flow. |
| 646 |
> |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 647 |
> |
|
| 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} |
| 687 |
|
A lot of well established and very effective numerical methods have |
| 688 |
|
been successful precisely because of their symplecticities even |
| 698 |
|
\end{enumerate} |
| 699 |
|
|
| 700 |
|
Generating function tends to lead to methods which are cumbersome |
| 701 |
< |
and difficult to use\cite{}. In dissipative systems, variational |
| 702 |
< |
methods can capture the decay of energy accurately\cite{}. Since |
| 703 |
< |
their geometrically unstable nature against non-Hamiltonian |
| 704 |
< |
perturbations, ordinary implicit Runge-Kutta methods are not |
| 705 |
< |
suitable for Hamiltonian system. Recently, various high-order |
| 706 |
< |
explicit 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}. The main idea behind splitting |
| 712 |
< |
methods is to decompose the discrete $\varphi_h$ as a composition of |
| 713 |
< |
simpler flows, |
| 701 |
> |
and difficult to use. In dissipative systems, variational methods |
| 702 |
> |
can capture the decay of energy accurately. Since their |
| 703 |
> |
geometrically unstable nature against non-Hamiltonian perturbations, |
| 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. 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 |
> |
|
| 715 |
> |
The main idea behind splitting methods is to decompose the discrete |
| 716 |
> |
$\varphi_h$ as a composition of simpler flows, |
| 717 |
|
\begin{equation} |
| 718 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 719 |
|
\varphi _{h_n } |
| 720 |
|
\label{introEquation:FlowDecomposition} |
| 721 |
|
\end{equation} |
| 722 |
|
where each of the sub-flow is chosen such that each represent a |
| 723 |
< |
simpler integration of the system. Let $\phi$ and $\psi$ both be |
| 724 |
< |
symplectic maps, it is easy to show that any composition of |
| 725 |
< |
symplectic flows yields a symplectic map, |
| 723 |
> |
simpler integration of the system. |
| 724 |
> |
|
| 725 |
> |
Suppose that a Hamiltonian system takes the form, |
| 726 |
> |
\[ |
| 727 |
> |
H = H_1 + H_2. |
| 728 |
> |
\] |
| 729 |
> |
Here, $H_1$ and $H_2$ may represent different physical processes of |
| 730 |
> |
the system. For instance, they may relate to kinetic and potential |
| 731 |
> |
energy respectively, which is a natural decomposition of the |
| 732 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
| 733 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
| 734 |
> |
order is then given by the Lie-Trotter formula |
| 735 |
|
\begin{equation} |
| 736 |
+ |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 737 |
+ |
\label{introEquation:firstOrderSplitting} |
| 738 |
+ |
\end{equation} |
| 739 |
+ |
where $\varphi _h$ is the result of applying the corresponding |
| 740 |
+ |
continuous $\varphi _i$ over a time $h$. By definition, as |
| 741 |
+ |
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
| 742 |
+ |
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
| 743 |
+ |
It is easy to show that any composition of symplectic flows yields a |
| 744 |
+ |
symplectic map, |
| 745 |
+ |
\begin{equation} |
| 746 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 747 |
< |
'\phi ' = \phi '^T J\phi ' = J. |
| 747 |
> |
'\phi ' = \phi '^T J\phi ' = J, |
| 748 |
|
\label{introEquation:SymplecticFlowComposition} |
| 749 |
|
\end{equation} |
| 750 |
< |
Suppose that a Hamiltonian system has a form with $H = T + V$ |
| 750 |
> |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
| 751 |
> |
splitting in this context automatically generates a symplectic map. |
| 752 |
> |
|
| 753 |
> |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
| 754 |
> |
introduces local errors proportional to $h^2$, while Strang |
| 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} , \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 |
> |
\label{introEquation:timeReversible} |
| 766 |
> |
\end{equation} |
| 767 |
> |
|
| 768 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
| 769 |
> |
The classical equation for a system consisting of interacting |
| 770 |
> |
particles can be written in Hamiltonian form, |
| 771 |
> |
\[ |
| 772 |
> |
H = T + V |
| 773 |
> |
\] |
| 774 |
> |
where $T$ is the kinetic energy and $V$ is the potential energy. |
| 775 |
> |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
| 776 |
> |
obtains the following: |
| 777 |
> |
\begin{align} |
| 778 |
> |
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 779 |
> |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
| 780 |
> |
\label{introEquation:Lp10a} \\% |
| 781 |
> |
% |
| 782 |
> |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
| 783 |
> |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
| 784 |
> |
\label{introEquation:Lp10b} |
| 785 |
> |
\end{align} |
| 786 |
> |
where $F(t)$ is the force at time $t$. This integration scheme is |
| 787 |
> |
known as \emph{velocity verlet} which is |
| 788 |
> |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
| 789 |
> |
time-reversible(\ref{introEquation:timeReversible}) and |
| 790 |
> |
volume-preserving (\ref{introEquation:volumePreserving}). These |
| 791 |
> |
geometric properties attribute to its long-time stability and its |
| 792 |
> |
popularity in the community. However, the most commonly used |
| 793 |
> |
velocity verlet integration scheme is written as below, |
| 794 |
> |
\begin{align} |
| 795 |
> |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
| 796 |
> |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
| 797 |
> |
% |
| 798 |
> |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
| 799 |
> |
\label{introEquation:Lp9b}\\% |
| 800 |
> |
% |
| 801 |
> |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 802 |
> |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
| 803 |
> |
\end{align} |
| 804 |
> |
From the preceding splitting, one can see that the integration of |
| 805 |
> |
the equations of motion would follow: |
| 806 |
> |
\begin{enumerate} |
| 807 |
> |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
| 808 |
|
|
| 809 |
+ |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 810 |
+ |
|
| 811 |
+ |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 812 |
+ |
|
| 813 |
+ |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 814 |
+ |
\end{enumerate} |
| 815 |
+ |
|
| 816 |
+ |
Simply switching the order of splitting and composing, a new |
| 817 |
+ |
integrator, the \emph{position verlet} integrator, can be generated, |
| 818 |
+ |
\begin{align} |
| 819 |
+ |
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
| 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 |
| 824 |
+ |
q(\Delta t)} \right]. % |
| 825 |
+ |
\label{introEquation:positionVerlet1} |
| 826 |
+ |
\end{align} |
| 827 |
+ |
|
| 828 |
+ |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
| 829 |
+ |
|
| 830 |
+ |
Baker-Campbell-Hausdorff formula can be used to determine the local |
| 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 |
| 835 |
+ |
\begin{equation} |
| 836 |
+ |
\exp (hX + hY) = \exp (hZ) |
| 837 |
+ |
\end{equation} |
| 838 |
+ |
where |
| 839 |
+ |
\begin{equation} |
| 840 |
+ |
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
| 841 |
+ |
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
| 842 |
+ |
\end{equation} |
| 843 |
+ |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
| 844 |
+ |
\[ |
| 845 |
+ |
[X,Y] = XY - YX . |
| 846 |
+ |
\] |
| 847 |
+ |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
+ |
can obtain |
| 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 |
+ |
& & \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 |
| 858 |
+ |
\begin{equation} |
| 859 |
+ |
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 860 |
+ |
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 861 |
+ |
\end{equation} |
| 862 |
+ |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
| 863 |
+ |
order method. Yoshida proposed an elegant way to compose higher |
| 864 |
+ |
order methods based on symmetric splitting. Given a symmetric second |
| 865 |
+ |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
| 866 |
+ |
method can be constructed by composing, |
| 867 |
+ |
\[ |
| 868 |
+ |
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
| 869 |
+ |
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
| 870 |
+ |
\] |
| 871 |
+ |
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
| 872 |
+ |
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
| 873 |
+ |
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
| 874 |
+ |
\begin{equation} |
| 875 |
+ |
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
| 876 |
+ |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
| 877 |
+ |
\end{equation} |
| 878 |
+ |
, if the weights are chosen as |
| 879 |
+ |
\[ |
| 880 |
+ |
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
| 881 |
+ |
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
| 882 |
+ |
\] |
| 883 |
+ |
|
| 884 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 885 |
|
|
| 886 |
|
As a special discipline of molecular modeling, Molecular dynamics |
| 890 |
|
|
| 891 |
|
\subsection{\label{introSec:mdInit}Initialization} |
| 892 |
|
|
| 893 |
+ |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 894 |
+ |
|
| 895 |
|
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 896 |
|
|
| 897 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 898 |
|
|
| 899 |
< |
A rigid body is a body in which the distance between any two given |
| 900 |
< |
points of a rigid body remains constant regardless of external |
| 901 |
< |
forces exerted on it. A rigid body therefore conserves its shape |
| 902 |
< |
during its motion. |
| 899 |
> |
Rigid bodies are frequently involved in the modeling of different |
| 900 |
> |
areas, from engineering, physics, to chemistry. For example, |
| 901 |
> |
missiles and vehicle are usually modeled by rigid bodies. The |
| 902 |
> |
movement of the objects in 3D gaming engine or other physics |
| 903 |
> |
simulator is governed by the rigid body dynamics. In molecular |
| 904 |
> |
simulation, rigid body is used to simplify the model in |
| 905 |
> |
protein-protein docking study{\cite{Gray03}}. |
| 906 |
|
|
| 907 |
< |
Applications of dynamics of rigid bodies. |
| 907 |
> |
It is very important to develop stable and efficient methods to |
| 908 |
> |
integrate the equations of motion of orientational degrees of |
| 909 |
> |
freedom. Euler angles are the nature choice to describe the |
| 910 |
> |
rotational degrees of freedom. However, due to its singularity, the |
| 911 |
> |
numerical integration of corresponding equations of motion is very |
| 912 |
> |
inefficient and inaccurate. Although an alternative integrator using |
| 913 |
> |
different sets of Euler angles can overcome this difficulty\cite{}, |
| 914 |
> |
the computational penalty and the lost of angular momentum |
| 915 |
> |
conservation still remain. A singularity free representation |
| 916 |
> |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
| 917 |
> |
this approach suffer from the nonseparable Hamiltonian resulted from |
| 918 |
> |
quaternion representation, which prevents the symplectic algorithm |
| 919 |
> |
to be utilized. Another different approach is to apply holonomic |
| 920 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
| 921 |
> |
moves independently under the normal forces deriving from potential |
| 922 |
> |
energy and constraint forces which are used to guarantee the |
| 923 |
> |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
| 924 |
> |
algorithm converge very slowly when the number of constraint |
| 925 |
> |
increases. |
| 926 |
|
|
| 927 |
< |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 927 |
> |
The break through in geometric literature suggests that, in order to |
| 928 |
> |
develop a long-term integration scheme, one should preserve the |
| 929 |
> |
symplectic structure of the flow. Introducing conjugate momentum to |
| 930 |
> |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
| 931 |
> |
symplectic integrator, RSHAKE, was proposed to evolve the |
| 932 |
> |
Hamiltonian system in a constraint manifold by iteratively |
| 933 |
> |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
| 934 |
> |
method using quaternion representation was developed by Omelyan. |
| 935 |
> |
However, both of these methods are iterative and inefficient. In |
| 936 |
> |
this section, we will present a symplectic Lie-Poisson integrator |
| 937 |
> |
for rigid body developed by Dullweber and his |
| 938 |
> |
coworkers\cite{Dullweber1997} in depth. |
| 939 |
|
|
| 940 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 940 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 941 |
> |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
| 942 |
> |
function |
| 943 |
> |
\begin{equation} |
| 944 |
> |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 945 |
> |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 946 |
> |
\label{introEquation:RBHamiltonian} |
| 947 |
> |
\end{equation} |
| 948 |
> |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 949 |
> |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 950 |
> |
$J$, a diagonal matrix, is defined by |
| 951 |
> |
\[ |
| 952 |
> |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 953 |
> |
\] |
| 954 |
> |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 955 |
> |
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 956 |
> |
\begin{equation} |
| 957 |
> |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 958 |
> |
\end{equation} |
| 959 |
> |
which is used to ensure rotation matrix's orthogonality. |
| 960 |
> |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 961 |
> |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 962 |
> |
\begin{equation} |
| 963 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 964 |
> |
\label{introEquation:RBFirstOrderConstraint} |
| 965 |
> |
\end{equation} |
| 966 |
|
|
| 967 |
< |
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
| 967 |
> |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 968 |
> |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 969 |
> |
the equations of motion, |
| 970 |
> |
\[ |
| 971 |
> |
\begin{array}{c} |
| 972 |
> |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 973 |
> |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 974 |
> |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 975 |
> |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 976 |
> |
\end{array} |
| 977 |
> |
\] |
| 978 |
|
|
| 979 |
< |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
| 979 |
> |
In general, there are two ways to satisfy the holonomic constraints. |
| 980 |
> |
We can use constraint force provided by lagrange multiplier on the |
| 981 |
> |
normal manifold to keep the motion on constraint space. Or we can |
| 982 |
> |
simply evolve the system in constraint manifold. The two method are |
| 983 |
> |
proved to be equivalent. The holonomic constraint and equations of |
| 984 |
> |
motions define a constraint manifold for rigid body |
| 985 |
> |
\[ |
| 986 |
> |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 987 |
> |
\right\}. |
| 988 |
> |
\] |
| 989 |
|
|
| 990 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 990 |
> |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 991 |
> |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 992 |
> |
transformation, the cotangent space and the phase space are |
| 993 |
> |
diffeomorphic. Introducing |
| 994 |
> |
\[ |
| 995 |
> |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 996 |
> |
\] |
| 997 |
> |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 998 |
> |
can be re-formulated as a Hamiltonian system on the cotangent space |
| 999 |
> |
\[ |
| 1000 |
> |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1001 |
> |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1002 |
> |
\] |
| 1003 |
|
|
| 1004 |
< |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1004 |
> |
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1005 |
> |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1006 |
> |
given as |
| 1007 |
> |
\begin{equation} |
| 1008 |
> |
X_i^{lab} = Q X_i + q. |
| 1009 |
> |
\end{equation} |
| 1010 |
> |
Therefore, potential energy $V(q,Q)$ is defined by |
| 1011 |
> |
\[ |
| 1012 |
> |
V(q,Q) = V(Q X_0 + q). |
| 1013 |
> |
\] |
| 1014 |
> |
Hence, the force and torque are given by |
| 1015 |
> |
\[ |
| 1016 |
> |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, |
| 1017 |
> |
\] |
| 1018 |
> |
and |
| 1019 |
> |
\[ |
| 1020 |
> |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1021 |
> |
\] |
| 1022 |
> |
respectively. |
| 1023 |
|
|
| 1024 |
< |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 1024 |
> |
As a common choice to describe the rotation dynamics of the rigid |
| 1025 |
> |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1026 |
> |
rewrite the equations of motion, |
| 1027 |
> |
\begin{equation} |
| 1028 |
> |
\begin{array}{l} |
| 1029 |
> |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1030 |
> |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1031 |
> |
\end{array} |
| 1032 |
> |
\label{introEqaution:RBMotionPI} |
| 1033 |
> |
\end{equation} |
| 1034 |
> |
, as well as holonomic constraints, |
| 1035 |
> |
\[ |
| 1036 |
> |
\begin{array}{l} |
| 1037 |
> |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1038 |
> |
Q^T Q = 1 \\ |
| 1039 |
> |
\end{array} |
| 1040 |
> |
\] |
| 1041 |
> |
|
| 1042 |
> |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1043 |
> |
so(3)^ \star$, the hat-map isomorphism, |
| 1044 |
> |
\begin{equation} |
| 1045 |
> |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1046 |
> |
{\begin{array}{*{20}c} |
| 1047 |
> |
0 & { - v_3 } & {v_2 } \\ |
| 1048 |
> |
{v_3 } & 0 & { - v_1 } \\ |
| 1049 |
> |
{ - v_2 } & {v_1 } & 0 \\ |
| 1050 |
> |
\end{array}} \right), |
| 1051 |
> |
\label{introEquation:hatmapIsomorphism} |
| 1052 |
> |
\end{equation} |
| 1053 |
> |
will let us associate the matrix products with traditional vector |
| 1054 |
> |
operations |
| 1055 |
> |
\[ |
| 1056 |
> |
\hat vu = v \times u |
| 1057 |
> |
\] |
| 1058 |
> |
|
| 1059 |
> |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1060 |
> |
matrix, |
| 1061 |
> |
\begin{equation} |
| 1062 |
> |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1063 |
> |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1064 |
> |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1065 |
> |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1066 |
> |
\end{equation} |
| 1067 |
> |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1068 |
> |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1069 |
> |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1070 |
> |
unique property eliminate the requirement of iterations which can |
| 1071 |
> |
not be avoided in other methods\cite{}. |
| 1072 |
> |
|
| 1073 |
> |
Applying hat-map isomorphism, we obtain the equation of motion for |
| 1074 |
> |
angular momentum on body frame |
| 1075 |
> |
\begin{equation} |
| 1076 |
> |
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1077 |
> |
F_i (r,Q)} \right) \times X_i }. |
| 1078 |
> |
\label{introEquation:bodyAngularMotion} |
| 1079 |
> |
\end{equation} |
| 1080 |
> |
In the same manner, the equation of motion for rotation matrix is |
| 1081 |
> |
given by |
| 1082 |
> |
\[ |
| 1083 |
> |
\dot Q = Qskew(I^{ - 1} \pi ) |
| 1084 |
> |
\] |
| 1085 |
> |
|
| 1086 |
> |
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1087 |
> |
Lie-Poisson Integrator for Free Rigid Body} |
| 1088 |
> |
|
| 1089 |
> |
If there is not external forces exerted on the rigid body, the only |
| 1090 |
> |
contribution to the rotational is from the kinetic potential (the |
| 1091 |
> |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
| 1092 |
> |
rigid body is an example of Lie-Poisson system with Hamiltonian |
| 1093 |
> |
function |
| 1094 |
> |
\begin{equation} |
| 1095 |
> |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1096 |
> |
\label{introEquation:rotationalKineticRB} |
| 1097 |
> |
\end{equation} |
| 1098 |
> |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
| 1099 |
> |
Lie-Poisson structure matrix, |
| 1100 |
> |
\begin{equation} |
| 1101 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 1102 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1103 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1104 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1105 |
> |
\end{array}} \right) |
| 1106 |
> |
\end{equation} |
| 1107 |
> |
Thus, the dynamics of free rigid body is governed by |
| 1108 |
> |
\begin{equation} |
| 1109 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1110 |
> |
\end{equation} |
| 1111 |
> |
|
| 1112 |
> |
One may notice that each $T_i^r$ in Equation |
| 1113 |
> |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1114 |
> |
instance, the equations of motion due to $T_1^r$ are given by |
| 1115 |
> |
\begin{equation} |
| 1116 |
> |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
| 1117 |
> |
\label{introEqaution:RBMotionSingleTerm} |
| 1118 |
> |
\end{equation} |
| 1119 |
> |
where |
| 1120 |
> |
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1121 |
> |
0 & 0 & 0 \\ |
| 1122 |
> |
0 & 0 & {\pi _1 } \\ |
| 1123 |
> |
0 & { - \pi _1 } & 0 \\ |
| 1124 |
> |
\end{array}} \right). |
| 1125 |
> |
\] |
| 1126 |
> |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1127 |
> |
\[ |
| 1128 |
> |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
| 1129 |
> |
Q(0)e^{\Delta tR_1 } |
| 1130 |
> |
\] |
| 1131 |
> |
with |
| 1132 |
> |
\[ |
| 1133 |
> |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
| 1134 |
> |
0 & 0 & 0 \\ |
| 1135 |
> |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
| 1136 |
> |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1137 |
> |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1138 |
> |
\] |
| 1139 |
> |
To reduce the cost of computing expensive functions in e^{\Delta |
| 1140 |
> |
tR_1 }, we can use Cayley transformation, |
| 1141 |
> |
\[ |
| 1142 |
> |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1143 |
> |
) |
| 1144 |
> |
\] |
| 1145 |
> |
|
| 1146 |
> |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
| 1147 |
> |
manner. |
| 1148 |
> |
|
| 1149 |
> |
In order to construct a second-order symplectic method, we split the |
| 1150 |
> |
angular kinetic Hamiltonian function can into five terms |
| 1151 |
> |
\[ |
| 1152 |
> |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1153 |
> |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1154 |
> |
(\pi _1 ) |
| 1155 |
> |
\]. |
| 1156 |
> |
Concatenating flows corresponding to these five terms, we can obtain |
| 1157 |
> |
an symplectic integrator, |
| 1158 |
> |
\[ |
| 1159 |
> |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1160 |
> |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1161 |
> |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1162 |
> |
_1 }. |
| 1163 |
> |
\] |
| 1164 |
> |
|
| 1165 |
> |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1166 |
> |
$F(\pi )$ and $G(\pi )$ is defined by |
| 1167 |
> |
\[ |
| 1168 |
> |
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1169 |
> |
) |
| 1170 |
> |
\] |
| 1171 |
> |
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1172 |
> |
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1173 |
> |
conserved quantity in Poisson system. We can easily verify that the |
| 1174 |
> |
norm of the angular momentum, $\parallel \pi |
| 1175 |
> |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1176 |
> |
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1177 |
> |
then by the chain rule |
| 1178 |
> |
\[ |
| 1179 |
> |
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1180 |
> |
}}{2})\pi |
| 1181 |
> |
\] |
| 1182 |
> |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1183 |
> |
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1184 |
> |
Lie-Poisson integrator is found to be extremely efficient and stable |
| 1185 |
> |
which can be explained by the fact the small angle approximation is |
| 1186 |
> |
used and the norm of the angular momentum is conserved. |
| 1187 |
> |
|
| 1188 |
> |
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
| 1189 |
> |
Splitting for Rigid Body} |
| 1190 |
> |
|
| 1191 |
> |
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1192 |
> |
energy and potential energy, |
| 1193 |
> |
\[ |
| 1194 |
> |
H = T(p,\pi ) + V(q,Q) |
| 1195 |
> |
\] |
| 1196 |
> |
The equations of motion corresponding to potential energy and |
| 1197 |
> |
kinetic energy are listed in the below table, |
| 1198 |
> |
\begin{center} |
| 1199 |
> |
\begin{tabular}{|l|l|} |
| 1200 |
> |
\hline |
| 1201 |
> |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 1202 |
> |
Potential & Kinetic \\ |
| 1203 |
> |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
| 1204 |
> |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
| 1205 |
> |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
| 1206 |
> |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
| 1207 |
> |
\hline |
| 1208 |
> |
\end{tabular} |
| 1209 |
> |
\end{center} |
| 1210 |
> |
A second-order symplectic method is now obtained by the composition |
| 1211 |
> |
of the flow maps, |
| 1212 |
> |
\[ |
| 1213 |
> |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1214 |
> |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1215 |
> |
\] |
| 1216 |
> |
Moreover, \varphi _{\Delta t/2,V} can be divided into two sub-flows |
| 1217 |
> |
which corresponding to force and torque respectively, |
| 1218 |
> |
\[ |
| 1219 |
> |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1220 |
> |
_{\Delta t/2,\tau }. |
| 1221 |
> |
\] |
| 1222 |
> |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1223 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 1224 |
> |
order inside \varphi _{\Delta t/2,V} does not matter. |
| 1225 |
> |
|
| 1226 |
> |
Furthermore, kinetic potential can be separated to translational |
| 1227 |
> |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1228 |
> |
\begin{equation} |
| 1229 |
> |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1230 |
> |
\end{equation} |
| 1231 |
> |
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1232 |
> |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1233 |
> |
corresponding flow maps are given by |
| 1234 |
> |
\[ |
| 1235 |
> |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1236 |
> |
_{\Delta t,T^r }. |
| 1237 |
> |
\] |
| 1238 |
> |
Finally, we obtain the overall symplectic flow maps for free moving |
| 1239 |
> |
rigid body |
| 1240 |
> |
\begin{equation} |
| 1241 |
> |
\begin{array}{c} |
| 1242 |
> |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1243 |
> |
\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 } \\ |
| 1244 |
> |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1245 |
> |
\end{array} |
| 1246 |
> |
\label{introEquation:overallRBFlowMaps} |
| 1247 |
> |
\end{equation} |
| 1248 |
> |
|
| 1249 |
> |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1250 |
> |
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1251 |
> |
mimics a simple heat bath with stochastic and dissipative forces, |
| 1252 |
> |
has been applied in a variety of studies. This section will review |
| 1253 |
> |
the theory of Langevin dynamics simulation. A brief derivation of |
| 1254 |
> |
generalized Langevin Dynamics will be given first. Follow that, we |
| 1255 |
> |
will discuss the physical meaning of the terms appearing in the |
| 1256 |
> |
equation as well as the calculation of friction tensor from |
| 1257 |
> |
hydrodynamics theory. |
| 1258 |
|
|
| 1259 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 1260 |
|
|
| 1300 |
|
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 1301 |
|
&= m\ddot x |
| 1302 |
|
&= - \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)} |
| 1303 |
< |
\label{introEq:Lp5} |
| 1303 |
> |
\label{introEquation:Lp5} |
| 1304 |
|
\end{align} |
| 1305 |
|
, and |
| 1306 |
|
\begin{align} |
| 1448 |
|
\label{introEquation:secondFluctuationDissipation} |
| 1449 |
|
\end{equation} |
| 1450 |
|
|
| 880 |
– |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
| 881 |
– |
|
| 1451 |
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1452 |
< |
\subsection{\label{introSection:analyticalApproach}Analytical |
| 1453 |
< |
Approach} |
| 1452 |
> |
Theoretically, the friction kernel can be determined using velocity |
| 1453 |
> |
autocorrelation function. However, this approach become impractical |
| 1454 |
> |
when the system become more and more complicate. Instead, various |
| 1455 |
> |
approaches based on hydrodynamics have been developed to calculate |
| 1456 |
> |
the friction coefficients. The friction effect is isotropic in |
| 1457 |
> |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1458 |
> |
tensor \Xi is a $6\times 6$ matrix given by |
| 1459 |
> |
\[ |
| 1460 |
> |
\Xi = \left( {\begin{array}{*{20}c} |
| 1461 |
> |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1462 |
> |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 1463 |
> |
\end{array}} \right). |
| 1464 |
> |
\] |
| 1465 |
> |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 1466 |
> |
tensor and rotational friction tensor respectively, while ${\Xi^{tr} |
| 1467 |
> |
}$ is translation-rotation coupling tensor and $ {\Xi^{rt} }$ is |
| 1468 |
> |
rotation-translation coupling tensor. |
| 1469 |
|
|
| 1470 |
< |
\subsection{\label{introSection:approximationApproach}Approximation |
| 1471 |
< |
Approach} |
| 1470 |
> |
\[ |
| 1471 |
> |
\left( \begin{array}{l} |
| 1472 |
> |
F_t \\ |
| 1473 |
> |
\tau \\ |
| 1474 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1475 |
> |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1476 |
> |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1477 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 1478 |
> |
v \\ |
| 1479 |
> |
w \\ |
| 1480 |
> |
\end{array} \right) |
| 1481 |
> |
\] |
| 1482 |
|
|
| 1483 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1484 |
< |
Body} |
| 1483 |
> |
\subsubsection{\label{introSection:analyticalApproach}The Friction Tensor for Regular Shape} |
| 1484 |
> |
For a spherical particle, the translational and rotational friction |
| 1485 |
> |
constant can be calculated from Stoke's law, |
| 1486 |
> |
\[ |
| 1487 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1488 |
> |
{6\pi \eta R} & 0 & 0 \\ |
| 1489 |
> |
0 & {6\pi \eta R} & 0 \\ |
| 1490 |
> |
0 & 0 & {6\pi \eta R} \\ |
| 1491 |
> |
\end{array}} \right) |
| 1492 |
> |
\] |
| 1493 |
> |
and |
| 1494 |
> |
\[ |
| 1495 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1496 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1497 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1498 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1499 |
> |
\end{array}} \right) |
| 1500 |
> |
\] |
| 1501 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1502 |
> |
hydrodynamics radius. |
| 1503 |
> |
|
| 1504 |
> |
Other non-spherical particles have more complex properties. |
| 1505 |
> |
|
| 1506 |
> |
\[ |
| 1507 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1508 |
> |
} }}{b} |
| 1509 |
> |
\] |
| 1510 |
> |
|
| 1511 |
> |
|
| 1512 |
> |
\[ |
| 1513 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1514 |
> |
}}{a} |
| 1515 |
> |
\] |
| 1516 |
> |
|
| 1517 |
> |
\[ |
| 1518 |
> |
\begin{array}{l} |
| 1519 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1520 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1521 |
> |
\end{array} |
| 1522 |
> |
\] |
| 1523 |
> |
|
| 1524 |
> |
\[ |
| 1525 |
> |
\begin{array}{l} |
| 1526 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1527 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1528 |
> |
\end{array} |
| 1529 |
> |
\] |
| 1530 |
> |
|
| 1531 |
> |
|
| 1532 |
> |
\subsubsection{\label{introSection:approximationApproach}The Friction Tensor for Arbitrary Shape} |
| 1533 |
> |
Unlike spherical and other regular shaped molecules, there is not |
| 1534 |
> |
analytical solution for friction tensor of any arbitrary shaped |
| 1535 |
> |
rigid molecules. The ellipsoid of revolution model and general |
| 1536 |
> |
triaxial ellipsoid model have been used to approximate the |
| 1537 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1538 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1539 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
| 1540 |
> |
unique\cite{Wegener79} as well as the intrinsic coupling between |
| 1541 |
> |
translational and rotational motion of rigid body\cite{}, general |
| 1542 |
> |
ellipsoid is not always suitable for modeling arbitrarily shaped |
| 1543 |
> |
rigid molecule. A number of studies have been devoted to determine |
| 1544 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
| 1545 |
> |
advanced method\cite{} where the molecule of interest was modeled by |
| 1546 |
> |
combinations of spheres(beads)\cite{} and the hydrodynamics |
| 1547 |
> |
properties of the molecule can be calculated using the hydrodynamic |
| 1548 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 1549 |
> |
immersed in a continuous medium. Due to hydrodynamics interaction, |
| 1550 |
> |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 1551 |
> |
unperturbed velocity $v_i$, |
| 1552 |
> |
\[ |
| 1553 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1554 |
> |
\] |
| 1555 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1556 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1557 |
> |
proportional to its ``net'' velocity |
| 1558 |
> |
\begin{equation} |
| 1559 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1560 |
> |
\label{introEquation:tensorExpression} |
| 1561 |
> |
\end{equation} |
| 1562 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1563 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1564 |
> |
\ref{introEquation:tensorExpression} |
| 1565 |
> |
\begin{equation} |
| 1566 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1567 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1568 |
> |
\label{introEquation:oseenTensor} |
| 1569 |
> |
\end{equation} |
| 1570 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1571 |
> |
A second order expression for element of different size was |
| 1572 |
> |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
| 1573 |
> |
la Torre and Bloomfield, |
| 1574 |
> |
\begin{equation} |
| 1575 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1576 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1577 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1578 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1579 |
> |
\label{introEquation:RPTensorNonOverlapped} |
| 1580 |
> |
\end{equation} |
| 1581 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1582 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1583 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1584 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
| 1585 |
> |
\begin{equation} |
| 1586 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1587 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1588 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1589 |
> |
\label{introEquation:RPTensorOverlapped} |
| 1590 |
> |
\end{equation} |
| 1591 |
> |
|
| 1592 |
> |
%Bead Modeling |
| 1593 |
> |
|
| 1594 |
> |
\[ |
| 1595 |
> |
B = \left( {\begin{array}{*{20}c} |
| 1596 |
> |
{T_{11} } & \ldots & {T_{1N} } \\ |
| 1597 |
> |
\vdots & \ddots & \vdots \\ |
| 1598 |
> |
{T_{N1} } & \cdots & {T_{NN} } \\ |
| 1599 |
> |
\end{array}} \right) |
| 1600 |
> |
\] |
| 1601 |
> |
|
| 1602 |
> |
\[ |
| 1603 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1604 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1605 |
> |
\vdots & \ddots & \vdots \\ |
| 1606 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1607 |
> |
\end{array}} \right) |
| 1608 |
> |
\] |
| 1609 |
> |
|
| 1610 |
> |
\begin{equation} |
| 1611 |
> |
\begin{array}{l} |
| 1612 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1613 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1614 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1615 |
> |
\end{array} |
| 1616 |
> |
\end{equation} |
| 1617 |
> |
where |
| 1618 |
> |
\[ |
| 1619 |
> |
U_i = \left( {\begin{array}{*{20}c} |
| 1620 |
> |
0 & { - z_i } & {y_i } \\ |
| 1621 |
> |
{z_i } & 0 & { - x_i } \\ |
| 1622 |
> |
{ - y_i } & {x_i } & 0 \\ |
| 1623 |
> |
\end{array}} \right) |
| 1624 |
> |
\] |
| 1625 |
> |
|
| 1626 |
> |
\[ |
| 1627 |
> |
r_{OR} = \left( \begin{array}{l} |
| 1628 |
> |
x_{OR} \\ |
| 1629 |
> |
y_{OR} \\ |
| 1630 |
> |
z_{OR} \\ |
| 1631 |
> |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
| 1632 |
> |
{\Xi _{yy}^{rr} + \Xi _{zz}^{rr} } & { - \Xi _{xy}^{rr} } & { - \Xi _{xz}^{rr} } \\ |
| 1633 |
> |
{ - \Xi _{yx}^{rr} } & {\Xi _{zz}^{rr} + \Xi _{xx}^{rr} } & { - \Xi _{yz}^{rr} } \\ |
| 1634 |
> |
{ - \Xi _{zx}^{rr} } & { - \Xi _{yz}^{rr} } & {\Xi _{xx}^{rr} + \Xi _{yy}^{rr} } \\ |
| 1635 |
> |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
| 1636 |
> |
\Xi _{yz}^{tr} - \Xi _{zy}^{tr} \\ |
| 1637 |
> |
\Xi _{zx}^{tr} - \Xi _{xz}^{tr} \\ |
| 1638 |
> |
\Xi _{xy}^{tr} - \Xi _{yx}^{tr} \\ |
| 1639 |
> |
\end{array} \right) |
| 1640 |
> |
\] |
| 1641 |
> |
|
| 1642 |
> |
\[ |
| 1643 |
> |
U_{OR} = \left( {\begin{array}{*{20}c} |
| 1644 |
> |
0 & { - z_{OR} } & {y_{OR} } \\ |
| 1645 |
> |
{z_i } & 0 & { - x_{OR} } \\ |
| 1646 |
> |
{ - y_{OR} } & {x_{OR} } & 0 \\ |
| 1647 |
> |
\end{array}} \right) |
| 1648 |
> |
\] |
| 1649 |
> |
|
| 1650 |
> |
\[ |
| 1651 |
> |
\begin{array}{l} |
| 1652 |
> |
\Xi _R^{tt} = \Xi _{}^{tt} \\ |
| 1653 |
> |
\Xi _R^{tr} = \Xi _R^{rt} = \Xi _{}^{tr} - U_{OR} \Xi _{}^{tt} \\ |
| 1654 |
> |
\Xi _R^{rr} = \Xi _{}^{rr} - U_{OR} \Xi _{}^{tt} U_{OR} + \Xi _{}^{tr} U_{OR} - U_{OR} \Xi _{}^{tr} ^{^T } \\ |
| 1655 |
> |
\end{array} |
| 1656 |
> |
\] |
| 1657 |
> |
|
| 1658 |
> |
\[ |
| 1659 |
> |
D_R = \left( {\begin{array}{*{20}c} |
| 1660 |
> |
{D_R^{tt} } & {D_R^{rt} } \\ |
| 1661 |
> |
{D_R^{tr} } & {D_R^{rr} } \\ |
| 1662 |
> |
\end{array}} \right) = k_b T\left( {\begin{array}{*{20}c} |
| 1663 |
> |
{\Xi _R^{tt} } & {\Xi _R^{rt} } \\ |
| 1664 |
> |
{\Xi _R^{tr} } & {\Xi _R^{rr} } \\ |
| 1665 |
> |
\end{array}} \right)^{ - 1} |
| 1666 |
> |
\] |
| 1667 |
> |
|
| 1668 |
> |
|
| 1669 |
> |
%Approximation Methods |
| 1670 |
> |
|
| 1671 |
> |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
