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. |
992 |
< |
|
993 |
< |
A natural approach to avoid system size issue is to represent the |
991 |
> |
algorithmic tricks. |
992 |
> |
|
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 |
|
\[ |
1112 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1113 |
< |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1113 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
1114 |
> |
(r)}{\rho}. |
1115 |
|
\] |
1116 |
|
Note that the delta function can be replaced by a histogram in |
1117 |
< |
computer simulation. Figure |
1118 |
< |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1119 |
< |
distribution function for the liquid argon system. The occurrence of |
1115 |
< |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1116 |
< |
to find particles at certain radial values than at others. This is a |
1117 |
< |
result of the attractive interaction at such distances. Because of |
1118 |
< |
the strong repulsive forces at short distance, the probability of |
1119 |
< |
locating particles at distances less than about 2.5{\AA} from each |
1120 |
< |
other is essentially zero. |
1117 |
> |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
1118 |
> |
the height of these peaks gradually decreases to 1 as the liquid of |
1119 |
> |
large distance approaches the bulk density. |
1120 |
|
|
1122 |
– |
%\begin{figure} |
1123 |
– |
%\centering |
1124 |
– |
%\includegraphics[width=\linewidth]{pdf.eps} |
1125 |
– |
%\caption[Pair distribution function for the liquid argon |
1126 |
– |
%]{Pair distribution function for the liquid argon} |
1127 |
– |
%\label{introFigure:pairDistributionFunction} |
1128 |
– |
%\end{figure} |
1121 |
|
|
1122 |
|
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
1123 |
|
Properties}} |
1124 |
|
|
1125 |
|
Time-dependent properties are usually calculated using \emph{time |
1126 |
< |
correlation function}, which correlates random variables $A$ and $B$ |
1127 |
< |
at two different time |
1126 |
> |
correlation functions}, which correlate random variables $A$ and $B$ |
1127 |
> |
at two different times, |
1128 |
|
\begin{equation} |
1129 |
|
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1130 |
|
\label{introEquation:timeCorrelationFunction} |
1131 |
|
\end{equation} |
1132 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
1133 |
< |
function is called \emph{auto correlation function}. One example of |
1134 |
< |
auto correlation function is velocity auto-correlation function |
1135 |
< |
which is directly related to transport properties of molecular |
1136 |
< |
liquids: |
1133 |
> |
function is called an \emph{autocorrelation function}. One example |
1134 |
> |
of an auto correlation function is the velocity auto-correlation |
1135 |
> |
function which is directly related to transport properties of |
1136 |
> |
molecular liquids: |
1137 |
|
\[ |
1138 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1139 |
|
\right\rangle } dt |
1140 |
|
\] |
1141 |
< |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1142 |
< |
function which is averaging over time origins and over all the |
1143 |
< |
atoms, dipole autocorrelation are calculated for the entire system. |
1144 |
< |
The dipole autocorrelation function is given by: |
1141 |
> |
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
1142 |
> |
function, which is averaging over time origins and over all the |
1143 |
> |
atoms, the dipole autocorrelation functions are calculated for the |
1144 |
> |
entire system. The dipole autocorrelation function is given by: |
1145 |
|
\[ |
1146 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1147 |
|
\right\rangle |
1167 |
|
areas, from engineering, physics, to chemistry. For example, |
1168 |
|
missiles and vehicle are usually modeled by rigid bodies. The |
1169 |
|
movement of the objects in 3D gaming engine or other physics |
1170 |
< |
simulator is governed by the rigid body dynamics. In molecular |
1171 |
< |
simulation, rigid body is used to simplify the model in |
1172 |
< |
protein-protein docking study\cite{Gray2003}. |
1170 |
> |
simulator is governed by rigid body dynamics. In molecular |
1171 |
> |
simulations, rigid bodies are used to simplify protein-protein |
1172 |
> |
docking studies\cite{Gray2003}. |
1173 |
|
|
1174 |
|
It is very important to develop stable and efficient methods to |
1175 |
< |
integrate the equations of motion of orientational degrees of |
1176 |
< |
freedom. Euler angles are the nature choice to describe the |
1177 |
< |
rotational degrees of freedom. However, due to its singularity, the |
1178 |
< |
numerical integration of corresponding equations of motion is very |
1179 |
< |
inefficient and inaccurate. Although an alternative integrator using |
1180 |
< |
different sets of Euler angles can overcome this |
1181 |
< |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
1182 |
< |
of angular momentum conservation still remain. A singularity free |
1183 |
< |
representation utilizing quaternions was developed by Evans in |
1184 |
< |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
1185 |
< |
nonseparable Hamiltonian resulted from quaternion representation, |
1186 |
< |
which prevents the symplectic algorithm to be utilized. Another |
1187 |
< |
different approach is to apply holonomic constraints to the atoms |
1188 |
< |
belonging to the rigid body. Each atom moves independently under the |
1189 |
< |
normal forces deriving from potential energy and constraint forces |
1190 |
< |
which are used to guarantee the rigidness. However, due to their |
1191 |
< |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
1192 |
< |
when the number of constraint increases\cite{Ryckaert1977, |
1193 |
< |
Andersen1983}. |
1175 |
> |
integrate the equations of motion for orientational degrees of |
1176 |
> |
freedom. Euler angles are the natural choice to describe the |
1177 |
> |
rotational degrees of freedom. However, due to $\frac {1}{sin |
1178 |
> |
\theta}$ singularities, the numerical integration of corresponding |
1179 |
> |
equations of motion is very inefficient and inaccurate. Although an |
1180 |
> |
alternative integrator using multiple sets of Euler angles can |
1181 |
> |
overcome this difficulty\cite{Barojas1973}, the computational |
1182 |
> |
penalty and the loss of angular momentum conservation still remain. |
1183 |
> |
A singularity-free representation utilizing quaternions was |
1184 |
> |
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
1185 |
> |
approach uses a nonseparable Hamiltonian resulting from the |
1186 |
> |
quaternion representation, which prevents the symplectic algorithm |
1187 |
> |
to be utilized. Another different approach is to apply holonomic |
1188 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
1189 |
> |
moves independently under the normal forces deriving from potential |
1190 |
> |
energy and constraint forces which are used to guarantee the |
1191 |
> |
rigidness. However, due to their iterative nature, the SHAKE and |
1192 |
> |
Rattle algorithms also converge very slowly when the number of |
1193 |
> |
constraints increases\cite{Ryckaert1977, Andersen1983}. |
1194 |
|
|
1195 |
< |
The break through in geometric literature suggests that, in order to |
1195 |
> |
A break-through in geometric literature suggests that, in order to |
1196 |
|
develop a long-term integration scheme, one should preserve the |
1197 |
< |
symplectic structure of the flow. Introducing conjugate momentum to |
1198 |
< |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
1199 |
< |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
1200 |
< |
the Hamiltonian system in a constraint manifold by iteratively |
1201 |
< |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
1202 |
< |
method using quaternion representation was developed by |
1203 |
< |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
1204 |
< |
iterative and inefficient. In this section, we will present a |
1197 |
> |
symplectic structure of the flow. By introducing a conjugate |
1198 |
> |
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
1199 |
> |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
1200 |
> |
proposed to evolve the Hamiltonian system in a constraint manifold |
1201 |
> |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
1202 |
> |
An alternative method using the quaternion representation was |
1203 |
> |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
1204 |
> |
methods are iterative and inefficient. In this section, we descibe a |
1205 |
|
symplectic Lie-Poisson integrator for rigid body developed by |
1206 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1207 |
|
|
1208 |
< |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
1209 |
< |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
1208 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
1209 |
> |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
1210 |
|
function |
1211 |
|
\begin{equation} |
1212 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
1220 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
1221 |
|
\] |
1222 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1223 |
< |
constrained Hamiltonian equation subjects to a holonomic constraint, |
1223 |
> |
constrained Hamiltonian equation is subjected to a holonomic |
1224 |
> |
constraint, |
1225 |
|
\begin{equation} |
1226 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1227 |
|
\end{equation} |
1228 |
< |
which is used to ensure rotation matrix's orthogonality. |
1229 |
< |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1230 |
< |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
1228 |
> |
which is used to ensure rotation matrix's unitarity. Differentiating |
1229 |
> |
\ref{introEquation:orthogonalConstraint} and using Equation |
1230 |
> |
\ref{introEquation:RBMotionMomentum}, one may obtain, |
1231 |
|
\begin{equation} |
1232 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1233 |
|
\label{introEquation:RBFirstOrderConstraint} |
1245 |
|
\end{eqnarray} |
1246 |
|
|
1247 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1248 |
< |
We can use constraint force provided by lagrange multiplier on the |
1249 |
< |
normal manifold to keep the motion on constraint space. Or we can |
1250 |
< |
simply evolve the system in constraint manifold. These two methods |
1251 |
< |
are proved to be equivalent. The holonomic constraint and equations |
1252 |
< |
of motions define a constraint manifold for rigid body |
1248 |
> |
We can use a constraint force provided by a Lagrange multiplier on |
1249 |
> |
the normal manifold to keep the motion on constraint space. Or we |
1250 |
> |
can simply evolve the system on the constraint manifold. These two |
1251 |
> |
methods have been proved to be equivalent. The holonomic constraint |
1252 |
> |
and equations of motions define a constraint manifold for rigid |
1253 |
> |
bodies |
1254 |
|
\[ |
1255 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1256 |
|
\right\}. |
1259 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
1260 |
|
$T_{\star}SO(3)$. However, it turns out that under symplectic |
1261 |
|
transformation, the cotangent space and the phase space are |
1262 |
< |
diffeomorphic. Introducing |
1262 |
> |
diffeomorphic. By introducing |
1263 |
|
\[ |
1264 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1265 |
|
\] |
1291 |
|
respectively. |
1292 |
|
|
1293 |
|
As a common choice to describe the rotation dynamics of the rigid |
1294 |
< |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
1295 |
< |
rewrite the equations of motion, |
1294 |
> |
body, the angular momentum on the body fixed frame $\Pi = Q^t P$ is |
1295 |
> |
introduced to rewrite the equations of motion, |
1296 |
|
\begin{equation} |
1297 |
|
\begin{array}{l} |
1298 |
|
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1335 |
|
Since $\Lambda$ is symmetric, the last term of Equation |
1336 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1337 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
1338 |
< |
unique property eliminate the requirement of iterations which can |
1338 |
> |
unique property eliminates the requirement of iterations which can |
1339 |
|
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
1340 |
|
|
1341 |
< |
Applying hat-map isomorphism, we obtain the equation of motion for |
1342 |
< |
angular momentum on body frame |
1341 |
> |
Applying the hat-map isomorphism, we obtain the equation of motion |
1342 |
> |
for angular momentum on body frame |
1343 |
|
\begin{equation} |
1344 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
1345 |
|
F_i (r,Q)} \right) \times X_i }. |
1354 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
1355 |
|
Lie-Poisson Integrator for Free Rigid Body} |
1356 |
|
|
1357 |
< |
If there is not external forces exerted on the rigid body, the only |
1358 |
< |
contribution to the rotational is from the kinetic potential (the |
1359 |
< |
first term of \ref{introEquation:bodyAngularMotion}). The free rigid |
1360 |
< |
body is an example of Lie-Poisson system with Hamiltonian function |
1357 |
> |
If there are no external forces exerted on the rigid body, the only |
1358 |
> |
contribution to the rotational motion is from the kinetic energy |
1359 |
> |
(the first term of \ref{introEquation:bodyAngularMotion}). The free |
1360 |
> |
rigid body is an example of a Lie-Poisson system with Hamiltonian |
1361 |
> |
function |
1362 |
|
\begin{equation} |
1363 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1364 |
|
\label{introEquation:rotationalKineticRB} |
1405 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1406 |
|
\] |
1407 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
1408 |
< |
tR_1 }$, we can use Cayley transformation, |
1408 |
> |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
1409 |
> |
propagator, |
1410 |
|
\[ |
1411 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1412 |
|
) |
1413 |
|
\] |
1414 |
|
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1415 |
< |
manner. |
1416 |
< |
|
1421 |
< |
In order to construct a second-order symplectic method, we split the |
1422 |
< |
angular kinetic Hamiltonian function can into five terms |
1415 |
> |
manner. In order to construct a second-order symplectic method, we |
1416 |
> |
split the angular kinetic Hamiltonian function can into five terms |
1417 |
|
\[ |
1418 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1419 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1420 |
< |
(\pi _1 ) |
1421 |
< |
\]. |
1422 |
< |
Concatenating flows corresponding to these five terms, we can obtain |
1423 |
< |
an symplectic integrator, |
1420 |
> |
(\pi _1 ). |
1421 |
> |
\] |
1422 |
> |
By concatenating the propagators corresponding to these five terms, |
1423 |
> |
we can obtain an symplectic integrator, |
1424 |
|
\[ |
1425 |
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
1426 |
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1447 |
|
\] |
1448 |
|
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
1449 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
1450 |
< |
Lie-Poisson integrator is found to be extremely efficient and stable |
1451 |
< |
which can be explained by the fact the small angle approximation is |
1452 |
< |
used and the norm of the angular momentum is conserved. |
1450 |
> |
Lie-Poisson integrator is found to be both extremely efficient and |
1451 |
> |
stable. These properties can be explained by the fact the small |
1452 |
> |
angle approximation is used and the norm of the angular momentum is |
1453 |
> |
conserved. |
1454 |
|
|
1455 |
|
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
1456 |
|
Splitting for Rigid Body} |
1477 |
|
\end{tabular} |
1478 |
|
\end{center} |
1479 |
|
\end{table} |
1480 |
< |
A second-order symplectic method is now obtained by the |
1481 |
< |
composition of the flow maps, |
1480 |
> |
A second-order symplectic method is now obtained by the composition |
1481 |
> |
of the position and velocity propagators, |
1482 |
|
\[ |
1483 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1484 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1485 |
|
\] |
1486 |
|
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
1487 |
< |
sub-flows which corresponding to force and torque respectively, |
1487 |
> |
sub-propagators which corresponding to force and torque |
1488 |
> |
respectively, |
1489 |
|
\[ |
1490 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1491 |
|
_{\Delta t/2,\tau }. |
1492 |
|
\] |
1493 |
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
1494 |
< |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
1495 |
< |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
1496 |
< |
|
1497 |
< |
Furthermore, kinetic potential can be separated to translational |
1502 |
< |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
1494 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order |
1495 |
> |
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the |
1496 |
> |
kinetic energy can be separated to translational kinetic term, $T^t |
1497 |
> |
(p)$, and rotational kinetic term, $T^r (\pi )$, |
1498 |
|
\begin{equation} |
1499 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
1500 |
|
\end{equation} |
1501 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
1502 |
|
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
1503 |
< |
corresponding flow maps are given by |
1503 |
> |
corresponding propagators are given by |
1504 |
|
\[ |
1505 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
1506 |
|
_{\Delta t,T^r }. |
1507 |
|
\] |
1508 |
< |
Finally, we obtain the overall symplectic flow maps for free moving |
1509 |
< |
rigid body |
1508 |
> |
Finally, we obtain the overall symplectic propagators for freely |
1509 |
> |
moving rigid bodies |
1510 |
|
\begin{equation} |
1511 |
|
\begin{array}{c} |
1512 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
1520 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
1521 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
1522 |
|
has been applied in a variety of studies. This section will review |
1523 |
< |
the theory of Langevin dynamics simulation. A brief derivation of |
1524 |
< |
generalized Langevin equation will be given first. Follow that, we |
1525 |
< |
will discuss the physical meaning of the terms appearing in the |
1526 |
< |
equation as well as the calculation of friction tensor from |
1527 |
< |
hydrodynamics theory. |
1523 |
> |
the theory of Langevin dynamics. A brief derivation of generalized |
1524 |
> |
Langevin equation will be given first. Following that, we will |
1525 |
> |
discuss the physical meaning of the terms appearing in the equation |
1526 |
> |
as well as the calculation of friction tensor from hydrodynamics |
1527 |
> |
theory. |
1528 |
|
|
1529 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
1530 |
|
|
1531 |
< |
Harmonic bath model, in which an effective set of harmonic |
1531 |
> |
A harmonic bath model, in which an effective set of harmonic |
1532 |
|
oscillators are used to mimic the effect of a linearly responding |
1533 |
|
environment, has been widely used in quantum chemistry and |
1534 |
|
statistical mechanics. One of the successful applications of |
1535 |
< |
Harmonic bath model is the derivation of Deriving Generalized |
1536 |
< |
Langevin Dynamics. Lets consider a system, in which the degree of |
1535 |
> |
Harmonic bath model is the derivation of the Generalized Langevin |
1536 |
> |
Dynamics (GLE). Lets consider a system, in which the degree of |
1537 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
1538 |
|
Hamiltonian of the form |
1539 |
|
\begin{equation} |
1540 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1541 |
|
\label{introEquation:bathGLE}. |
1542 |
|
\end{equation} |
1543 |
< |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
1544 |
< |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
1543 |
> |
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated |
1544 |
> |
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
1545 |
|
\[ |
1546 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1547 |
|
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
1549 |
|
\] |
1550 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
1551 |
|
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
1552 |
< |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
1552 |
> |
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath |
1553 |
|
coupling, |
1554 |
|
\[ |
1555 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1556 |
|
\] |
1557 |
< |
where $g_\alpha$ are the coupling constants between the bath and the |
1558 |
< |
coordinate $x$. Introducing |
1557 |
> |
where $g_\alpha$ are the coupling constants between the bath |
1558 |
> |
coordinates ($x_ \alpha$) and the system coordinate ($x$). |
1559 |
> |
Introducing |
1560 |
|
\[ |
1561 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1562 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1571 |
|
\] |
1572 |
|
Since the first two terms of the new Hamiltonian depend only on the |
1573 |
|
system coordinates, we can get the equations of motion for |
1574 |
< |
Generalized Langevin Dynamics by Hamilton's equations |
1579 |
< |
\ref{introEquation:motionHamiltonianCoordinate, |
1580 |
< |
introEquation:motionHamiltonianMomentum}, |
1574 |
> |
Generalized Langevin Dynamics by Hamilton's equations, |
1575 |
|
\begin{equation} |
1576 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
1577 |
|
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
1588 |
|
In order to derive an equation for $x$, the dynamics of the bath |
1589 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
1590 |
|
transform which is particularly useful in solving linear ordinary |
1591 |
< |
differential equations, Laplace transform is the appropriate tool to |
1592 |
< |
solve this problem. The basic idea is to transform the difficult |
1591 |
> |
differential equations,the Laplace transform is the appropriate tool |
1592 |
> |
to solve this problem. The basic idea is to transform the difficult |
1593 |
|
differential equations into simple algebra problems which can be |
1594 |
< |
solved easily. Then applying inverse Laplace transform, also known |
1595 |
< |
as the Bromwich integral, we can retrieve the solutions of the |
1594 |
> |
solved easily. Then, by applying the inverse Laplace transform, also |
1595 |
> |
known as the Bromwich integral, we can retrieve the solutions of the |
1596 |
|
original problems. |
1597 |
|
|
1598 |
|
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
1612 |
|
\end{eqnarray*} |
1613 |
|
|
1614 |
|
|
1615 |
< |
Applying Laplace transform to the bath coordinates, we obtain |
1615 |
> |
Applying the Laplace transform to the bath coordinates, we obtain |
1616 |
|
\begin{eqnarray*} |
1617 |
|
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) \\ |
1618 |
|
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1689 |
|
\end{array} |
1690 |
|
\] |
1691 |
|
This property is what we expect from a truly random process. As long |
1692 |
< |
as the model, which is gaussian distribution in general, chosen for |
1693 |
< |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
1700 |
< |
still remains. |
1692 |
> |
as the model chosen for $R(t)$ was a gaussian distribution in |
1693 |
> |
general, the stochastic nature of the GLE still remains. |
1694 |
|
|
1695 |
|
%dynamic friction kernel |
1696 |
|
The convolution integral |
1711 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
1712 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
1713 |
|
\] |
1714 |
< |
which can be used to describe dynamic caging effect. The other |
1715 |
< |
extreme is the bath that responds infinitely quickly to motions in |
1716 |
< |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
1717 |
< |
time: |
1714 |
> |
which can be used to describe the effect of dynamic caging in |
1715 |
> |
viscous solvents. The other extreme is the bath that responds |
1716 |
> |
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
1717 |
> |
taken as a $delta$ function in time: |
1718 |
|
\[ |
1719 |
|
\xi (t) = 2\xi _0 \delta (t) |
1720 |
|
\] |