| 496 |
|
geometric integrators, which preserve various phase-flow invariants |
| 497 |
|
such as symplectic structure, volume and time reversal symmetry, are |
| 498 |
|
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
| 499 |
< |
Leimkuhler1999}. The velocity verlet method, which happens to be a |
| 499 |
> |
Leimkuhler1999}. The velocity Verlet method, which happens to be a |
| 500 |
|
simple example of symplectic integrator, continues to gain |
| 501 |
|
popularity in the molecular dynamics community. This fact can be |
| 502 |
|
partly explained by its geometric nature. |
| 591 |
|
\end{equation} |
| 592 |
|
|
| 593 |
|
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
| 594 |
< |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
| 594 |
> |
Instead, we use an approximate map, $\psi_\tau$, which is usually |
| 595 |
|
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
| 596 |
|
the Taylor series of $\psi_\tau$ agree to order $p$, |
| 597 |
|
\begin{equation} |
| 598 |
< |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 598 |
> |
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 599 |
|
\end{equation} |
| 600 |
|
|
| 601 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 602 |
|
|
| 603 |
< |
The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE |
| 604 |
< |
and its flow play important roles in numerical studies. Many of them |
| 605 |
< |
can be found in systems which occur naturally in applications. |
| 603 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
| 604 |
> |
ODE and its flow play important roles in numerical studies. Many of |
| 605 |
> |
them can be found in systems which occur naturally in applications. |
| 606 |
|
|
| 607 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 608 |
|
a \emph{symplectic} flow if it satisfies, |
| 617 |
|
\begin{equation} |
| 618 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 619 |
|
\end{equation} |
| 620 |
< |
is the property must be preserved by the integrator. |
| 620 |
> |
is the property that must be preserved by the integrator. |
| 621 |
|
|
| 622 |
|
It is possible to construct a \emph{volume-preserving} flow for a |
| 623 |
< |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
| 623 |
> |
source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ |
| 624 |
|
\det d\varphi = 1$. One can show easily that a symplectic flow will |
| 625 |
|
be volume-preserving. |
| 626 |
|
|
| 627 |
< |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
| 628 |
< |
will result in a new system, |
| 627 |
> |
Changing the variables $y = h(x)$ in an ODE |
| 628 |
> |
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
| 629 |
|
\[ |
| 630 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 631 |
|
\] |
| 675 |
|
A lot of well established and very effective numerical methods have |
| 676 |
|
been successful precisely because of their symplecticities even |
| 677 |
|
though this fact was not recognized when they were first |
| 678 |
< |
constructed. The most famous example is the Verlet-leapfrog methods |
| 678 |
> |
constructed. The most famous example is the Verlet-leapfrog method |
| 679 |
|
in molecular dynamics. In general, symplectic integrators can be |
| 680 |
|
constructed using one of four different methods. |
| 681 |
|
\begin{enumerate} |
| 754 |
|
\label{introEquation:timeReversible} |
| 755 |
|
\end{equation},appendixFig:architecture |
| 756 |
|
|
| 757 |
< |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Example of Splitting Method}} |
| 757 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
| 758 |
|
The classical equation for a system consisting of interacting |
| 759 |
|
particles can be written in Hamiltonian form, |
| 760 |
|
\[ |
| 761 |
|
H = T + V |
| 762 |
|
\] |
| 763 |
|
where $T$ is the kinetic energy and $V$ is the potential energy. |
| 764 |
< |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
| 764 |
> |
Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one |
| 765 |
|
obtains the following: |
| 766 |
|
\begin{align} |
| 767 |
|
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 788 |
|
\label{introEquation:Lp9b}\\% |
| 789 |
|
% |
| 790 |
|
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 791 |
< |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
| 791 |
> |
\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c} |
| 792 |
|
\end{align} |
| 793 |
|
From the preceding splitting, one can see that the integration of |
| 794 |
|
the equations of motion would follow: |
| 797 |
|
|
| 798 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 799 |
|
|
| 800 |
< |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 800 |
> |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
| 801 |
|
|
| 802 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 803 |
|
\end{enumerate} |
| 804 |
|
|
| 805 |
< |
Simply switching the order of splitting and composing, a new |
| 806 |
< |
integrator, the \emph{position verlet} integrator, can be generated, |
| 805 |
> |
By simply switching the order of the propagators in the splitting |
| 806 |
> |
and composing a new integrator, the \emph{position verlet} |
| 807 |
> |
integrator, can be generated, |
| 808 |
|
\begin{align} |
| 809 |
|
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
| 810 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
| 817 |
|
|
| 818 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 819 |
|
|
| 820 |
< |
Baker-Campbell-Hausdorff formula can be used to determine the local |
| 821 |
< |
error of splitting method in terms of commutator of the |
| 820 |
> |
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 821 |
> |
local error of splitting method in terms of the commutator of the |
| 822 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
| 823 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
| 823 |
> |
the sub-flow. For operators $hX$ and $hY$ which are associated with |
| 824 |
|
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 825 |
|
\begin{equation} |
| 826 |
|
\exp (hX + hY) = \exp (hZ) |
| 834 |
|
\[ |
| 835 |
|
[X,Y] = XY - YX . |
| 836 |
|
\] |
| 837 |
< |
Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to |
| 838 |
< |
Sprang splitting, we can obtain |
| 837 |
> |
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
| 838 |
> |
to the Sprang splitting, we can obtain |
| 839 |
|
\begin{eqnarray*} |
| 840 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
| 841 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 842 |
|
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
| 843 |
|
\end{eqnarray*} |
| 844 |
< |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 844 |
> |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0,\] the dominant local |
| 845 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 846 |
< |
procedure can be applied to general splitting, of the form |
| 846 |
> |
procedure can be applied to a general splitting, of the form |
| 847 |
|
\begin{equation} |
| 848 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 849 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 850 |
|
\end{equation} |
| 851 |
< |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 852 |
< |
order method. Yoshida proposed an elegant way to compose higher |
| 851 |
> |
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 852 |
> |
order methods. Yoshida proposed an elegant way to compose higher |
| 853 |
|
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 854 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 855 |
|
fourth-order symmetric method can be constructed by composing, |
| 862 |
|
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
| 863 |
|
\begin{equation} |
| 864 |
|
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
| 865 |
< |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
| 865 |
> |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)}, |
| 866 |
|
\end{equation} |
| 867 |
< |
, if the weights are chosen as |
| 867 |
> |
if the weights are chosen as |
| 868 |
|
\[ |
| 869 |
|
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
| 870 |
|
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
| 902 |
|
These three individual steps will be covered in the following |
| 903 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 904 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
| 905 |
< |
will discusses issues in production run. |
| 905 |
> |
will discusse issues in production run. |
| 906 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 907 |
|
trajectory analysis. |
| 908 |
|
|
| 915 |
|
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
| 916 |
|
thousands of crystal structures of molecules are discovered every |
| 917 |
|
year, many more remain unknown due to the difficulties of |
| 918 |
< |
purification and crystallization. Even for the molecule with known |
| 919 |
< |
structure, some important information is missing. For example, the |
| 918 |
> |
purification and crystallization. Even for molecules with known |
| 919 |
> |
structure, some important information is missing. For example, a |
| 920 |
|
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 921 |
|
be added. Moreover, in order to include electrostatic interaction, |
| 922 |
|
one may need to specify the partial charges for individual atoms. |
| 923 |
|
Under some circumstances, we may even need to prepare the system in |
| 924 |
< |
a special setup. For instance, when studying transport phenomenon in |
| 925 |
< |
membrane system, we may prepare the lipids in bilayer structure |
| 926 |
< |
instead of placing lipids randomly in solvent, since we are not |
| 927 |
< |
interested in self-aggregation and it takes a long time to happen. |
| 924 |
> |
a special configuration. For instance, when studying transport |
| 925 |
> |
phenomenon in membrane systems, we may prepare the lipids in a |
| 926 |
> |
bilayer structure instead of placing lipids randomly in solvent, |
| 927 |
> |
since we are not interested in the slow self-aggregation process. |
| 928 |
|
|
| 929 |
|
\subsubsection{\textbf{Minimization}} |
| 930 |
|
|
| 931 |
|
It is quite possible that some of molecules in the system from |
| 932 |
< |
preliminary preparation may be overlapped with each other. This |
| 933 |
< |
close proximity leads to high potential energy which consequently |
| 934 |
< |
jeopardizes any molecular dynamics simulations. To remove these |
| 935 |
< |
steric overlaps, one typically performs energy minimization to find |
| 936 |
< |
a more reasonable conformation. Several energy minimization methods |
| 937 |
< |
have been developed to exploit the energy surface and to locate the |
| 938 |
< |
local minimum. While converging slowly near the minimum, steepest |
| 939 |
< |
descent method is extremely robust when systems are far from |
| 940 |
< |
harmonic. Thus, it is often used to refine structure from |
| 941 |
< |
crystallographic data. Relied on the gradient or hessian, advanced |
| 942 |
< |
methods like conjugate gradient and Newton-Raphson converge rapidly |
| 943 |
< |
to a local minimum, while become unstable if the energy surface is |
| 944 |
< |
far from quadratic. Another factor must be taken into account, when |
| 932 |
> |
preliminary preparation may be overlapping with each other. This |
| 933 |
> |
close proximity leads to high initial potential energy which |
| 934 |
> |
consequently jeopardizes any molecular dynamics simulations. To |
| 935 |
> |
remove these steric overlaps, one typically performs energy |
| 936 |
> |
minimization to find a more reasonable conformation. Several energy |
| 937 |
> |
minimization methods have been developed to exploit the energy |
| 938 |
> |
surface and to locate the local minimum. While converging slowly |
| 939 |
> |
near the minimum, steepest descent method is extremely robust when |
| 940 |
> |
systems are strongly anharmonic. Thus, it is often used to refine |
| 941 |
> |
structure from crystallographic data. Relied on the gradient or |
| 942 |
> |
hessian, advanced methods like Newton-Raphson converge rapidly to a |
| 943 |
> |
local minimum, but become unstable if the energy surface is far from |
| 944 |
> |
quadratic. Another factor that must be taken into account, when |
| 945 |
|
choosing energy minimization method, is the size of the system. |
| 946 |
|
Steepest descent and conjugate gradient can deal with models of any |
| 947 |
< |
size. Because of the limit of computation power to calculate hessian |
| 948 |
< |
matrix and insufficient storage capacity to store them, most |
| 949 |
< |
Newton-Raphson methods can not be used with very large models. |
| 947 |
> |
size. Because of the limits on computer memory to store the hessian |
| 948 |
> |
matrix and the computing power needed to diagonalized these |
| 949 |
> |
matrices, most Newton-Raphson methods can not be used with very |
| 950 |
> |
large systems. |
| 951 |
|
|
| 952 |
|
\subsubsection{\textbf{Heating}} |
| 953 |
|
|
| 954 |
|
Typically, Heating is performed by assigning random velocities |
| 955 |
< |
according to a Gaussian distribution for a temperature. Beginning at |
| 956 |
< |
a lower temperature and gradually increasing the temperature by |
| 957 |
< |
assigning greater random velocities, we end up with setting the |
| 958 |
< |
temperature of the system to a final temperature at which the |
| 959 |
< |
simulation will be conducted. In heating phase, we should also keep |
| 960 |
< |
the system from drifting or rotating as a whole. Equivalently, the |
| 961 |
< |
net linear momentum and angular momentum of the system should be |
| 962 |
< |
shifted to zero. |
| 955 |
> |
according to a Maxwell-Boltzman distribution for a desired |
| 956 |
> |
temperature. Beginning at a lower temperature and gradually |
| 957 |
> |
increasing the temperature by assigning larger random velocities, we |
| 958 |
> |
end up with setting the temperature of the system to a final |
| 959 |
> |
temperature at which the simulation will be conducted. In heating |
| 960 |
> |
phase, we should also keep the system from drifting or rotating as a |
| 961 |
> |
whole. To do this, the net linear momentum and angular momentum of |
| 962 |
> |
the system is shifted to zero after each resampling from the Maxwell |
| 963 |
> |
-Boltzman distribution. |
| 964 |
|
|
| 965 |
|
\subsubsection{\textbf{Equilibration}} |
| 966 |
|
|
| 976 |
|
|
| 977 |
|
\subsection{\label{introSection:production}Production} |
| 978 |
|
|
| 979 |
< |
Production run is the most important step of the simulation, in |
| 979 |
> |
The production run is the most important step of the simulation, in |
| 980 |
|
which the equilibrated structure is used as a starting point and the |
| 981 |
|
motions of the molecules are collected for later analysis. In order |
| 982 |
|
to capture the macroscopic properties of the system, the molecular |
| 983 |
< |
dynamics simulation must be performed in correct and efficient way. |
| 983 |
> |
dynamics simulation must be performed by sampling correctly and |
| 984 |
> |
efficiently from the relevant thermodynamic ensemble. |
| 985 |
|
|
| 986 |
|
The most expensive part of a molecular dynamics simulation is the |
| 987 |
|
calculation of non-bonded forces, such as van der Waals force and |
| 988 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
| 989 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
| 990 |
|
which making large simulations prohibitive in the absence of any |
| 991 |
< |
computation saving techniques. |
| 991 |
> |
algorithmic tricks. |
| 992 |
|
|
| 993 |
< |
A natural approach to avoid system size issue is to represent the |
| 993 |
> |
A natural approach to avoid system size issues is to represent the |
| 994 |
|
bulk behavior by a finite number of the particles. However, this |
| 995 |
< |
approach will suffer from the surface effect. To offset this, |
| 996 |
< |
\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc}) |
| 997 |
< |
is developed to simulate bulk properties with a relatively small |
| 998 |
< |
number of particles. In this method, the simulation box is |
| 999 |
< |
replicated throughout space to form an infinite lattice. During the |
| 1000 |
< |
simulation, when a particle moves in the primary cell, its image in |
| 1001 |
< |
other cells move in exactly the same direction with exactly the same |
| 1002 |
< |
orientation. Thus, as a particle leaves the primary cell, one of its |
| 1003 |
< |
images will enter through the opposite face. |
| 995 |
> |
approach will suffer from the surface effect at the edges of the |
| 996 |
> |
simulation. To offset this, \textit{Periodic boundary conditions} |
| 997 |
> |
(see Fig.~\ref{introFig:pbc}) is developed to simulate bulk |
| 998 |
> |
properties with a relatively small number of particles. In this |
| 999 |
> |
method, the simulation box is replicated throughout space to form an |
| 1000 |
> |
infinite lattice. During the simulation, when a particle moves in |
| 1001 |
> |
the primary cell, its image in other cells move in exactly the same |
| 1002 |
> |
direction with exactly the same orientation. Thus, as a particle |
| 1003 |
> |
leaves the primary cell, one of its images will enter through the |
| 1004 |
> |
opposite face. |
| 1005 |
|
\begin{figure} |
| 1006 |
|
\centering |
| 1007 |
|
\includegraphics[width=\linewidth]{pbc.eps} |
| 1013 |
|
|
| 1014 |
|
%cutoff and minimum image convention |
| 1015 |
|
Another important technique to improve the efficiency of force |
| 1016 |
< |
evaluation is to apply cutoff where particles farther than a |
| 1017 |
< |
predetermined distance, are not included in the calculation |
| 1016 |
> |
evaluation is to apply spherical cutoff where particles farther than |
| 1017 |
> |
a predetermined distance are not included in the calculation |
| 1018 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 1019 |
|
discontinuity in the potential energy curve. Fortunately, one can |
| 1020 |
< |
shift the potential to ensure the potential curve go smoothly to |
| 1021 |
< |
zero at the cutoff radius. Cutoff strategy works pretty well for |
| 1022 |
< |
Lennard-Jones interaction because of its short range nature. |
| 1023 |
< |
However, simply truncating the electrostatic interaction with the |
| 1024 |
< |
use of cutoff has been shown to lead to severe artifacts in |
| 1025 |
< |
simulations. Ewald summation, in which the slowly conditionally |
| 1026 |
< |
convergent Coulomb potential is transformed into direct and |
| 1027 |
< |
reciprocal sums with rapid and absolute convergence, has proved to |
| 1028 |
< |
minimize the periodicity artifacts in liquid simulations. Taking the |
| 1029 |
< |
advantages of the fast Fourier transform (FFT) for calculating |
| 1030 |
< |
discrete Fourier transforms, the particle mesh-based |
| 1020 |
> |
shift simple radial potential to ensure the potential curve go |
| 1021 |
> |
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 1022 |
> |
well for Lennard-Jones interaction because of its short range |
| 1023 |
> |
nature. However, simply truncating the electrostatic interaction |
| 1024 |
> |
with the use of cutoffs has been shown to lead to severe artifacts |
| 1025 |
> |
in simulations. The Ewald summation, in which the slowly decaying |
| 1026 |
> |
Coulomb potential is transformed into direct and reciprocal sums |
| 1027 |
> |
with rapid and absolute convergence, has proved to minimize the |
| 1028 |
> |
periodicity artifacts in liquid simulations. Taking the advantages |
| 1029 |
> |
of the fast Fourier transform (FFT) for calculating discrete Fourier |
| 1030 |
> |
transforms, the particle mesh-based |
| 1031 |
|
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 1032 |
< |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast |
| 1033 |
< |
multipole method}\cite{Greengard1987, Greengard1994}, which treats |
| 1034 |
< |
Coulombic interaction exactly at short range, and approximate the |
| 1035 |
< |
potential at long range through multipolar expansion. In spite of |
| 1036 |
< |
their wide acceptances at the molecular simulation community, these |
| 1037 |
< |
two methods are hard to be implemented correctly and efficiently. |
| 1038 |
< |
Instead, we use a damped and charge-neutralized Coulomb potential |
| 1039 |
< |
method developed by Wolf and his coworkers\cite{Wolf1999}. The |
| 1040 |
< |
shifted Coulomb potential for particle $i$ and particle $j$ at |
| 1041 |
< |
distance $r_{rj}$ is given by: |
| 1032 |
> |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
| 1033 |
> |
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
| 1034 |
> |
which treats Coulombic interactions exactly at short range, and |
| 1035 |
> |
approximate the potential at long range through multipolar |
| 1036 |
> |
expansion. In spite of their wide acceptance at the molecular |
| 1037 |
> |
simulation community, these two methods are difficult to implement |
| 1038 |
> |
correctly and efficiently. Instead, we use a damped and |
| 1039 |
> |
charge-neutralized Coulomb potential method developed by Wolf and |
| 1040 |
> |
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
| 1041 |
> |
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 1042 |
|
\begin{equation} |
| 1043 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 1044 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 1060 |
|
|
| 1061 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 1062 |
|
|
| 1063 |
< |
Recently, advanced visualization technique are widely applied to |
| 1063 |
> |
Recently, advanced visualization technique have become applied to |
| 1064 |
|
monitor the motions of molecules. Although the dynamics of the |
| 1065 |
|
system can be described qualitatively from animation, quantitative |
| 1066 |
< |
trajectory analysis are more appreciable. According to the |
| 1067 |
< |
principles of Statistical Mechanics, |
| 1068 |
< |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 1069 |
< |
thermodynamics properties, analyze fluctuations of structural |
| 1070 |
< |
parameters, and investigate time-dependent processes of the molecule |
| 1066 |
< |
from the trajectories. |
| 1066 |
> |
trajectory analysis are more useful. According to the principles of |
| 1067 |
> |
Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics}, |
| 1068 |
> |
one can compute thermodynamic properties, analyze fluctuations of |
| 1069 |
> |
structural parameters, and investigate time-dependent processes of |
| 1070 |
> |
the molecule from the trajectories. |
| 1071 |
|
|
| 1072 |
< |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamics Properties}} |
| 1072 |
> |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
| 1073 |
|
|
| 1074 |
< |
Thermodynamics properties, which can be expressed in terms of some |
| 1074 |
> |
Thermodynamic properties, which can be expressed in terms of some |
| 1075 |
|
function of the coordinates and momenta of all particles in the |
| 1076 |
|
system, can be directly computed from molecular dynamics. The usual |
| 1077 |
|
way to measure the pressure is based on virial theorem of Clausius |
| 1094 |
|
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
| 1095 |
|
|
| 1096 |
|
Structural Properties of a simple fluid can be described by a set of |
| 1097 |
< |
distribution functions. Among these functions,\emph{pair |
| 1097 |
> |
distribution functions. Among these functions,the \emph{pair |
| 1098 |
|
distribution function}, also known as \emph{radial distribution |
| 1099 |
< |
function}, is of most fundamental importance to liquid-state theory. |
| 1100 |
< |
Pair distribution function can be gathered by Fourier transforming |
| 1101 |
< |
raw data from a series of neutron diffraction experiments and |
| 1102 |
< |
integrating over the surface factor \cite{Powles1973}. The |
| 1103 |
< |
experiment result can serve as a criterion to justify the |
| 1104 |
< |
correctness of the theory. Moreover, various equilibrium |
| 1105 |
< |
thermodynamic and structural properties can also be expressed in |
| 1106 |
< |
terms of radial distribution function \cite{Allen1987}. |
| 1099 |
> |
function}, is of most fundamental importance to liquid theory. |
| 1100 |
> |
Experimentally, pair distribution function can be gathered by |
| 1101 |
> |
Fourier transforming raw data from a series of neutron diffraction |
| 1102 |
> |
experiments and integrating over the surface factor |
| 1103 |
> |
\cite{Powles1973}. The experimental results can serve as a criterion |
| 1104 |
> |
to justify the correctness of a liquid model. Moreover, various |
| 1105 |
> |
equilibrium thermodynamic and structural properties can also be |
| 1106 |
> |
expressed in terms of radial distribution function \cite{Allen1987}. |
| 1107 |
|
|
| 1108 |
< |
A pair distribution functions $g(r)$ gives the probability that a |
| 1108 |
> |
The pair distribution functions $g(r)$ gives the probability that a |
| 1109 |
|
particle $i$ will be located at a distance $r$ from a another |
| 1110 |
|
particle $j$ in the system |
| 1111 |
|
\[ |
| 1112 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1113 |
< |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
| 1113 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \fract{\rho |
| 1114 |
> |
(r)}{\rho}. |
| 1115 |
|
\] |
| 1116 |
|
Note that the delta function can be replaced by a histogram in |
| 1117 |
|
computer simulation. Figure |
| 1121 |
|
to find particles at certain radial values than at others. This is a |
| 1122 |
|
result of the attractive interaction at such distances. Because of |
| 1123 |
|
the strong repulsive forces at short distance, the probability of |
| 1124 |
< |
locating particles at distances less than about 2.5{\AA} from each |
| 1124 |
> |
locating particles at distances less than about 3.7{\AA} from each |
| 1125 |
|
other is essentially zero. |
| 1126 |
|
|
| 1127 |
|
%\begin{figure} |
| 1136 |
|
Properties}} |
| 1137 |
|
|
| 1138 |
|
Time-dependent properties are usually calculated using \emph{time |
| 1139 |
< |
correlation function}, which correlates random variables $A$ and $B$ |
| 1140 |
< |
at two different time |
| 1139 |
> |
correlation functions}, which correlate random variables $A$ and $B$ |
| 1140 |
> |
at two different times, |
| 1141 |
|
\begin{equation} |
| 1142 |
|
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
| 1143 |
|
\label{introEquation:timeCorrelationFunction} |
| 1144 |
|
\end{equation} |
| 1145 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1146 |
< |
function is called \emph{auto correlation function}. One example of |
| 1147 |
< |
auto correlation function is velocity auto-correlation function |
| 1148 |
< |
which is directly related to transport properties of molecular |
| 1149 |
< |
liquids: |
| 1146 |
> |
function is called an \emph{autocorrelation function}. One example |
| 1147 |
> |
of an auto correlation function is the velocity auto-correlation |
| 1148 |
> |
function which is directly related to transport properties of |
| 1149 |
> |
molecular liquids: |
| 1150 |
|
\[ |
| 1151 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
| 1152 |
|
\right\rangle } dt |
| 1153 |
|
\] |
| 1154 |
< |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
| 1155 |
< |
function which is averaging over time origins and over all the |
| 1156 |
< |
atoms, dipole autocorrelation are calculated for the entire system. |
| 1157 |
< |
The dipole autocorrelation function is given by: |
| 1154 |
> |
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1155 |
> |
function, which is averaging over time origins and over all the |
| 1156 |
> |
atoms, the dipole autocorrelation functions are calculated for the |
| 1157 |
> |
entire system. The dipole autocorrelation function is given by: |
| 1158 |
|
\[ |
| 1159 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1160 |
|
\right\rangle |
| 1180 |
|
areas, from engineering, physics, to chemistry. For example, |
| 1181 |
|
missiles and vehicle are usually modeled by rigid bodies. The |
| 1182 |
|
movement of the objects in 3D gaming engine or other physics |
| 1183 |
< |
simulator is governed by the rigid body dynamics. In molecular |
| 1184 |
< |
simulation, rigid body is used to simplify the model in |
| 1185 |
< |
protein-protein docking study\cite{Gray2003}. |
| 1183 |
> |
simulator is governed by rigid body dynamics. In molecular |
| 1184 |
> |
simulations, rigid bodies are used to simplify protein-protein |
| 1185 |
> |
docking studies\cite{Gray2003}. |
| 1186 |
|
|
| 1187 |
|
It is very important to develop stable and efficient methods to |
| 1188 |
< |
integrate the equations of motion of orientational degrees of |
| 1189 |
< |
freedom. Euler angles are the nature choice to describe the |
| 1190 |
< |
rotational degrees of freedom. However, due to its singularity, the |
| 1191 |
< |
numerical integration of corresponding equations of motion is very |
| 1192 |
< |
inefficient and inaccurate. Although an alternative integrator using |
| 1193 |
< |
different sets of Euler angles can overcome this |
| 1194 |
< |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
| 1195 |
< |
of angular momentum conservation still remain. A singularity free |
| 1196 |
< |
representation utilizing quaternions was developed by Evans in |
| 1197 |
< |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
| 1198 |
< |
nonseparable Hamiltonian resulted from quaternion representation, |
| 1199 |
< |
which prevents the symplectic algorithm to be utilized. Another |
| 1200 |
< |
different approach is to apply holonomic constraints to the atoms |
| 1201 |
< |
belonging to the rigid body. Each atom moves independently under the |
| 1202 |
< |
normal forces deriving from potential energy and constraint forces |
| 1203 |
< |
which are used to guarantee the rigidness. However, due to their |
| 1204 |
< |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
| 1205 |
< |
when the number of constraint increases\cite{Ryckaert1977, |
| 1206 |
< |
Andersen1983}. |
| 1188 |
> |
integrate the equations of motion for orientational degrees of |
| 1189 |
> |
freedom. Euler angles are the natural choice to describe the |
| 1190 |
> |
rotational degrees of freedom. However, due to $\frac {1}{sin |
| 1191 |
> |
\theta}$ singularities, the numerical integration of corresponding |
| 1192 |
> |
equations of motion is very inefficient and inaccurate. Although an |
| 1193 |
> |
alternative integrator using multiple sets of Euler angles can |
| 1194 |
> |
overcome this difficulty\cite{Barojas1973}, the computational |
| 1195 |
> |
penalty and the loss of angular momentum conservation still remain. |
| 1196 |
> |
A singularity-free representation utilizing quaternions was |
| 1197 |
> |
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
| 1198 |
> |
approach uses a nonseparable Hamiltonian resulting from the |
| 1199 |
> |
quaternion representation, which prevents the symplectic algorithm |
| 1200 |
> |
to be utilized. Another different approach is to apply holonomic |
| 1201 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
| 1202 |
> |
moves independently under the normal forces deriving from potential |
| 1203 |
> |
energy and constraint forces which are used to guarantee the |
| 1204 |
> |
rigidness. However, due to their iterative nature, the SHAKE and |
| 1205 |
> |
Rattle algorithms also converge very slowly when the number of |
| 1206 |
> |
constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1207 |
|
|
| 1208 |
< |
The break through in geometric literature suggests that, in order to |
| 1208 |
> |
A break-through in geometric literature suggests that, in order to |
| 1209 |
|
develop a long-term integration scheme, one should preserve the |
| 1210 |
< |
symplectic structure of the flow. Introducing conjugate momentum to |
| 1211 |
< |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
| 1212 |
< |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
| 1213 |
< |
the Hamiltonian system in a constraint manifold by iteratively |
| 1214 |
< |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
| 1215 |
< |
method using quaternion representation was developed by |
| 1216 |
< |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
| 1217 |
< |
iterative and inefficient. In this section, we will present a |
| 1210 |
> |
symplectic structure of the flow. By introducing a conjugate |
| 1211 |
> |
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
| 1212 |
> |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
| 1213 |
> |
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1214 |
> |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 1215 |
> |
An alternative method using the quaternion representation was |
| 1216 |
> |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
| 1217 |
> |
methods are iterative and inefficient. In this section, we descibe a |
| 1218 |
|
symplectic Lie-Poisson integrator for rigid body developed by |
| 1219 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1220 |
|
|
| 1221 |
< |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 1222 |
< |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
| 1221 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1222 |
> |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
| 1223 |
|
function |
| 1224 |
|
\begin{equation} |
| 1225 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1233 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1234 |
|
\] |
| 1235 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1236 |
< |
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 1236 |
> |
constrained Hamiltonian equation is subjected to a holonomic |
| 1237 |
> |
constraint, |
| 1238 |
|
\begin{equation} |
| 1239 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1240 |
|
\end{equation} |
| 1241 |
< |
which is used to ensure rotation matrix's orthogonality. |
| 1242 |
< |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 1243 |
< |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1241 |
> |
which is used to ensure rotation matrix's unitarity. Differentiating |
| 1242 |
> |
\ref{introEquation:orthogonalConstraint} and using Equation |
| 1243 |
> |
\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1244 |
|
\begin{equation} |
| 1245 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1246 |
|
\label{introEquation:RBFirstOrderConstraint} |
| 1258 |
|
\end{eqnarray} |
| 1259 |
|
|
| 1260 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1261 |
< |
We can use constraint force provided by lagrange multiplier on the |
| 1262 |
< |
normal manifold to keep the motion on constraint space. Or we can |
| 1263 |
< |
simply evolve the system in constraint manifold. These two methods |
| 1264 |
< |
are proved to be equivalent. The holonomic constraint and equations |
| 1265 |
< |
of motions define a constraint manifold for rigid body |
| 1261 |
> |
We can use a constraint force provided by a Lagrange multiplier on |
| 1262 |
> |
the normal manifold to keep the motion on constraint space. Or we |
| 1263 |
> |
can simply evolve the system on the constraint manifold. These two |
| 1264 |
> |
methods have been proved to be equivalent. The holonomic constraint |
| 1265 |
> |
and equations of motions define a constraint manifold for rigid |
| 1266 |
> |
bodies |
| 1267 |
|
\[ |
| 1268 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1269 |
|
\right\}. |
| 1272 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1273 |
|
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 1274 |
|
transformation, the cotangent space and the phase space are |
| 1275 |
< |
diffeomorphic. Introducing |
| 1275 |
> |
diffeomorphic. By introducing |
| 1276 |
|
\[ |
| 1277 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1278 |
|
\] |
| 1304 |
|
respectively. |
| 1305 |
|
|
| 1306 |
|
As a common choice to describe the rotation dynamics of the rigid |
| 1307 |
< |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1308 |
< |
rewrite the equations of motion, |
| 1307 |
> |
body, the angular momentum on the body fixed frame $\Pi = Q^t P$ is |
| 1308 |
> |
introduced to rewrite the equations of motion, |
| 1309 |
|
\begin{equation} |
| 1310 |
|
\begin{array}{l} |
| 1311 |
|
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1348 |
|
Since $\Lambda$ is symmetric, the last term of Equation |
| 1349 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1350 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1351 |
< |
unique property eliminate the requirement of iterations which can |
| 1351 |
> |
unique property eliminates the requirement of iterations which can |
| 1352 |
|
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
| 1353 |
|
|
| 1354 |
< |
Applying hat-map isomorphism, we obtain the equation of motion for |
| 1355 |
< |
angular momentum on body frame |
| 1354 |
> |
Applying the hat-map isomorphism, we obtain the equation of motion |
| 1355 |
> |
for angular momentum on body frame |
| 1356 |
|
\begin{equation} |
| 1357 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1358 |
|
F_i (r,Q)} \right) \times X_i }. |
| 1367 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1368 |
|
Lie-Poisson Integrator for Free Rigid Body} |
| 1369 |
|
|
| 1370 |
< |
If there is not external forces exerted on the rigid body, the only |
| 1371 |
< |
contribution to the rotational is from the kinetic potential (the |
| 1372 |
< |
first term of \ref{introEquation:bodyAngularMotion}). The free rigid |
| 1373 |
< |
body is an example of Lie-Poisson system with Hamiltonian function |
| 1370 |
> |
If there are no external forces exerted on the rigid body, the only |
| 1371 |
> |
contribution to the rotational motion is from the kinetic energy |
| 1372 |
> |
(the first term of \ref{introEquation:bodyAngularMotion}). The free |
| 1373 |
> |
rigid body is an example of a Lie-Poisson system with Hamiltonian |
| 1374 |
> |
function |
| 1375 |
|
\begin{equation} |
| 1376 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1377 |
|
\label{introEquation:rotationalKineticRB} |
| 1418 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1419 |
|
\] |
| 1420 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1421 |
< |
tR_1 }$, we can use Cayley transformation, |
| 1421 |
> |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
| 1422 |
> |
propagator, |
| 1423 |
|
\[ |
| 1424 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1425 |
|
) |
| 1426 |
|
\] |
| 1427 |
|
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1428 |
< |
manner. |
| 1429 |
< |
|
| 1421 |
< |
In order to construct a second-order symplectic method, we split the |
| 1422 |
< |
angular kinetic Hamiltonian function can into five terms |
| 1428 |
> |
manner. In order to construct a second-order symplectic method, we |
| 1429 |
> |
split the angular kinetic Hamiltonian function can into five terms |
| 1430 |
|
\[ |
| 1431 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1432 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1433 |
< |
(\pi _1 ) |
| 1434 |
< |
\]. |
| 1435 |
< |
Concatenating flows corresponding to these five terms, we can obtain |
| 1436 |
< |
an symplectic integrator, |
| 1433 |
> |
(\pi _1 ). |
| 1434 |
> |
\] |
| 1435 |
> |
By concatenating the propagators corresponding to these five terms, |
| 1436 |
> |
we can obtain an symplectic integrator, |
| 1437 |
|
\[ |
| 1438 |
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1439 |
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1460 |
|
\] |
| 1461 |
|
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1462 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1463 |
< |
Lie-Poisson integrator is found to be extremely efficient and stable |
| 1464 |
< |
which can be explained by the fact the small angle approximation is |
| 1465 |
< |
used and the norm of the angular momentum is conserved. |
| 1463 |
> |
Lie-Poisson integrator is found to be both extremely efficient and |
| 1464 |
> |
stable. These properties can be explained by the fact the small |
| 1465 |
> |
angle approximation is used and the norm of the angular momentum is |
| 1466 |
> |
conserved. |
| 1467 |
|
|
| 1468 |
|
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
| 1469 |
|
Splitting for Rigid Body} |
| 1490 |
|
\end{tabular} |
| 1491 |
|
\end{center} |
| 1492 |
|
\end{table} |
| 1493 |
< |
A second-order symplectic method is now obtained by the |
| 1494 |
< |
composition of the flow maps, |
| 1493 |
> |
A second-order symplectic method is now obtained by the composition |
| 1494 |
> |
of the position and velocity propagators, |
| 1495 |
|
\[ |
| 1496 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1497 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1498 |
|
\] |
| 1499 |
|
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 1500 |
< |
sub-flows which corresponding to force and torque respectively, |
| 1500 |
> |
sub-propagators which corresponding to force and torque |
| 1501 |
> |
respectively, |
| 1502 |
|
\[ |
| 1503 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1504 |
|
_{\Delta t/2,\tau }. |
| 1505 |
|
\] |
| 1506 |
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1507 |
< |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 1508 |
< |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
| 1509 |
< |
|
| 1510 |
< |
Furthermore, kinetic potential can be separated to translational |
| 1502 |
< |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1507 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order |
| 1508 |
> |
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the |
| 1509 |
> |
kinetic energy can be separated to translational kinetic term, $T^t |
| 1510 |
> |
(p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1511 |
|
\begin{equation} |
| 1512 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1513 |
|
\end{equation} |
| 1514 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1515 |
|
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1516 |
< |
corresponding flow maps are given by |
| 1516 |
> |
corresponding propagators are given by |
| 1517 |
|
\[ |
| 1518 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1519 |
|
_{\Delta t,T^r }. |
| 1520 |
|
\] |
| 1521 |
< |
Finally, we obtain the overall symplectic flow maps for free moving |
| 1522 |
< |
rigid body |
| 1521 |
> |
Finally, we obtain the overall symplectic propagators for freely |
| 1522 |
> |
moving rigid bodies |
| 1523 |
|
\begin{equation} |
| 1524 |
|
\begin{array}{c} |
| 1525 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1533 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1534 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
| 1535 |
|
has been applied in a variety of studies. This section will review |
| 1536 |
< |
the theory of Langevin dynamics simulation. A brief derivation of |
| 1537 |
< |
generalized Langevin equation will be given first. Follow that, we |
| 1538 |
< |
will discuss the physical meaning of the terms appearing in the |
| 1539 |
< |
equation as well as the calculation of friction tensor from |
| 1540 |
< |
hydrodynamics theory. |
| 1536 |
> |
the theory of Langevin dynamics. A brief derivation of generalized |
| 1537 |
> |
Langevin equation will be given first. Following that, we will |
| 1538 |
> |
discuss the physical meaning of the terms appearing in the equation |
| 1539 |
> |
as well as the calculation of friction tensor from hydrodynamics |
| 1540 |
> |
theory. |
| 1541 |
|
|
| 1542 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1543 |
|
|
| 1544 |
< |
Harmonic bath model, in which an effective set of harmonic |
| 1544 |
> |
A harmonic bath model, in which an effective set of harmonic |
| 1545 |
|
oscillators are used to mimic the effect of a linearly responding |
| 1546 |
|
environment, has been widely used in quantum chemistry and |
| 1547 |
|
statistical mechanics. One of the successful applications of |
| 1548 |
< |
Harmonic bath model is the derivation of Deriving Generalized |
| 1549 |
< |
Langevin Dynamics. Lets consider a system, in which the degree of |
| 1548 |
> |
Harmonic bath model is the derivation of the Generalized Langevin |
| 1549 |
> |
Dynamics (GLE). Lets consider a system, in which the degree of |
| 1550 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1551 |
|
Hamiltonian of the form |
| 1552 |
|
\begin{equation} |
| 1553 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1554 |
|
\label{introEquation:bathGLE}. |
| 1555 |
|
\end{equation} |
| 1556 |
< |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
| 1557 |
< |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
| 1556 |
> |
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated |
| 1557 |
> |
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
| 1558 |
|
\[ |
| 1559 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1560 |
|
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1562 |
|
\] |
| 1563 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1564 |
|
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
| 1565 |
< |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
| 1565 |
> |
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath |
| 1566 |
|
coupling, |
| 1567 |
|
\[ |
| 1568 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1569 |
|
\] |
| 1570 |
< |
where $g_\alpha$ are the coupling constants between the bath and the |
| 1571 |
< |
coordinate $x$. Introducing |
| 1570 |
> |
where $g_\alpha$ are the coupling constants between the bath |
| 1571 |
> |
coordinates ($x_ \apha$) and the system coordinate ($x$). |
| 1572 |
> |
Introducing |
| 1573 |
|
\[ |
| 1574 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1575 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1584 |
|
\] |
| 1585 |
|
Since the first two terms of the new Hamiltonian depend only on the |
| 1586 |
|
system coordinates, we can get the equations of motion for |
| 1587 |
< |
Generalized Langevin Dynamics by Hamilton's equations |
| 1579 |
< |
\ref{introEquation:motionHamiltonianCoordinate, |
| 1580 |
< |
introEquation:motionHamiltonianMomentum}, |
| 1587 |
> |
Generalized Langevin Dynamics by Hamilton's equations, |
| 1588 |
|
\begin{equation} |
| 1589 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1590 |
|
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
| 1601 |
|
In order to derive an equation for $x$, the dynamics of the bath |
| 1602 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1603 |
|
transform which is particularly useful in solving linear ordinary |
| 1604 |
< |
differential equations, Laplace transform is the appropriate tool to |
| 1605 |
< |
solve this problem. The basic idea is to transform the difficult |
| 1604 |
> |
differential equations,the Laplace transform is the appropriate tool |
| 1605 |
> |
to solve this problem. The basic idea is to transform the difficult |
| 1606 |
|
differential equations into simple algebra problems which can be |
| 1607 |
< |
solved easily. Then applying inverse Laplace transform, also known |
| 1608 |
< |
as the Bromwich integral, we can retrieve the solutions of the |
| 1607 |
> |
solved easily. Then, by applying the inverse Laplace transform, also |
| 1608 |
> |
known as the Bromwich integral, we can retrieve the solutions of the |
| 1609 |
|
original problems. |
| 1610 |
|
|
| 1611 |
|
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1625 |
|
\end{eqnarray*} |
| 1626 |
|
|
| 1627 |
|
|
| 1628 |
< |
Applying Laplace transform to the bath coordinates, we obtain |
| 1628 |
> |
Applying the Laplace transform to the bath coordinates, we obtain |
| 1629 |
|
\begin{eqnarray*} |
| 1630 |
|
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
| 1631 |
|
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1702 |
|
\end{array} |
| 1703 |
|
\] |
| 1704 |
|
This property is what we expect from a truly random process. As long |
| 1705 |
< |
as the model, which is gaussian distribution in general, chosen for |
| 1706 |
< |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
| 1700 |
< |
still remains. |
| 1705 |
> |
as the model chosen for $R(t)$ was a gaussian distribution in |
| 1706 |
> |
general, the stochastic nature of the GLE still remains. |
| 1707 |
|
|
| 1708 |
|
%dynamic friction kernel |
| 1709 |
|
The convolution integral |
| 1724 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1725 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1726 |
|
\] |
| 1727 |
< |
which can be used to describe dynamic caging effect. The other |
| 1728 |
< |
extreme is the bath that responds infinitely quickly to motions in |
| 1729 |
< |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
| 1730 |
< |
time: |
| 1727 |
> |
which can be used to describe the effect of dynamic caging in |
| 1728 |
> |
viscous solvents. The other extreme is the bath that responds |
| 1729 |
> |
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
| 1730 |
> |
taken as a $delta$ function in time: |
| 1731 |
|
\[ |
| 1732 |
|
\xi (t) = 2\xi _0 \delta (t) |
| 1733 |
|
\] |
