| 67 |
|
\begin{equation}E = T + V. \label{introEquation:energyConservation} |
| 68 |
|
\end{equation} |
| 69 |
|
All of these conserved quantities are important factors to determine |
| 70 |
< |
the quality of numerical integration schemes for rigid bodies |
| 71 |
< |
\cite{Dullweber1997}. |
| 70 |
> |
the quality of numerical integration schemes for rigid |
| 71 |
> |
bodies.\cite{Dullweber1997} |
| 72 |
|
|
| 73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 74 |
|
|
| 178 |
|
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 179 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
| 180 |
|
equation of motion. Due to their symmetrical formula, they are also |
| 181 |
< |
known as the canonical equations of motions \cite{Goldstein2001}. |
| 181 |
> |
known as the canonical equations of motions.\cite{Goldstein2001} |
| 182 |
|
|
| 183 |
|
An important difference between Lagrangian approach and the |
| 184 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
| 188 |
|
Hamiltonian Mechanics is more appropriate for application to |
| 189 |
|
statistical mechanics and quantum mechanics, since it treats the |
| 190 |
|
coordinate and its time derivative as independent variables and it |
| 191 |
< |
only works with 1st-order differential equations\cite{Marion1990}. |
| 191 |
> |
only works with 1st-order differential equations.\cite{Marion1990} |
| 192 |
|
In Newtonian Mechanics, a system described by conservative forces |
| 193 |
|
conserves the total energy |
| 194 |
|
(Eq.~\ref{introEquation:energyConservation}). It follows that |
| 208 |
|
The thermodynamic behaviors and properties of Molecular Dynamics |
| 209 |
|
simulation are governed by the principle of Statistical Mechanics. |
| 210 |
|
The following section will give a brief introduction to some of the |
| 211 |
< |
Statistical Mechanics concepts and theorem presented in this |
| 211 |
> |
Statistical Mechanics concepts and theorems presented in this |
| 212 |
|
dissertation. |
| 213 |
|
|
| 214 |
|
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
| 337 |
|
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
| 338 |
|
distribution, |
| 339 |
|
\begin{equation} |
| 340 |
< |
\rho \propto e^{ - \beta H} |
| 340 |
> |
\rho \propto e^{ - \beta H}. |
| 341 |
|
\label{introEquation:densityAndHamiltonian} |
| 342 |
|
\end{equation} |
| 343 |
|
|
| 349 |
|
If this region is small enough, the density $\rho$ can be regarded |
| 350 |
|
as uniform over the whole integral. Thus, the number of phase points |
| 351 |
|
inside this region is given by, |
| 352 |
< |
\begin{equation} |
| 353 |
< |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
| 354 |
< |
dp_1 } ..dp_f. |
| 355 |
< |
\end{equation} |
| 356 |
< |
|
| 357 |
< |
\begin{equation} |
| 358 |
< |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
| 352 |
> |
\begin{eqnarray} |
| 353 |
> |
\delta N &=& \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f,\\ |
| 354 |
> |
\frac{{d(\delta N)}}{{dt}} &=& \frac{{d\rho }}{{dt}}\delta v + \rho |
| 355 |
|
\frac{d}{{dt}}(\delta v) = 0. |
| 356 |
< |
\end{equation} |
| 356 |
> |
\end{eqnarray} |
| 357 |
|
With the help of the stationary assumption |
| 358 |
|
(Eq.~\ref{introEquation:stationary}), we obtain the principle of |
| 359 |
|
\emph{conservation of volume in phase space}, |
| 368 |
|
Liouville's theorem can be expressed in a variety of different forms |
| 369 |
|
which are convenient within different contexts. For any two function |
| 370 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
| 371 |
< |
bracket ${F, G}$ is defined as |
| 371 |
> |
bracket $\{F,G\}$ is defined as |
| 372 |
|
\begin{equation} |
| 373 |
|
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
| 374 |
|
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
| 412 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 413 |
|
Hypothesis makes a connection between time average and the ensemble |
| 414 |
|
average. It states that the time average and average over the |
| 415 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
| 415 |
> |
statistical ensemble are identical:\cite{Frenkel1996, Leach2001} |
| 416 |
|
\begin{equation} |
| 417 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 418 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 430 |
|
utilized. Or if the system lends itself to a time averaging |
| 431 |
|
approach, the Molecular Dynamics techniques in |
| 432 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 433 |
< |
choice\cite{Frenkel1996}. |
| 433 |
> |
choice.\cite{Frenkel1996} |
| 434 |
|
|
| 435 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 436 |
|
A variety of numerical integrators have been proposed to simulate |
| 437 |
|
the motions of atoms in MD simulation. They usually begin with |
| 438 |
< |
initial conditionals and move the objects in the direction governed |
| 439 |
< |
by the differential equations. However, most of them ignore the |
| 440 |
< |
hidden physical laws contained within the equations. Since 1990, |
| 441 |
< |
geometric integrators, which preserve various phase-flow invariants |
| 442 |
< |
such as symplectic structure, volume and time reversal symmetry, |
| 443 |
< |
were developed to address this issue\cite{Dullweber1997, |
| 444 |
< |
McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which |
| 445 |
< |
happens to be a simple example of symplectic integrator, continues |
| 446 |
< |
to gain popularity in the molecular dynamics community. This fact |
| 447 |
< |
can be partly explained by its geometric nature. |
| 438 |
> |
initial conditions and move the objects in the direction governed by |
| 439 |
> |
the differential equations. However, most of them ignore the hidden |
| 440 |
> |
physical laws contained within the equations. Since 1990, geometric |
| 441 |
> |
integrators, which preserve various phase-flow invariants such as |
| 442 |
> |
symplectic structure, volume and time reversal symmetry, were |
| 443 |
> |
developed to address this issue.\cite{Dullweber1997, McLachlan1998, |
| 444 |
> |
Leimkuhler1999} The velocity Verlet method, which happens to be a |
| 445 |
> |
simple example of symplectic integrator, continues to gain |
| 446 |
> |
popularity in the molecular dynamics community. This fact can be |
| 447 |
> |
partly explained by its geometric nature. |
| 448 |
|
|
| 449 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 449 |
> |
\subsection{\label{introSection:symplecticManifold}Manifolds and Bundles} |
| 450 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
| 451 |
|
locally like Euclidean space, but when viewed globally, it may have |
| 452 |
|
more complicated structure. A good example of manifold is the |
| 453 |
|
surface of Earth. It seems to be flat locally, but it is round if |
| 454 |
|
viewed as a whole. A \emph{differentiable manifold} (also known as |
| 455 |
|
\emph{smooth manifold}) is a manifold on which it is possible to |
| 456 |
< |
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
| 456 |
> |
apply calculus.\cite{Hirsch1997} A \emph{symplectic manifold} is |
| 457 |
|
defined as a pair $(M, \omega)$ which consists of a |
| 458 |
< |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 458 |
> |
\emph{differentiable manifold} $M$ and a close, non-degenerate, |
| 459 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 460 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
| 461 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 462 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 463 |
< |
$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in |
| 464 |
< |
vector field is an example of symplectic form. One of the |
| 465 |
< |
motivations to study \emph{symplectic manifolds} in Hamiltonian |
| 466 |
< |
Mechanics is that a symplectic manifold can represent all possible |
| 467 |
< |
configurations of the system and the phase space of the system can |
| 468 |
< |
be described by it's cotangent bundle\cite{Jost2002}. Every |
| 469 |
< |
symplectic manifold is even dimensional. For instance, in Hamilton |
| 470 |
< |
equations, coordinate and momentum always appear in pairs. |
| 463 |
> |
$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in |
| 464 |
> |
vector field is an example of symplectic form. |
| 465 |
> |
Given vector spaces $V$ and $W$ over same field $F$, $f: V \to W$ is a linear transformation if |
| 466 |
> |
\begin{eqnarray*} |
| 467 |
> |
f(x+y) & = & f(x) + f(y) \\ |
| 468 |
> |
f(ax) & = & af(x) |
| 469 |
> |
\end{eqnarray*} |
| 470 |
> |
are always satisfied for any two vectors $x$ and $y$ in $V$ and any scalar $a$ in $F$. One can define the dual vector space $V^*$ of $V$ if any two built-in linear transformations $\phi$ and $\psi$ in $V^*$ satisfy the following definition of addition and scalar multiplication: |
| 471 |
> |
\begin{eqnarray*} |
| 472 |
> |
(\phi+\psi)(x) & = & \phi(x)+\psi(x) \\ |
| 473 |
> |
(a\phi)(x) & = & a \phi(x) |
| 474 |
> |
\end{eqnarray*} |
| 475 |
> |
for all $a$ in $F$ and $x$ in $V$. For a manifold $M$, one can define a tangent vector of a tangent space $TM_q$ at every point $q$ |
| 476 |
> |
\begin{equation} |
| 477 |
> |
\dot q = \mathop {\lim }\limits_{t \to 0} \frac{{\phi (t) - \phi (0)}}{t} |
| 478 |
> |
\end{equation} |
| 479 |
> |
where $\phi(0)=q$ and $\phi(t) \in M$. One may also define a cotangent space $T^*M_q$ as the dual space of the tangent space $TM_q$. The tangent space and the cotangent space are isomorphic to each other, since they are both real vector spaces with same dimension. |
| 480 |
> |
The union of tangent spaces at every point of $M$ is called the tangent bundle of $M$ and is denoted by $TM$, while cotangent bundle $T^*M$ is defined as the union of the cotangent spaces to $M$.\cite{Jost2002} For a Hamiltonian system with configuration manifold $V$, the $(q,\dot q)$ phase space is the tangent bundle of the configuration manifold $V$, while the cotangent bundle is represented by $(q,p)$. |
| 481 |
|
|
| 482 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 483 |
|
|
| 485 |
|
\begin{equation} |
| 486 |
|
\dot x = f(x) |
| 487 |
|
\end{equation} |
| 488 |
< |
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
| 488 |
> |
where $x = x(q,p)$, this system is a canonical Hamiltonian, if |
| 489 |
|
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian |
| 490 |
|
function and $J$ is the skew-symmetric matrix |
| 491 |
|
\begin{equation} |
| 502 |
|
\label{introEquation:compactHamiltonian} |
| 503 |
|
\end{equation}In this case, $f$ is |
| 504 |
|
called a \emph{Hamiltonian vector field}. Another generalization of |
| 505 |
< |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
| 505 |
> |
Hamiltonian dynamics is Poisson Dynamics,\cite{Olver1986} |
| 506 |
|
\begin{equation} |
| 507 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 508 |
|
\end{equation} |
| 509 |
< |
The most obvious change being that matrix $J$ now depends on $x$. |
| 509 |
> |
where the most obvious change being that matrix $J$ now depends on |
| 510 |
> |
$x$. |
| 511 |
|
|
| 512 |
|
\subsection{\label{introSection:exactFlow}Exact Propagator} |
| 513 |
|
|
| 534 |
|
\begin{equation} |
| 535 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 536 |
|
\end{equation} |
| 530 |
– |
The exact propagator can also be written in terms of operator, |
| 531 |
– |
\begin{equation} |
| 532 |
– |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 533 |
– |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 534 |
– |
\label{introEquation:exponentialOperator} |
| 535 |
– |
\end{equation} |
| 537 |
|
In most cases, it is not easy to find the exact propagator |
| 538 |
|
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
| 539 |
|
which is usually called an integrator. The order of an integrator |
| 621 |
|
Generating functions\cite{Channell1990} tend to lead to methods |
| 622 |
|
which are cumbersome and difficult to use. In dissipative systems, |
| 623 |
|
variational methods can capture the decay of energy |
| 624 |
< |
accurately\cite{Kane2000}. Since they are geometrically unstable |
| 624 |
> |
accurately.\cite{Kane2000} Since they are geometrically unstable |
| 625 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 626 |
< |
methods are not suitable for Hamiltonian system. Recently, various |
| 627 |
< |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
| 628 |
< |
have been developed to overcome this instability. However, due to |
| 629 |
< |
computational penalty involved in implementing the Runge-Kutta |
| 630 |
< |
methods, they have not attracted much attention from the Molecular |
| 631 |
< |
Dynamics community. Instead, splitting methods have been widely |
| 632 |
< |
accepted since they exploit natural decompositions of the |
| 633 |
< |
system\cite{Tuckerman1992, McLachlan1998}. |
| 626 |
> |
methods are not suitable for Hamiltonian |
| 627 |
> |
system.\cite{Cartwright1992} Recently, various high-order explicit |
| 628 |
> |
Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to |
| 629 |
> |
overcome this instability. However, due to computational penalty |
| 630 |
> |
involved in implementing the Runge-Kutta methods, they have not |
| 631 |
> |
attracted much attention from the Molecular Dynamics community. |
| 632 |
> |
Instead, splitting methods have been widely accepted since they |
| 633 |
> |
exploit natural decompositions of the system.\cite{McLachlan1998, |
| 634 |
> |
Tuckerman1992} |
| 635 |
|
|
| 636 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 637 |
|
|
| 654 |
|
problem. If $H_1$ and $H_2$ can be integrated using exact |
| 655 |
|
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
| 656 |
|
simple first order expression is then given by the Lie-Trotter |
| 657 |
< |
formula |
| 657 |
> |
formula\cite{Trotter1959} |
| 658 |
|
\begin{equation} |
| 659 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 660 |
|
\label{introEquation:firstOrderSplitting} |
| 675 |
|
The Lie-Trotter |
| 676 |
|
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
| 677 |
|
local errors proportional to $h^2$, while the Strang splitting gives |
| 678 |
< |
a second-order decomposition, |
| 678 |
> |
a second-order decomposition,\cite{Strang1968} |
| 679 |
|
\begin{equation} |
| 680 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 681 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 731 |
|
|
| 732 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 733 |
|
|
| 734 |
< |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
| 734 |
> |
\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move. |
| 735 |
|
|
| 736 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 737 |
|
\end{enumerate} |
| 750 |
|
|
| 751 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 752 |
|
|
| 753 |
< |
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 754 |
< |
local error of a splitting method in terms of the commutator of the |
| 755 |
< |
operators(Eq.~\ref{introEquation:exponentialOperator}) associated with |
| 756 |
< |
the sub-propagator. For operators $hX$ and $hY$ which are associated |
| 757 |
< |
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 753 |
> |
The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used |
| 754 |
> |
to determine the local error of a splitting method in terms of the |
| 755 |
> |
commutator of the |
| 756 |
> |
operators associated |
| 757 |
> |
with the sub-propagator. For operators $hX$ and $hY$ which are |
| 758 |
> |
associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we |
| 759 |
> |
have |
| 760 |
|
\begin{equation} |
| 761 |
|
\exp (hX + hY) = \exp (hZ) |
| 762 |
|
\end{equation} |
| 786 |
|
\end{equation} |
| 787 |
|
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 788 |
|
order methods. Yoshida proposed an elegant way to compose higher |
| 789 |
< |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 789 |
> |
order methods based on symmetric splitting.\cite{Yoshida1990} Given |
| 790 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 791 |
|
fourth-order symmetric method can be constructed by composing, |
| 792 |
|
\[ |
| 872 |
|
minimization to find a more reasonable conformation. Several energy |
| 873 |
|
minimization methods have been developed to exploit the energy |
| 874 |
|
surface and to locate the local minimum. While converging slowly |
| 875 |
< |
near the minimum, steepest descent method is extremely robust when |
| 875 |
> |
near the minimum, the steepest descent method is extremely robust when |
| 876 |
|
systems are strongly anharmonic. Thus, it is often used to refine |
| 877 |
|
structures from crystallographic data. Relying on the Hessian, |
| 878 |
|
advanced methods like Newton-Raphson converge rapidly to a local |
| 891 |
|
temperature. Beginning at a lower temperature and gradually |
| 892 |
|
increasing the temperature by assigning larger random velocities, we |
| 893 |
|
end up setting the temperature of the system to a final temperature |
| 894 |
< |
at which the simulation will be conducted. In heating phase, we |
| 894 |
> |
at which the simulation will be conducted. In the heating phase, we |
| 895 |
|
should also keep the system from drifting or rotating as a whole. To |
| 896 |
|
do this, the net linear momentum and angular momentum of the system |
| 897 |
|
is shifted to zero after each resampling from the Maxwell -Boltzman |
| 947 |
|
%cutoff and minimum image convention |
| 948 |
|
Another important technique to improve the efficiency of force |
| 949 |
|
evaluation is to apply spherical cutoffs where particles farther |
| 950 |
< |
than a predetermined distance are not included in the calculation |
| 951 |
< |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 952 |
< |
discontinuity in the potential energy curve. Fortunately, one can |
| 950 |
> |
than a predetermined distance are not included in the |
| 951 |
> |
calculation.\cite{Frenkel1996} The use of a cutoff radius will cause |
| 952 |
> |
a discontinuity in the potential energy curve. Fortunately, one can |
| 953 |
|
shift a simple radial potential to ensure the potential curve go |
| 954 |
|
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 955 |
|
well for Lennard-Jones interaction because of its short range |
| 958 |
|
in simulations. The Ewald summation, in which the slowly decaying |
| 959 |
|
Coulomb potential is transformed into direct and reciprocal sums |
| 960 |
|
with rapid and absolute convergence, has proved to minimize the |
| 961 |
< |
periodicity artifacts in liquid simulations. Taking the advantages |
| 962 |
< |
of the fast Fourier transform (FFT) for calculating discrete Fourier |
| 963 |
< |
transforms, the particle mesh-based |
| 961 |
> |
periodicity artifacts in liquid simulations. Taking advantage of |
| 962 |
> |
fast Fourier transform (FFT) techniques for calculating discrete |
| 963 |
> |
Fourier transforms, the particle mesh-based |
| 964 |
|
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 965 |
|
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
| 966 |
|
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
| 970 |
|
simulation community, these two methods are difficult to implement |
| 971 |
|
correctly and efficiently. Instead, we use a damped and |
| 972 |
|
charge-neutralized Coulomb potential method developed by Wolf and |
| 973 |
< |
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
| 973 |
> |
his coworkers.\cite{Wolf1999} The shifted Coulomb potential for |
| 974 |
|
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 975 |
|
\begin{equation} |
| 976 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 989 |
|
\label{introFigure:shiftedCoulomb} |
| 990 |
|
\end{figure} |
| 991 |
|
|
| 988 |
– |
%multiple time step |
| 989 |
– |
|
| 992 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 993 |
|
|
| 994 |
< |
Recently, advanced visualization techniques have been applied to |
| 993 |
< |
monitor the motions of molecules. Although the dynamics of the |
| 994 |
< |
system can be described qualitatively from animation, quantitative |
| 995 |
< |
trajectory analysis is more useful. According to the principles of |
| 994 |
> |
According to the principles of |
| 995 |
|
Statistical Mechanics in |
| 996 |
|
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 997 |
|
thermodynamic properties, analyze fluctuations of structural |
| 1028 |
|
function}, is of most fundamental importance to liquid theory. |
| 1029 |
|
Experimentally, pair distribution functions can be gathered by |
| 1030 |
|
Fourier transforming raw data from a series of neutron diffraction |
| 1031 |
< |
experiments and integrating over the surface factor |
| 1032 |
< |
\cite{Powles1973}. The experimental results can serve as a criterion |
| 1033 |
< |
to justify the correctness of a liquid model. Moreover, various |
| 1034 |
< |
equilibrium thermodynamic and structural properties can also be |
| 1035 |
< |
expressed in terms of the radial distribution function |
| 1036 |
< |
\cite{Allen1987}. The pair distribution functions $g(r)$ gives the |
| 1037 |
< |
probability that a particle $i$ will be located at a distance $r$ |
| 1038 |
< |
from a another particle $j$ in the system |
| 1031 |
> |
experiments and integrating over the surface |
| 1032 |
> |
factor.\cite{Powles1973} The experimental results can serve as a |
| 1033 |
> |
criterion to justify the correctness of a liquid model. Moreover, |
| 1034 |
> |
various equilibrium thermodynamic and structural properties can also |
| 1035 |
> |
be expressed in terms of the radial distribution |
| 1036 |
> |
function.\cite{Allen1987} The pair distribution functions $g(r)$ |
| 1037 |
> |
gives the probability that a particle $i$ will be located at a |
| 1038 |
> |
distance $r$ from a another particle $j$ in the system |
| 1039 |
|
\begin{equation} |
| 1040 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1041 |
|
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
| 1058 |
|
\label{introEquation:timeCorrelationFunction} |
| 1059 |
|
\end{equation} |
| 1060 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1061 |
< |
functions are called \emph{autocorrelation functions}. One example |
| 1063 |
< |
of auto correlation function is the velocity auto-correlation |
| 1061 |
> |
functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation |
| 1062 |
|
function which is directly related to transport properties of |
| 1063 |
|
molecular liquids: |
| 1064 |
< |
\[ |
| 1064 |
> |
\begin{equation} |
| 1065 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
| 1066 |
|
\right\rangle } dt |
| 1067 |
< |
\] |
| 1067 |
> |
\end{equation} |
| 1068 |
|
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1069 |
|
function, which is averaged over time origins and over all the |
| 1070 |
|
atoms, the dipole autocorrelation functions is calculated for the |
| 1071 |
|
entire system. The dipole autocorrelation function is given by: |
| 1072 |
< |
\[ |
| 1072 |
> |
\begin{equation} |
| 1073 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1074 |
|
\right\rangle |
| 1075 |
< |
\] |
| 1075 |
> |
\end{equation} |
| 1076 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
| 1077 |
|
by |
| 1078 |
< |
\[ |
| 1078 |
> |
\begin{equation} |
| 1079 |
|
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
| 1080 |
< |
\] |
| 1080 |
> |
\end{equation} |
| 1081 |
|
In principle, many time correlation functions can be related to |
| 1082 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1083 |
|
scattering spectra of molecular liquids. In practice, one can |
| 1084 |
|
extract the IR spectrum from the intensity of the molecular dipole |
| 1085 |
|
fluctuation at each frequency using the following relationship: |
| 1086 |
< |
\[ |
| 1086 |
> |
\begin{equation} |
| 1087 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1088 |
|
i2\pi vt} dt}. |
| 1089 |
< |
\] |
| 1089 |
> |
\end{equation} |
| 1090 |
|
|
| 1091 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1092 |
|
|
| 1093 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1094 |
< |
areas, from engineering, physics, to chemistry. For example, |
| 1094 |
> |
areas, including engineering, physics and chemistry. For example, |
| 1095 |
|
missiles and vehicles are usually modeled by rigid bodies. The |
| 1096 |
|
movement of the objects in 3D gaming engines or other physics |
| 1097 |
|
simulators is governed by rigid body dynamics. In molecular |
| 1098 |
|
simulations, rigid bodies are used to simplify protein-protein |
| 1099 |
< |
docking studies\cite{Gray2003}. |
| 1099 |
> |
docking studies.\cite{Gray2003} |
| 1100 |
|
|
| 1101 |
|
It is very important to develop stable and efficient methods to |
| 1102 |
|
integrate the equations of motion for orientational degrees of |
| 1108 |
|
angles can overcome this difficulty\cite{Barojas1973}, the |
| 1109 |
|
computational penalty and the loss of angular momentum conservation |
| 1110 |
|
still remain. A singularity-free representation utilizing |
| 1111 |
< |
quaternions was developed by Evans in 1977\cite{Evans1977}. |
| 1111 |
> |
quaternions was developed by Evans in 1977.\cite{Evans1977} |
| 1112 |
|
Unfortunately, this approach used a nonseparable Hamiltonian |
| 1113 |
|
resulting from the quaternion representation, which prevented the |
| 1114 |
|
symplectic algorithm from being utilized. Another different approach |
| 1117 |
|
deriving from potential energy and constraint forces which are used |
| 1118 |
|
to guarantee the rigidness. However, due to their iterative nature, |
| 1119 |
|
the SHAKE and Rattle algorithms also converge very slowly when the |
| 1120 |
< |
number of constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1120 |
> |
number of constraints increases.\cite{Ryckaert1977, Andersen1983} |
| 1121 |
|
|
| 1122 |
|
A break-through in geometric literature suggests that, in order to |
| 1123 |
|
develop a long-term integration scheme, one should preserve the |
| 1127 |
|
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1128 |
|
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 1129 |
|
An alternative method using the quaternion representation was |
| 1130 |
< |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
| 1130 |
> |
developed by Omelyan.\cite{Omelyan1998} However, both of these |
| 1131 |
|
methods are iterative and inefficient. In this section, we descibe a |
| 1132 |
|
symplectic Lie-Poisson integrator for rigid bodies developed by |
| 1133 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1134 |
|
|
| 1135 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1136 |
< |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
| 1139 |
< |
function |
| 1136 |
> |
The Hamiltonian of a rigid body is given by |
| 1137 |
|
\begin{equation} |
| 1138 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1139 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1192 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1193 |
|
\] |
| 1194 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1195 |
< |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1195 |
> |
the rigid body, its corresponding lab fixed vector $X_i^{lab}$ is |
| 1196 |
|
given as |
| 1197 |
|
\begin{equation} |
| 1198 |
|
X_i^{lab} = Q X_i + q. |
| 1247 |
|
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
| 1248 |
|
Lagrange multiplier $\Lambda$ is absent from the equations of |
| 1249 |
|
motion. This unique property eliminates the requirement of |
| 1250 |
< |
iterations which can not be avoided in other methods\cite{Kol1997, |
| 1251 |
< |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
| 1252 |
< |
equation of motion for angular momentum in the body frame |
| 1250 |
> |
iterations which can not be avoided in other methods.\cite{Kol1997, |
| 1251 |
> |
Omelyan1998} Applying the hat-map isomorphism, we obtain the |
| 1252 |
> |
equation of motion for angular momentum |
| 1253 |
|
\begin{equation} |
| 1254 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1255 |
|
F_i (r,Q)} \right) \times X_i }. |
| 1345 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1346 |
|
_1 }. |
| 1347 |
|
\] |
| 1348 |
< |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1352 |
< |
$F(\pi )$ and $G(\pi )$ is defined by |
| 1348 |
> |
The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by |
| 1349 |
|
\[ |
| 1350 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1351 |
|
). |
| 1354 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1355 |
|
conserved quantity in Poisson system. We can easily verify that the |
| 1356 |
|
norm of the angular momentum, $\parallel \pi |
| 1357 |
< |
\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1357 |
> |
\parallel$, is a \emph{Casimir}.\cite{McLachlan1993} Let $F(\pi ) = S(\frac{{\parallel |
| 1358 |
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1359 |
|
then by the chain rule |
| 1360 |
|
\[ |
| 1375 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1376 |
|
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations |
| 1377 |
|
of motion corresponding to potential energy and kinetic energy are |
| 1378 |
< |
listed in Table~\ref{introTable:rbEquations} |
| 1378 |
> |
listed in Table~\ref{introTable:rbEquations}. |
| 1379 |
|
\begin{table} |
| 1380 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1381 |
|
\label{introTable:rbEquations} |
| 1433 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1434 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
| 1435 |
|
has been applied in a variety of studies. This section will review |
| 1436 |
< |
the theory of Langevin dynamics. A brief derivation of generalized |
| 1436 |
> |
the theory of Langevin dynamics. A brief derivation of the generalized |
| 1437 |
|
Langevin equation will be given first. Following that, we will |
| 1438 |
< |
discuss the physical meaning of the terms appearing in the equation |
| 1443 |
< |
as well as the calculation of friction tensor from hydrodynamics |
| 1444 |
< |
theory. |
| 1438 |
> |
discuss the physical meaning of the terms appearing in the equation. |
| 1439 |
|
|
| 1440 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1441 |
|
|
| 1444 |
|
environment, has been widely used in quantum chemistry and |
| 1445 |
|
statistical mechanics. One of the successful applications of |
| 1446 |
|
Harmonic bath model is the derivation of the Generalized Langevin |
| 1447 |
< |
Dynamics (GLE). Lets consider a system, in which the degree of |
| 1447 |
> |
Dynamics (GLE). Consider a system, in which the degree of |
| 1448 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1449 |
|
Hamiltonian of the form |
| 1450 |
|
\begin{equation} |
| 1455 |
|
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
| 1456 |
|
\[ |
| 1457 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1458 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1458 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha x_\alpha ^2 } |
| 1459 |
|
\right\}} |
| 1460 |
|
\] |
| 1461 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1508 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1509 |
|
\] |
| 1510 |
|
where $p$ is real and $L$ is called the Laplace Transform |
| 1511 |
< |
Operator. Below are some important properties of Laplace transform |
| 1511 |
> |
Operator. Below are some important properties of the Laplace transform |
| 1512 |
|
\begin{eqnarray*} |
| 1513 |
|
L(x + y) & = & L(x) + L(y) \\ |
| 1514 |
|
L(ax) & = & aL(x) \\ |
| 1577 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1578 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1579 |
|
\end{equation} |
| 1580 |
< |
which is known as the \emph{generalized Langevin equation}. |
| 1580 |
> |
which is known as the \emph{generalized Langevin equation} (GLE). |
| 1581 |
|
|
| 1582 |
|
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
| 1583 |
|
|
| 1584 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
| 1585 |
|
implies it is completely deterministic within the context of a |
| 1586 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1587 |
< |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
| 1587 |
> |
uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)} |
| 1588 |
|
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
| 1589 |
|
0.$ This property is what we expect from a truly random process. As |
| 1590 |
|
long as the model chosen for $R(t)$ was a gaussian distribution in |
| 1613 |
|
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
| 1614 |
|
taken as a $delta$ function in time: |
| 1615 |
|
\[ |
| 1616 |
< |
\xi (t) = 2\xi _0 \delta (t) |
| 1616 |
> |
\xi (t) = 2\xi _0 \delta (t). |
| 1617 |
|
\] |
| 1618 |
|
Hence, the convolution integral becomes |
| 1619 |
|
\[ |
| 1620 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1621 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1622 |
|
\] |
| 1623 |
< |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1623 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes the |
| 1624 |
> |
Langevin equation |
| 1625 |
|
\begin{equation} |
| 1626 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1627 |
< |
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1627 |
> |
x(t) + R(t) \label{introEquation:LangevinEquation}. |
| 1628 |
|
\end{equation} |
| 1629 |
< |
which is known as the Langevin equation. The static friction |
| 1630 |
< |
coefficient $\xi _0$ can either be calculated from spectral density |
| 1631 |
< |
or be determined by Stokes' law for regular shaped particles. A |
| 1632 |
< |
brief review on calculating friction tensors for arbitrary shaped |
| 1633 |
< |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1629 |
> |
The static friction coefficient $\xi _0$ can either be calculated |
| 1630 |
> |
from spectral density or be determined by Stokes' law for regular |
| 1631 |
> |
shaped particles. A brief review on calculating friction tensors for |
| 1632 |
> |
arbitrary shaped particles is given in |
| 1633 |
> |
Sec.~\ref{introSection:frictionTensor}. |
| 1634 |
|
|
| 1635 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1636 |
|
|
| 1639 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1640 |
|
^2 }}x(0), |
| 1641 |
|
\] |
| 1642 |
< |
we can rewrite $R(T)$ as |
| 1642 |
> |
we can rewrite $R(t)$ as |
| 1643 |
|
\[ |
| 1644 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1645 |
|
\] |
