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 |
|
|
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 |
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 $p$, |
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. |
639 |
|
\end{equation} |
645 |
|
\begin{equation} |
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 |
+ |
When designing any numerical methods, one should always try to |
665 |
+ |
preserve the structural properties of the original ODE and its flow. |
666 |
+ |
|
667 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
668 |
|
A lot of well established and very effective numerical methods have |
669 |
|
been successful precisely because of their symplecticities even |
679 |
|
\end{enumerate} |
680 |
|
|
681 |
|
Generating function tends to lead to methods which are cumbersome |
682 |
< |
and difficult to use\cite{}. In dissipative systems, variational |
683 |
< |
methods can capture the decay of energy accurately\cite{}. Since |
684 |
< |
their geometrically unstable nature against non-Hamiltonian |
685 |
< |
perturbations, ordinary implicit Runge-Kutta methods are not |
686 |
< |
suitable for Hamiltonian system. Recently, various high-order |
687 |
< |
explicit Runge--Kutta methods have been developed to overcome this |
682 |
> |
and difficult to use. In dissipative systems, variational methods |
683 |
> |
can capture the decay of energy accurately. Since their |
684 |
> |
geometrically unstable nature against non-Hamiltonian perturbations, |
685 |
> |
ordinary implicit Runge-Kutta methods are not suitable for |
686 |
> |
Hamiltonian system. Recently, various high-order explicit |
687 |
> |
Runge--Kutta methods have been developed to overcome this |
688 |
|
instability \cite{}. However, due to computational penalty involved |
689 |
|
in implementing the Runge-Kutta methods, they do not attract too |
690 |
|
much attention from Molecular Dynamics community. Instead, splitting |
691 |
|
have been widely accepted since they exploit natural decompositions |
692 |
< |
of the system\cite{Tuckerman92}. The main idea behind splitting |
693 |
< |
methods is to decompose the discrete $\varphi_h$ as a composition of |
694 |
< |
simpler flows, |
692 |
> |
of the system\cite{Tuckerman92}. |
693 |
> |
|
694 |
> |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
695 |
> |
|
696 |
> |
The main idea behind splitting methods is to decompose the discrete |
697 |
> |
$\varphi_h$ as a composition of simpler flows, |
698 |
|
\begin{equation} |
699 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
700 |
|
\varphi _{h_n } |
701 |
|
\label{introEquation:FlowDecomposition} |
702 |
|
\end{equation} |
703 |
|
where each of the sub-flow is chosen such that each represent a |
704 |
< |
simpler integration of the system. Let $\phi$ and $\psi$ both be |
705 |
< |
symplectic maps, it is easy to show that any composition of |
706 |
< |
symplectic flows yields a symplectic map, |
704 |
> |
simpler integration of the system. |
705 |
> |
|
706 |
> |
Suppose that a Hamiltonian system takes the form, |
707 |
> |
\[ |
708 |
> |
H = H_1 + H_2. |
709 |
> |
\] |
710 |
> |
Here, $H_1$ and $H_2$ may represent different physical processes of |
711 |
> |
the system. For instance, they may relate to kinetic and potential |
712 |
> |
energy respectively, which is a natural decomposition of the |
713 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
714 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
715 |
> |
order is then given by the Lie-Trotter formula |
716 |
|
\begin{equation} |
717 |
+ |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
718 |
+ |
\label{introEquation:firstOrderSplitting} |
719 |
+ |
\end{equation} |
720 |
+ |
where $\varphi _h$ is the result of applying the corresponding |
721 |
+ |
continuous $\varphi _i$ over a time $h$. By definition, as |
722 |
+ |
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
723 |
+ |
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
724 |
+ |
It is easy to show that any composition of symplectic flows yields a |
725 |
+ |
symplectic map, |
726 |
+ |
\begin{equation} |
727 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
728 |
< |
'\phi ' = \phi '^T J\phi ' = J. |
728 |
> |
'\phi ' = \phi '^T J\phi ' = J, |
729 |
|
\label{introEquation:SymplecticFlowComposition} |
730 |
|
\end{equation} |
731 |
< |
Suppose that a Hamiltonian system has a form with $H = T + V$ |
731 |
> |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
732 |
> |
splitting in this context automatically generates a symplectic map. |
733 |
|
|
734 |
+ |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
735 |
+ |
introduces local errors proportional to $h^2$, while Strang |
736 |
+ |
splitting gives a second-order decomposition, |
737 |
+ |
\begin{equation} |
738 |
+ |
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
739 |
+ |
_{1,h/2} , |
740 |
+ |
\label{introEqaution:secondOrderSplitting} |
741 |
+ |
\end{equation} |
742 |
+ |
which has a local error proportional to $h^3$. Sprang splitting's |
743 |
+ |
popularity in molecular simulation community attribute to its |
744 |
+ |
symmetric property, |
745 |
+ |
\begin{equation} |
746 |
+ |
\varphi _h^{ - 1} = \varphi _{ - h}. |
747 |
+ |
\lable{introEquation:timeReversible} |
748 |
+ |
\end{equation} |
749 |
+ |
|
750 |
+ |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
751 |
+ |
The classical equation for a system consisting of interacting |
752 |
+ |
particles can be written in Hamiltonian form, |
753 |
+ |
\[ |
754 |
+ |
H = T + V |
755 |
+ |
\] |
756 |
+ |
where $T$ is the kinetic energy and $V$ is the potential energy. |
757 |
+ |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
758 |
+ |
obtains the following: |
759 |
+ |
\begin{align} |
760 |
+ |
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
761 |
+ |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
762 |
+ |
\label{introEquation:Lp10a} \\% |
763 |
+ |
% |
764 |
+ |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
765 |
+ |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
766 |
+ |
\label{introEquation:Lp10b} |
767 |
+ |
\end{align} |
768 |
+ |
where $F(t)$ is the force at time $t$. This integration scheme is |
769 |
+ |
known as \emph{velocity verlet} which is |
770 |
+ |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
771 |
+ |
time-reversible(\ref{introEquation:timeReversible}) and |
772 |
+ |
volume-preserving (\ref{introEquation:volumePreserving}). These |
773 |
+ |
geometric properties attribute to its long-time stability and its |
774 |
+ |
popularity in the community. However, the most commonly used |
775 |
+ |
velocity verlet integration scheme is written as below, |
776 |
+ |
\begin{align} |
777 |
+ |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
778 |
+ |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
779 |
+ |
% |
780 |
+ |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
781 |
+ |
\label{introEquation:Lp9b}\\% |
782 |
+ |
% |
783 |
+ |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
784 |
+ |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
785 |
+ |
\end{align} |
786 |
+ |
From the preceding splitting, one can see that the integration of |
787 |
+ |
the equations of motion would follow: |
788 |
+ |
\begin{enumerate} |
789 |
+ |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
790 |
+ |
|
791 |
+ |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
792 |
+ |
|
793 |
+ |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
794 |
+ |
|
795 |
+ |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
796 |
+ |
\end{enumerate} |
797 |
+ |
|
798 |
+ |
Simply switching the order of splitting and composing, a new |
799 |
+ |
integrator, the \emph{position verlet} integrator, can be generated, |
800 |
+ |
\begin{align} |
801 |
+ |
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
802 |
+ |
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
803 |
+ |
\label{introEquation:positionVerlet1} \\% |
804 |
+ |
% |
805 |
+ |
q(\Delta t) = q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
806 |
+ |
q(\Delta t)} \right]. % |
807 |
+ |
\label{introEquation:positionVerlet1} |
808 |
+ |
\end{align} |
809 |
+ |
|
810 |
+ |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
811 |
+ |
|
812 |
+ |
Baker-Campbell-Hausdorff formula can be used to determine the local |
813 |
+ |
error of splitting method in terms of commutator of the |
814 |
+ |
operators(\ref{introEquation:exponentialOperator}) associated with |
815 |
+ |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
816 |
+ |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
817 |
+ |
\begin{equation} |
818 |
+ |
\exp (hX + hY) = \exp (hZ) |
819 |
+ |
\end{equation} |
820 |
+ |
where |
821 |
+ |
\begin{equation} |
822 |
+ |
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
823 |
+ |
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
824 |
+ |
\end{equation} |
825 |
+ |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
826 |
+ |
\[ |
827 |
+ |
[X,Y] = XY - YX . |
828 |
+ |
\] |
829 |
+ |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
830 |
+ |
can obtain |
831 |
+ |
\begin{eqnarray} |
832 |
+ |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
833 |
+ |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 + |
834 |
+ |
h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 + \ldots ) |
835 |
+ |
\end{eqnarray} |
836 |
+ |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
837 |
+ |
error of Spring splitting is proportional to $h^3$. The same |
838 |
+ |
procedure can be applied to general splitting, of the form |
839 |
+ |
\begin{equation} |
840 |
+ |
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
841 |
+ |
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
842 |
+ |
\end{equation} |
843 |
+ |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
844 |
+ |
order method. Yoshida proposed an elegant way to compose higher |
845 |
+ |
order methods based on symmetric splitting. Given a symmetric second |
846 |
+ |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
847 |
+ |
method can be constructed by composing, |
848 |
+ |
\[ |
849 |
+ |
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
850 |
+ |
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
851 |
+ |
\] |
852 |
+ |
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
853 |
+ |
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
854 |
+ |
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
855 |
+ |
\begin{equation} |
856 |
+ |
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
857 |
+ |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
858 |
+ |
\end{equation} |
859 |
+ |
, if the weights are chosen as |
860 |
+ |
\[ |
861 |
+ |
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
862 |
+ |
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
863 |
+ |
\] |
864 |
+ |
|
865 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
866 |
|
|
867 |
|
As a special discipline of molecular modeling, Molecular dynamics |
888 |
|
|
889 |
|
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
890 |
|
|
680 |
– |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
681 |
– |
|
891 |
|
\section{\label{introSection:correlationFunctions}Correlation Functions} |
892 |
|
|
893 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
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{introEq:Lp5} |
941 |
> |
\label{introEquation:Lp5} |
942 |
|
\end{align} |
943 |
|
, and |
944 |
|
\begin{align} |