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 |
|
|
394 |
|
\begin{equation} |
395 |
|
\rho \propto e^{ - \beta H} |
396 |
|
\label{introEquation:densityAndHamiltonian} |
397 |
+ |
\end{equation} |
398 |
+ |
|
399 |
+ |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
400 |
+ |
Lets consider a region in the phase space, |
401 |
+ |
\begin{equation} |
402 |
+ |
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
403 |
+ |
\end{equation} |
404 |
+ |
If this region is small enough, the density $\rho$ can be regarded |
405 |
+ |
as uniform over the whole phase space. Thus, the number of phase |
406 |
+ |
points inside this region is given by, |
407 |
+ |
\begin{equation} |
408 |
+ |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
409 |
+ |
dp_1 } ..dp_f. |
410 |
|
\end{equation} |
411 |
|
|
412 |
+ |
\begin{equation} |
413 |
+ |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
414 |
+ |
\frac{d}{{dt}}(\delta v) = 0. |
415 |
+ |
\end{equation} |
416 |
+ |
With the help of stationary assumption |
417 |
+ |
(\ref{introEquation:stationary}), we obtain the principle of the |
418 |
+ |
\emph{conservation of extension in phase space}, |
419 |
+ |
\begin{equation} |
420 |
+ |
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
421 |
+ |
...dq_f dp_1 } ..dp_f = 0. |
422 |
+ |
\label{introEquation:volumePreserving} |
423 |
+ |
\end{equation} |
424 |
+ |
|
425 |
+ |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
426 |
+ |
|
427 |
|
Liouville's theorem can be expresses in a variety of different forms |
428 |
|
which are convenient within different contexts. For any two function |
429 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
459 |
|
\label{introEquation:liouvilleTheoremInOperator} |
460 |
|
\end{equation} |
461 |
|
|
434 |
– |
|
462 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
463 |
|
|
464 |
|
Various thermodynamic properties can be calculated from Molecular |
587 |
|
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
588 |
|
\end{equation} |
589 |
|
|
590 |
< |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
590 |
> |
\subsection{\label{introSection:exactFlow}Exact Flow} |
591 |
> |
|
592 |
|
Let $x(t)$ be the exact solution of the ODE system, |
593 |
|
\begin{equation} |
594 |
|
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
598 |
|
x(t+\tau) =\varphi_\tau(x(t)) |
599 |
|
\] |
600 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
601 |
< |
space to itself. In most cases, it is not easy to find the exact |
602 |
< |
flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$, |
603 |
< |
which is usually called integrator. The order of an integrator |
604 |
< |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
605 |
< |
order $p$, |
601 |
> |
space to itself. The flow has the continuous group property, |
602 |
> |
\begin{equation} |
603 |
> |
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
604 |
> |
+ \tau _2 } . |
605 |
> |
\end{equation} |
606 |
> |
In particular, |
607 |
> |
\begin{equation} |
608 |
> |
\varphi _\tau \circ \varphi _{ - \tau } = I |
609 |
> |
\end{equation} |
610 |
> |
Therefore, the exact flow is self-adjoint, |
611 |
|
\begin{equation} |
612 |
+ |
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
613 |
+ |
\end{equation} |
614 |
+ |
The exact flow can also be written in terms of the of an operator, |
615 |
+ |
\begin{equation} |
616 |
+ |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
617 |
+ |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
618 |
+ |
\label{introEquation:exponentialOperator} |
619 |
+ |
\end{equation} |
620 |
+ |
|
621 |
+ |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
622 |
+ |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
623 |
+ |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
624 |
+ |
the Taylor series of $\psi_\tau$ agree to order $p$, |
625 |
+ |
\begin{equation} |
626 |
|
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
627 |
|
\end{equation} |
628 |
|
|
629 |
+ |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
630 |
+ |
|
631 |
|
The hidden geometric properties of ODE and its flow play important |
632 |
< |
roles in numerical studies. Let $\varphi$ be the flow of Hamiltonian |
633 |
< |
vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies, |
632 |
> |
roles in numerical studies. Many of them can be found in systems |
633 |
> |
which occur naturally in applications. |
634 |
> |
|
635 |
> |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
636 |
> |
a \emph{symplectic} flow if it satisfies, |
637 |
|
\begin{equation} |
638 |
< |
'\varphi^T J '\varphi = J. |
638 |
> |
{\varphi '}^T J \varphi ' = J. |
639 |
|
\end{equation} |
640 |
|
According to Liouville's theorem, the symplectic volume is invariant |
641 |
|
under a Hamiltonian flow, which is the basis for classical |
643 |
|
field on a symplectic manifold can be shown to be a |
644 |
|
symplectomorphism. As to the Poisson system, |
645 |
|
\begin{equation} |
646 |
< |
'\varphi ^T J '\varphi = J \circ \varphi |
646 |
> |
{\varphi '}^T J \varphi ' = J \circ \varphi |
647 |
|
\end{equation} |
648 |
< |
is the property must be preserved by the integrator. It is possible |
649 |
< |
to construct a \emph{volume-preserving} flow for a source free($ |
650 |
< |
\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi = |
651 |
< |
1$. Changing the variables $y = h(x)$ in a |
652 |
< |
ODE\ref{introEquation:ODE} will result in a new system, |
648 |
> |
is the property must be preserved by the integrator. |
649 |
> |
|
650 |
> |
It is possible to construct a \emph{volume-preserving} flow for a |
651 |
> |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
652 |
> |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
653 |
> |
be volume-preserving. |
654 |
> |
|
655 |
> |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
656 |
> |
will result in a new system, |
657 |
|
\[ |
658 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
659 |
|
\] |
660 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
661 |
|
In other words, the flow of this vector field is reversible if and |
662 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. When |
607 |
< |
designing any numerical methods, one should always try to preserve |
608 |
< |
the structural properties of the original ODE and its flow. |
662 |
> |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
663 |
|
|
664 |
+ |
A \emph{first integral}, or conserved quantity of a general |
665 |
+ |
differential function is a function $ G:R^{2d} \to R^d $ which is |
666 |
+ |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
667 |
+ |
\[ |
668 |
+ |
\frac{{dG(x(t))}}{{dt}} = 0. |
669 |
+ |
\] |
670 |
+ |
Using chain rule, one may obtain, |
671 |
+ |
\[ |
672 |
+ |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
673 |
+ |
\] |
674 |
+ |
which is the condition for conserving \emph{first integral}. For a |
675 |
+ |
canonical Hamiltonian system, the time evolution of an arbitrary |
676 |
+ |
smooth function $G$ is given by, |
677 |
+ |
\begin{equation} |
678 |
+ |
\begin{array}{c} |
679 |
+ |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
680 |
+ |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
681 |
+ |
\end{array} |
682 |
+ |
\label{introEquation:firstIntegral1} |
683 |
+ |
\end{equation} |
684 |
+ |
Using poisson bracket notion, Equation |
685 |
+ |
\ref{introEquation:firstIntegral1} can be rewritten as |
686 |
+ |
\[ |
687 |
+ |
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
688 |
+ |
\] |
689 |
+ |
Therefore, the sufficient condition for $G$ to be the \emph{first |
690 |
+ |
integral} of a Hamiltonian system is |
691 |
+ |
\[ |
692 |
+ |
\left\{ {G,H} \right\} = 0. |
693 |
+ |
\] |
694 |
+ |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
695 |
+ |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
696 |
+ |
0$. |
697 |
+ |
|
698 |
+ |
|
699 |
+ |
When designing any numerical methods, one should always try to |
700 |
+ |
preserve the structural properties of the original ODE and its flow. |
701 |
+ |
|
702 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
703 |
|
A lot of well established and very effective numerical methods have |
704 |
|
been successful precisely because of their symplecticities even |
714 |
|
\end{enumerate} |
715 |
|
|
716 |
|
Generating function tends to lead to methods which are cumbersome |
717 |
< |
and difficult to use\cite{}. In dissipative systems, variational |
718 |
< |
methods can capture the decay of energy accurately\cite{}. Since |
719 |
< |
their geometrically unstable nature against non-Hamiltonian |
720 |
< |
perturbations, ordinary implicit Runge-Kutta methods are not |
721 |
< |
suitable for Hamiltonian system. Recently, various high-order |
722 |
< |
explicit Runge--Kutta methods have been developed to overcome this |
723 |
< |
instability \cite{}. However, due to computational penalty involved |
724 |
< |
in implementing the Runge-Kutta methods, they do not attract too |
725 |
< |
much attention from Molecular Dynamics community. Instead, splitting |
726 |
< |
have been widely accepted since they exploit natural decompositions |
727 |
< |
of the system\cite{Tuckerman92}. The main idea behind splitting |
728 |
< |
methods is to decompose the discrete $\varphi_h$ as a composition of |
729 |
< |
simpler flows, |
717 |
> |
and difficult to use. In dissipative systems, variational methods |
718 |
> |
can capture the decay of energy accurately. Since their |
719 |
> |
geometrically unstable nature against non-Hamiltonian perturbations, |
720 |
> |
ordinary implicit Runge-Kutta methods are not suitable for |
721 |
> |
Hamiltonian system. Recently, various high-order explicit |
722 |
> |
Runge--Kutta methods have been developed to overcome this |
723 |
> |
instability. However, due to computational penalty involved in |
724 |
> |
implementing the Runge-Kutta methods, they do not attract too much |
725 |
> |
attention from Molecular Dynamics community. Instead, splitting have |
726 |
> |
been widely accepted since they exploit natural decompositions of |
727 |
> |
the system\cite{Tuckerman92}. |
728 |
> |
|
729 |
> |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
730 |
> |
|
731 |
> |
The main idea behind splitting methods is to decompose the discrete |
732 |
> |
$\varphi_h$ as a composition of simpler flows, |
733 |
|
\begin{equation} |
734 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
735 |
|
\varphi _{h_n } |
736 |
|
\label{introEquation:FlowDecomposition} |
737 |
|
\end{equation} |
738 |
|
where each of the sub-flow is chosen such that each represent a |
739 |
< |
simpler integration of the system. Let $\phi$ and $\psi$ both be |
740 |
< |
symplectic maps, it is easy to show that any composition of |
741 |
< |
symplectic flows yields a symplectic map, |
739 |
> |
simpler integration of the system. |
740 |
> |
|
741 |
> |
Suppose that a Hamiltonian system takes the form, |
742 |
> |
\[ |
743 |
> |
H = H_1 + H_2. |
744 |
> |
\] |
745 |
> |
Here, $H_1$ and $H_2$ may represent different physical processes of |
746 |
> |
the system. For instance, they may relate to kinetic and potential |
747 |
> |
energy respectively, which is a natural decomposition of the |
748 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
749 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
750 |
> |
order is then given by the Lie-Trotter formula |
751 |
> |
\begin{equation} |
752 |
> |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
753 |
> |
\label{introEquation:firstOrderSplitting} |
754 |
> |
\end{equation} |
755 |
> |
where $\varphi _h$ is the result of applying the corresponding |
756 |
> |
continuous $\varphi _i$ over a time $h$. By definition, as |
757 |
> |
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
758 |
> |
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
759 |
> |
It is easy to show that any composition of symplectic flows yields a |
760 |
> |
symplectic map, |
761 |
|
\begin{equation} |
762 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
763 |
< |
'\phi ' = \phi '^T J\phi ' = J. |
763 |
> |
'\phi ' = \phi '^T J\phi ' = J, |
764 |
|
\label{introEquation:SymplecticFlowComposition} |
765 |
|
\end{equation} |
766 |
< |
Suppose that a Hamiltonian system has a form with $H = T + V$ |
766 |
> |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
767 |
> |
splitting in this context automatically generates a symplectic map. |
768 |
> |
|
769 |
> |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
770 |
> |
introduces local errors proportional to $h^2$, while Strang |
771 |
> |
splitting gives a second-order decomposition, |
772 |
> |
\begin{equation} |
773 |
> |
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
774 |
> |
_{1,h/2} , |
775 |
> |
\label{introEqaution:secondOrderSplitting} |
776 |
> |
\end{equation} |
777 |
> |
which has a local error proportional to $h^3$. Sprang splitting's |
778 |
> |
popularity in molecular simulation community attribute to its |
779 |
> |
symmetric property, |
780 |
> |
\begin{equation} |
781 |
> |
\varphi _h^{ - 1} = \varphi _{ - h}. |
782 |
> |
\label{introEquation:timeReversible} |
783 |
> |
\end{equation} |
784 |
> |
|
785 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
786 |
> |
The classical equation for a system consisting of interacting |
787 |
> |
particles can be written in Hamiltonian form, |
788 |
> |
\[ |
789 |
> |
H = T + V |
790 |
> |
\] |
791 |
> |
where $T$ is the kinetic energy and $V$ is the potential energy. |
792 |
> |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
793 |
> |
obtains the following: |
794 |
> |
\begin{align} |
795 |
> |
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
796 |
> |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
797 |
> |
\label{introEquation:Lp10a} \\% |
798 |
> |
% |
799 |
> |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
800 |
> |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
801 |
> |
\label{introEquation:Lp10b} |
802 |
> |
\end{align} |
803 |
> |
where $F(t)$ is the force at time $t$. This integration scheme is |
804 |
> |
known as \emph{velocity verlet} which is |
805 |
> |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
806 |
> |
time-reversible(\ref{introEquation:timeReversible}) and |
807 |
> |
volume-preserving (\ref{introEquation:volumePreserving}). These |
808 |
> |
geometric properties attribute to its long-time stability and its |
809 |
> |
popularity in the community. However, the most commonly used |
810 |
> |
velocity verlet integration scheme is written as below, |
811 |
> |
\begin{align} |
812 |
> |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
813 |
> |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
814 |
> |
% |
815 |
> |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
816 |
> |
\label{introEquation:Lp9b}\\% |
817 |
> |
% |
818 |
> |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
819 |
> |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
820 |
> |
\end{align} |
821 |
> |
From the preceding splitting, one can see that the integration of |
822 |
> |
the equations of motion would follow: |
823 |
> |
\begin{enumerate} |
824 |
> |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
825 |
> |
|
826 |
> |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
827 |
> |
|
828 |
> |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
829 |
> |
|
830 |
> |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
831 |
> |
\end{enumerate} |
832 |
> |
|
833 |
> |
Simply switching the order of splitting and composing, a new |
834 |
> |
integrator, the \emph{position verlet} integrator, can be generated, |
835 |
> |
\begin{align} |
836 |
> |
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
837 |
> |
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
838 |
> |
\label{introEquation:positionVerlet1} \\% |
839 |
> |
% |
840 |
> |
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
841 |
> |
q(\Delta t)} \right]. % |
842 |
> |
\label{introEquation:positionVerlet1} |
843 |
> |
\end{align} |
844 |
> |
|
845 |
> |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
846 |
> |
|
847 |
> |
Baker-Campbell-Hausdorff formula can be used to determine the local |
848 |
> |
error of splitting method in terms of commutator of the |
849 |
> |
operators(\ref{introEquation:exponentialOperator}) associated with |
850 |
> |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
851 |
> |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
852 |
> |
\begin{equation} |
853 |
> |
\exp (hX + hY) = \exp (hZ) |
854 |
> |
\end{equation} |
855 |
> |
where |
856 |
> |
\begin{equation} |
857 |
> |
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
858 |
> |
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
859 |
> |
\end{equation} |
860 |
> |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
861 |
> |
\[ |
862 |
> |
[X,Y] = XY - YX . |
863 |
> |
\] |
864 |
> |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
865 |
> |
can obtain |
866 |
> |
\begin{eqnarray*} |
867 |
> |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
868 |
> |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
869 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
870 |
> |
\ldots ) |
871 |
> |
\end{eqnarray*} |
872 |
> |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
873 |
> |
error of Spring splitting is proportional to $h^3$. The same |
874 |
> |
procedure can be applied to general splitting, of the form |
875 |
> |
\begin{equation} |
876 |
> |
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
877 |
> |
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
878 |
> |
\end{equation} |
879 |
> |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
880 |
> |
order method. Yoshida proposed an elegant way to compose higher |
881 |
> |
order methods based on symmetric splitting. Given a symmetric second |
882 |
> |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
883 |
> |
method can be constructed by composing, |
884 |
> |
\[ |
885 |
> |
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
886 |
> |
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
887 |
> |
\] |
888 |
> |
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
889 |
> |
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
890 |
> |
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
891 |
> |
\begin{equation} |
892 |
> |
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
893 |
> |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
894 |
> |
\end{equation} |
895 |
> |
, if the weights are chosen as |
896 |
> |
\[ |
897 |
> |
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
898 |
> |
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
899 |
> |
\] |
900 |
|
|
901 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
902 |
|
|
907 |
|
|
908 |
|
\subsection{\label{introSec:mdInit}Initialization} |
909 |
|
|
910 |
+ |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
911 |
+ |
|
912 |
|
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
913 |
|
|
914 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
915 |
|
|
916 |
< |
A rigid body is a body in which the distance between any two given |
917 |
< |
points of a rigid body remains constant regardless of external |
918 |
< |
forces exerted on it. A rigid body therefore conserves its shape |
919 |
< |
during its motion. |
916 |
> |
Rigid bodies are frequently involved in the modeling of different |
917 |
> |
areas, from engineering, physics, to chemistry. For example, |
918 |
> |
missiles and vehicle are usually modeled by rigid bodies. The |
919 |
> |
movement of the objects in 3D gaming engine or other physics |
920 |
> |
simulator is governed by the rigid body dynamics. In molecular |
921 |
> |
simulation, rigid body is used to simplify the model in |
922 |
> |
protein-protein docking study{\cite{Gray03}}. |
923 |
|
|
924 |
< |
Applications of dynamics of rigid bodies. |
924 |
> |
It is very important to develop stable and efficient methods to |
925 |
> |
integrate the equations of motion of orientational degrees of |
926 |
> |
freedom. Euler angles are the nature choice to describe the |
927 |
> |
rotational degrees of freedom. However, due to its singularity, the |
928 |
> |
numerical integration of corresponding equations of motion is very |
929 |
> |
inefficient and inaccurate. Although an alternative integrator using |
930 |
> |
different sets of Euler angles can overcome this difficulty\cite{}, |
931 |
> |
the computational penalty and the lost of angular momentum |
932 |
> |
conservation still remain. A singularity free representation |
933 |
> |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
934 |
> |
this approach suffer from the nonseparable Hamiltonian resulted from |
935 |
> |
quaternion representation, which prevents the symplectic algorithm |
936 |
> |
to be utilized. Another different approach is to apply holonomic |
937 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
938 |
> |
moves independently under the normal forces deriving from potential |
939 |
> |
energy and constraint forces which are used to guarantee the |
940 |
> |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
941 |
> |
algorithm converge very slowly when the number of constraint |
942 |
> |
increases. |
943 |
|
|
944 |
+ |
The break through in geometric literature suggests that, in order to |
945 |
+ |
develop a long-term integration scheme, one should preserve the |
946 |
+ |
symplectic structure of the flow. Introducing conjugate momentum to |
947 |
+ |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
948 |
+ |
symplectic integrator, RSHAKE, was proposed to evolve the |
949 |
+ |
Hamiltonian system in a constraint manifold by iteratively |
950 |
+ |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
951 |
+ |
method using quaternion representation was developed by Omelyan. |
952 |
+ |
However, both of these methods are iterative and inefficient. In |
953 |
+ |
this section, we will present a symplectic Lie-Poisson integrator |
954 |
+ |
for rigid body developed by Dullweber and his coworkers\cite{}. |
955 |
+ |
|
956 |
|
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
957 |
|
|
958 |
|
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
959 |
|
|
960 |
|
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
961 |
|
|
680 |
– |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
681 |
– |
|
962 |
|
\section{\label{introSection:correlationFunctions}Correlation Functions} |
963 |
|
|
964 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1009 |
|
\dot p &= - \frac{{\partial H}}{{\partial x}} |
1010 |
|
&= m\ddot x |
1011 |
|
&= - \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)} |
1012 |
< |
\label{introEq:Lp5} |
1012 |
> |
\label{introEquation:Lp5} |
1013 |
|
\end{align} |
1014 |
|
, and |
1015 |
|
\begin{align} |