| 3 |
|
\section{\label{introSection:classicalMechanics}Classical |
| 4 |
|
Mechanics} |
| 5 |
|
|
| 6 |
< |
Closely related to Classical Mechanics, Molecular Dynamics |
| 7 |
< |
simulations are carried out by integrating the equations of motion |
| 8 |
< |
for a given system of particles. There are three fundamental ideas |
| 9 |
< |
behind classical mechanics. Firstly, one can determine the state of |
| 10 |
< |
a mechanical system at any time of interest; Secondly, all the |
| 11 |
< |
mechanical properties of the system at that time can be determined |
| 12 |
< |
by combining the knowledge of the properties of the system with the |
| 13 |
< |
specification of this state; Finally, the specification of the state |
| 14 |
< |
when further combine with the laws of mechanics will also be |
| 15 |
< |
sufficient to predict the future behavior of the system. |
| 6 |
> |
Using equations of motion derived from Classical Mechanics, |
| 7 |
> |
Molecular Dynamics simulations are carried out by integrating the |
| 8 |
> |
equations of motion for a given system of particles. There are three |
| 9 |
> |
fundamental ideas behind classical mechanics. Firstly, one can |
| 10 |
> |
determine the state of a mechanical system at any time of interest; |
| 11 |
> |
Secondly, all the mechanical properties of the system at that time |
| 12 |
> |
can be determined by combining the knowledge of the properties of |
| 13 |
> |
the system with the specification of this state; Finally, the |
| 14 |
> |
specification of the state when further combined with the laws of |
| 15 |
> |
mechanics will also be sufficient to predict the future behavior of |
| 16 |
> |
the system. |
| 17 |
|
|
| 18 |
|
\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
| 19 |
|
The discovery of Newton's three laws of mechanics which govern the |
| 72 |
|
|
| 73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 74 |
|
|
| 75 |
< |
Newtonian Mechanics suffers from a important limitation: motions can |
| 75 |
> |
Newtonian Mechanics suffers from an important limitation: motion can |
| 76 |
|
only be described in cartesian coordinate systems which make it |
| 77 |
|
impossible to predict analytically the properties of the system even |
| 78 |
|
if we know all of the details of the interaction. In order to |
| 79 |
|
overcome some of the practical difficulties which arise in attempts |
| 80 |
< |
to apply Newton's equation to complex system, approximate numerical |
| 80 |
> |
to apply Newton's equation to complex systems, approximate numerical |
| 81 |
|
procedures may be developed. |
| 82 |
|
|
| 83 |
|
\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
| 85 |
|
|
| 86 |
|
Hamilton introduced the dynamical principle upon which it is |
| 87 |
|
possible to base all of mechanics and most of classical physics. |
| 88 |
< |
Hamilton's Principle may be stated as follows: the actual |
| 89 |
< |
trajectory, along which a dynamical system may move from one point |
| 90 |
< |
to another within a specified time, is derived by finding the path |
| 91 |
< |
which minimizes the time integral of the difference between the |
| 92 |
< |
kinetic $K$, and potential energies $U$, |
| 88 |
> |
Hamilton's Principle may be stated as follows: the trajectory, along |
| 89 |
> |
which a dynamical system may move from one point to another within a |
| 90 |
> |
specified time, is derived by finding the path which minimizes the |
| 91 |
> |
time integral of the difference between the kinetic $K$, and |
| 92 |
> |
potential energies $U$, |
| 93 |
|
\begin{equation} |
| 94 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}. |
| 95 |
|
\label{introEquation:halmitonianPrinciple1} |
| 214 |
|
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
| 215 |
|
|
| 216 |
|
Mathematically, phase space is the space which represents all |
| 217 |
< |
possible states. Each possible state of the system corresponds to |
| 218 |
< |
one unique point in the phase space. For mechanical systems, the |
| 219 |
< |
phase space usually consists of all possible values of position and |
| 220 |
< |
momentum variables. Consider a dynamic system of $f$ particles in a |
| 221 |
< |
cartesian space, where each of the $6f$ coordinates and momenta is |
| 222 |
< |
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
| 223 |
< |
this system is a $6f$ dimensional space. A point, $x = |
| 217 |
> |
possible states of a system. Each possible state of the system |
| 218 |
> |
corresponds to one unique point in the phase space. For mechanical |
| 219 |
> |
systems, the phase space usually consists of all possible values of |
| 220 |
> |
position and momentum variables. Consider a dynamic system of $f$ |
| 221 |
> |
particles in a cartesian space, where each of the $6f$ coordinates |
| 222 |
> |
and momenta is assigned to one of $6f$ mutually orthogonal axes, the |
| 223 |
> |
phase space of this system is a $6f$ dimensional space. A point, $x |
| 224 |
> |
= |
| 225 |
|
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 226 |
|
\over q} _1 , \ldots |
| 227 |
|
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 243 |
|
\label{introEquation:densityDistribution} |
| 244 |
|
\end{equation} |
| 245 |
|
Governed by the principles of mechanics, the phase points change |
| 246 |
< |
their locations which would change the density at any time at phase |
| 246 |
> |
their locations which changes the density at any time at phase |
| 247 |
|
space. Hence, the density distribution is also to be taken as a |
| 248 |
|
function of the time. The number of systems $\delta N$ at time $t$ |
| 249 |
|
can be determined by, |
| 251 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
| 252 |
|
\label{introEquation:deltaN} |
| 253 |
|
\end{equation} |
| 254 |
< |
Assuming a large enough population of systems, we can sufficiently |
| 254 |
> |
Assuming enough copies of the systems, we can sufficiently |
| 255 |
|
approximate $\delta N$ without introducing discontinuity when we go |
| 256 |
|
from one region in the phase space to another. By integrating over |
| 257 |
|
the whole phase space, |
| 259 |
|
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
| 260 |
|
\label{introEquation:totalNumberSystem} |
| 261 |
|
\end{equation} |
| 262 |
< |
gives us an expression for the total number of the systems. Hence, |
| 263 |
< |
the probability per unit in the phase space can be obtained by, |
| 262 |
> |
gives us an expression for the total number of copies. Hence, the |
| 263 |
> |
probability per unit volume in the phase space can be obtained by, |
| 264 |
|
\begin{equation} |
| 265 |
|
\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int |
| 266 |
|
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 269 |
|
With the help of Eq.~\ref{introEquation:unitProbability} and the |
| 270 |
|
knowledge of the system, it is possible to calculate the average |
| 271 |
|
value of any desired quantity which depends on the coordinates and |
| 272 |
< |
momenta of the system. Even when the dynamics of the real system is |
| 272 |
> |
momenta of the system. Even when the dynamics of the real system are |
| 273 |
|
complex, or stochastic, or even discontinuous, the average |
| 274 |
< |
properties of the ensemble of possibilities as a whole remaining |
| 275 |
< |
well defined. For a classical system in thermal equilibrium with its |
| 274 |
> |
properties of the ensemble of possibilities as a whole remain well |
| 275 |
> |
defined. For a classical system in thermal equilibrium with its |
| 276 |
|
environment, the ensemble average of a mechanical quantity, $\langle |
| 277 |
|
A(q , p) \rangle_t$, takes the form of an integral over the phase |
| 278 |
|
space of the system, |
| 357 |
|
\end{equation} |
| 358 |
|
Liouville's theorem states that the distribution function is |
| 359 |
|
constant along any trajectory in phase space. In classical |
| 360 |
< |
statistical mechanics, since the number of members in an ensemble is |
| 361 |
< |
huge and constant, we can assume the local density has no reason |
| 362 |
< |
(other than classical mechanics) to change, |
| 360 |
> |
statistical mechanics, since the number of system copies in an |
| 361 |
> |
ensemble is huge and constant, we can assume the local density has |
| 362 |
> |
no reason (other than classical mechanics) to change, |
| 363 |
|
\begin{equation} |
| 364 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
| 365 |
|
\label{introEquation:stationary} |
| 389 |
|
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
| 390 |
|
\frac{d}{{dt}}(\delta v) = 0. |
| 391 |
|
\end{equation} |
| 392 |
< |
With the help of stationary assumption |
| 393 |
< |
(\ref{introEquation:stationary}), we obtain the principle of the |
| 392 |
> |
With the help of the stationary assumption |
| 393 |
> |
(Eq.~\ref{introEquation:stationary}), we obtain the principle of |
| 394 |
|
\emph{conservation of volume in phase space}, |
| 395 |
|
\begin{equation} |
| 396 |
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
| 400 |
|
|
| 401 |
|
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
| 402 |
|
|
| 403 |
< |
Liouville's theorem can be expresses in a variety of different forms |
| 403 |
> |
Liouville's theorem can be expressed in a variety of different forms |
| 404 |
|
which are convenient within different contexts. For any two function |
| 405 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
| 406 |
|
bracket ${F, G}$ is defined as |
| 434 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
| 435 |
|
\label{introEquation:liouvilleTheoremInOperator} |
| 436 |
|
\end{equation} |
| 437 |
< |
|
| 437 |
> |
which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$. |
| 438 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
| 439 |
|
|
| 440 |
|
Various thermodynamic properties can be calculated from Molecular |
| 443 |
|
simulation and the quality of the underlying model. However, both |
| 444 |
|
experiments and computer simulations are usually performed during a |
| 445 |
|
certain time interval and the measurements are averaged over a |
| 446 |
< |
period of them which is different from the average behavior of |
| 446 |
> |
period of time which is different from the average behavior of |
| 447 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 448 |
|
Hypothesis makes a connection between time average and the ensemble |
| 449 |
|
average. It states that the time average and average over the |
| 456 |
|
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
| 457 |
|
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
| 458 |
|
distribution function. If an observation is averaged over a |
| 459 |
< |
sufficiently long time (longer than relaxation time), all accessible |
| 460 |
< |
microstates in phase space are assumed to be equally probed, giving |
| 461 |
< |
a properly weighted statistical average. This allows the researcher |
| 462 |
< |
freedom of choice when deciding how best to measure a given |
| 463 |
< |
observable. In case an ensemble averaged approach sounds most |
| 459 |
> |
sufficiently long time (longer than the relaxation time), all |
| 460 |
> |
accessible microstates in phase space are assumed to be equally |
| 461 |
> |
probed, giving a properly weighted statistical average. This allows |
| 462 |
> |
the researcher freedom of choice when deciding how best to measure a |
| 463 |
> |
given observable. In case an ensemble averaged approach sounds most |
| 464 |
|
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
| 465 |
|
utilized. Or if the system lends itself to a time averaging |
| 466 |
|
approach, the Molecular Dynamics techniques in |
| 474 |
|
by the differential equations. However, most of them ignore the |
| 475 |
|
hidden physical laws contained within the equations. Since 1990, |
| 476 |
|
geometric integrators, which preserve various phase-flow invariants |
| 477 |
< |
such as symplectic structure, volume and time reversal symmetry, are |
| 478 |
< |
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
| 479 |
< |
Leimkuhler1999}. The velocity Verlet method, which happens to be a |
| 480 |
< |
simple example of symplectic integrator, continues to gain |
| 481 |
< |
popularity in the molecular dynamics community. This fact can be |
| 482 |
< |
partly explained by its geometric nature. |
| 477 |
> |
such as symplectic structure, volume and time reversal symmetry, |
| 478 |
> |
were developed to address this issue\cite{Dullweber1997, |
| 479 |
> |
McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which |
| 480 |
> |
happens to be a simple example of symplectic integrator, continues |
| 481 |
> |
to gain popularity in the molecular dynamics community. This fact |
| 482 |
> |
can be partly explained by its geometric nature. |
| 483 |
|
|
| 484 |
|
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 485 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
| 488 |
|
surface of Earth. It seems to be flat locally, but it is round if |
| 489 |
|
viewed as a whole. A \emph{differentiable manifold} (also known as |
| 490 |
|
\emph{smooth manifold}) is a manifold on which it is possible to |
| 491 |
< |
apply calculus on \emph{differentiable manifold}. A \emph{symplectic |
| 492 |
< |
manifold} is defined as a pair $(M, \omega)$ which consists of a |
| 491 |
> |
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
| 492 |
> |
defined as a pair $(M, \omega)$ which consists of a |
| 493 |
|
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 494 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 495 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
| 496 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 497 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 498 |
< |
$\omega(x, x) = 0$. The cross product operation in vector field is |
| 499 |
< |
an example of symplectic form. One of the motivations to study |
| 500 |
< |
\emph{symplectic manifolds} in Hamiltonian Mechanics is that a |
| 501 |
< |
symplectic manifold can represent all possible configurations of the |
| 502 |
< |
system and the phase space of the system can be described by it's |
| 503 |
< |
cotangent bundle. Every symplectic manifold is even dimensional. For |
| 504 |
< |
instance, in Hamilton equations, coordinate and momentum always |
| 505 |
< |
appear in pairs. |
| 498 |
> |
$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in |
| 499 |
> |
vector field is an example of symplectic form. One of the |
| 500 |
> |
motivations to study \emph{symplectic manifolds} in Hamiltonian |
| 501 |
> |
Mechanics is that a symplectic manifold can represent all possible |
| 502 |
> |
configurations of the system and the phase space of the system can |
| 503 |
> |
be described by it's cotangent bundle\cite{Jost2002}. Every |
| 504 |
> |
symplectic manifold is even dimensional. For instance, in Hamilton |
| 505 |
> |
equations, coordinate and momentum always appear in pairs. |
| 506 |
|
|
| 507 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 508 |
|
|
| 511 |
|
\dot x = f(x) |
| 512 |
|
\end{equation} |
| 513 |
|
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
| 514 |
< |
$f(r) = J\nabla _x H(r)$. Here, $H = H (q, p)$ is Hamiltonian |
| 514 |
> |
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian |
| 515 |
|
function and $J$ is the skew-symmetric matrix |
| 516 |
|
\begin{equation} |
| 517 |
|
J = \left( {\begin{array}{*{20}c} |
| 533 |
|
\end{equation} |
| 534 |
|
The most obvious change being that matrix $J$ now depends on $x$. |
| 535 |
|
|
| 536 |
< |
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 536 |
> |
\subsection{\label{introSection:exactFlow}Exact Propagator} |
| 537 |
|
|
| 538 |
|
Let $x(t)$ be the exact solution of the ODE |
| 539 |
|
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
| 540 |
< |
define its exact flow(solution) $\varphi_\tau$ |
| 540 |
> |
define its exact propagator(solution) $\varphi_\tau$ |
| 541 |
|
\[ x(t+\tau) |
| 542 |
|
=\varphi_\tau(x(t)) |
| 543 |
|
\] |
| 544 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 545 |
< |
space to itself. The flow has the continuous group property, |
| 545 |
> |
space to itself. The propagator has the continuous group property, |
| 546 |
|
\begin{equation} |
| 547 |
|
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
| 548 |
|
+ \tau _2 } . |
| 551 |
|
\begin{equation} |
| 552 |
|
\varphi _\tau \circ \varphi _{ - \tau } = I |
| 553 |
|
\end{equation} |
| 554 |
< |
Therefore, the exact flow is self-adjoint, |
| 554 |
> |
Therefore, the exact propagator is self-adjoint, |
| 555 |
|
\begin{equation} |
| 556 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 557 |
|
\end{equation} |
| 558 |
< |
The exact flow can also be written in terms of the of an operator, |
| 558 |
> |
The exact propagator can also be written in terms of operator, |
| 559 |
|
\begin{equation} |
| 560 |
|
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 561 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 562 |
|
\label{introEquation:exponentialOperator} |
| 563 |
|
\end{equation} |
| 564 |
< |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
| 565 |
< |
Instead, we use an approximate map, $\psi_\tau$, which is usually |
| 566 |
< |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
| 567 |
< |
the Taylor series of $\psi_\tau$ agree to order $p$, |
| 564 |
> |
In most cases, it is not easy to find the exact propagator |
| 565 |
> |
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
| 566 |
> |
which is usually called an integrator. The order of an integrator |
| 567 |
> |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
| 568 |
> |
order $p$, |
| 569 |
|
\begin{equation} |
| 570 |
|
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 571 |
|
\end{equation} |
| 573 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 574 |
|
|
| 575 |
|
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
| 576 |
< |
ODE and its flow play important roles in numerical studies. Many of |
| 577 |
< |
them can be found in systems which occur naturally in applications. |
| 578 |
< |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 579 |
< |
a \emph{symplectic} flow if it satisfies, |
| 576 |
> |
ODE and its propagator play important roles in numerical studies. |
| 577 |
> |
Many of them can be found in systems which occur naturally in |
| 578 |
> |
applications. Let $\varphi$ be the propagator of Hamiltonian vector |
| 579 |
> |
field, $\varphi$ is a \emph{symplectic} propagator if it satisfies, |
| 580 |
|
\begin{equation} |
| 581 |
|
{\varphi '}^T J \varphi ' = J. |
| 582 |
|
\end{equation} |
| 583 |
|
According to Liouville's theorem, the symplectic volume is invariant |
| 584 |
< |
under a Hamiltonian flow, which is the basis for classical |
| 585 |
< |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
| 586 |
< |
field on a symplectic manifold can be shown to be a |
| 584 |
> |
under a Hamiltonian propagator, which is the basis for classical |
| 585 |
> |
statistical mechanics. Furthermore, the propagator of a Hamiltonian |
| 586 |
> |
vector field on a symplectic manifold can be shown to be a |
| 587 |
|
symplectomorphism. As to the Poisson system, |
| 588 |
|
\begin{equation} |
| 589 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 590 |
|
\end{equation} |
| 591 |
|
is the property that must be preserved by the integrator. It is |
| 592 |
< |
possible to construct a \emph{volume-preserving} flow for a source |
| 593 |
< |
free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det |
| 594 |
< |
d\varphi = 1$. One can show easily that a symplectic flow will be |
| 595 |
< |
volume-preserving. Changing the variables $y = h(x)$ in an ODE |
| 596 |
< |
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
| 592 |
> |
possible to construct a \emph{volume-preserving} propagator for a |
| 593 |
> |
source free ODE ($ \nabla \cdot f = 0 $), if the propagator |
| 594 |
> |
satisfies $ \det d\varphi = 1$. One can show easily that a |
| 595 |
> |
symplectic propagator will be volume-preserving. Changing the |
| 596 |
> |
variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will |
| 597 |
> |
result in a new system, |
| 598 |
|
\[ |
| 599 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 600 |
|
\] |
| 601 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 602 |
< |
In other words, the flow of this vector field is reversible if and |
| 603 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
| 604 |
< |
\emph{first integral}, or conserved quantity of a general |
| 605 |
< |
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 606 |
< |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 602 |
> |
In other words, the propagator of this vector field is reversible if |
| 603 |
> |
and only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
| 604 |
> |
conserved quantity of a general differential function is a function |
| 605 |
> |
$ G:R^{2d} \to R^d $ which is constant for all solutions of the ODE |
| 606 |
> |
$\frac{{dx}}{{dt}} = f(x)$ , |
| 607 |
|
\[ |
| 608 |
|
\frac{{dG(x(t))}}{{dt}} = 0. |
| 609 |
|
\] |
| 610 |
< |
Using chain rule, one may obtain, |
| 610 |
> |
Using the chain rule, one may obtain, |
| 611 |
|
\[ |
| 612 |
|
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
| 613 |
|
\] |
| 614 |
< |
which is the condition for conserving \emph{first integral}. For a |
| 615 |
< |
canonical Hamiltonian system, the time evolution of an arbitrary |
| 616 |
< |
smooth function $G$ is given by, |
| 614 |
> |
which is the condition for conserved quantities. For a canonical |
| 615 |
> |
Hamiltonian system, the time evolution of an arbitrary smooth |
| 616 |
> |
function $G$ is given by, |
| 617 |
|
\begin{eqnarray} |
| 618 |
|
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
| 619 |
|
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
| 624 |
|
\[ |
| 625 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 626 |
|
\] |
| 627 |
< |
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 628 |
< |
integral} of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ |
| 629 |
< |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 630 |
< |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 631 |
< |
0$. When designing any numerical methods, one should always try to |
| 632 |
< |
preserve the structural properties of the original ODE and its flow. |
| 627 |
> |
Therefore, the sufficient condition for $G$ to be a conserved |
| 628 |
> |
quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As |
| 629 |
> |
is well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 630 |
> |
is a conserved quantity, which is due to the fact $\{ H,H\} = 0$. |
| 631 |
> |
When designing any numerical methods, one should always try to |
| 632 |
> |
preserve the structural properties of the original ODE and its |
| 633 |
> |
propagator. |
| 634 |
|
|
| 635 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 636 |
|
A lot of well established and very effective numerical methods have |
| 637 |
< |
been successful precisely because of their symplecticities even |
| 637 |
> |
been successful precisely because of their symplectic nature even |
| 638 |
|
though this fact was not recognized when they were first |
| 639 |
|
constructed. The most famous example is the Verlet-leapfrog method |
| 640 |
|
in molecular dynamics. In general, symplectic integrators can be |
| 645 |
|
\item Runge-Kutta methods |
| 646 |
|
\item Splitting methods |
| 647 |
|
\end{enumerate} |
| 648 |
< |
Generating function\cite{Channell1990} tends to lead to methods |
| 648 |
> |
Generating functions\cite{Channell1990} tend to lead to methods |
| 649 |
|
which are cumbersome and difficult to use. In dissipative systems, |
| 650 |
|
variational methods can capture the decay of energy |
| 651 |
< |
accurately\cite{Kane2000}. Since their geometrically unstable nature |
| 651 |
> |
accurately\cite{Kane2000}. Since they are geometrically unstable |
| 652 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 653 |
|
methods are not suitable for Hamiltonian system. Recently, various |
| 654 |
|
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
| 662 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 663 |
|
|
| 664 |
|
The main idea behind splitting methods is to decompose the discrete |
| 665 |
< |
$\varphi_h$ as a composition of simpler flows, |
| 665 |
> |
$\varphi_h$ as a composition of simpler propagators, |
| 666 |
|
\begin{equation} |
| 667 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 668 |
|
\varphi _{h_n } |
| 669 |
|
\label{introEquation:FlowDecomposition} |
| 670 |
|
\end{equation} |
| 671 |
< |
where each of the sub-flow is chosen such that each represent a |
| 672 |
< |
simpler integration of the system. Suppose that a Hamiltonian system |
| 673 |
< |
takes the form, |
| 671 |
> |
where each of the sub-propagator is chosen such that each represent |
| 672 |
> |
a simpler integration of the system. Suppose that a Hamiltonian |
| 673 |
> |
system takes the form, |
| 674 |
|
\[ |
| 675 |
|
H = H_1 + H_2. |
| 676 |
|
\] |
| 677 |
|
Here, $H_1$ and $H_2$ may represent different physical processes of |
| 678 |
|
the system. For instance, they may relate to kinetic and potential |
| 679 |
|
energy respectively, which is a natural decomposition of the |
| 680 |
< |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
| 681 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
| 682 |
< |
order expression is then given by the Lie-Trotter formula |
| 680 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact |
| 681 |
> |
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
| 682 |
> |
simple first order expression is then given by the Lie-Trotter |
| 683 |
> |
formula |
| 684 |
|
\begin{equation} |
| 685 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 686 |
|
\label{introEquation:firstOrderSplitting} |
| 689 |
|
continuous $\varphi _i$ over a time $h$. By definition, as |
| 690 |
|
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
| 691 |
|
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
| 692 |
< |
It is easy to show that any composition of symplectic flows yields a |
| 693 |
< |
symplectic map, |
| 692 |
> |
It is easy to show that any composition of symplectic propagators |
| 693 |
> |
yields a symplectic map, |
| 694 |
|
\begin{equation} |
| 695 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 696 |
|
'\phi ' = \phi '^T J\phi ' = J, |
| 698 |
|
\end{equation} |
| 699 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
| 700 |
|
splitting in this context automatically generates a symplectic map. |
| 695 |
– |
|
| 701 |
|
The Lie-Trotter |
| 702 |
|
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
| 703 |
< |
local errors proportional to $h^2$, while Strang splitting gives a |
| 704 |
< |
second-order decomposition, |
| 703 |
> |
local errors proportional to $h^2$, while the Strang splitting gives |
| 704 |
> |
a second-order decomposition, |
| 705 |
|
\begin{equation} |
| 706 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 707 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 708 |
|
\end{equation} |
| 709 |
< |
which has a local error proportional to $h^3$. The Sprang |
| 709 |
> |
which has a local error proportional to $h^3$. The Strang |
| 710 |
|
splitting's popularity in molecular simulation community attribute |
| 711 |
|
to its symmetric property, |
| 712 |
|
\begin{equation} |
| 777 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 778 |
|
|
| 779 |
|
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 780 |
< |
local error of splitting method in terms of the commutator of the |
| 780 |
> |
local error of a splitting method in terms of the commutator of the |
| 781 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
| 782 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associated with |
| 783 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 782 |
> |
the sub-propagator. For operators $hX$ and $hY$ which are associated |
| 783 |
> |
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 784 |
|
\begin{equation} |
| 785 |
|
\exp (hX + hY) = \exp (hZ) |
| 786 |
|
\end{equation} |
| 789 |
|
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
| 790 |
|
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
| 791 |
|
\end{equation} |
| 792 |
< |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
| 792 |
> |
Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by |
| 793 |
|
\[ |
| 794 |
|
[X,Y] = XY - YX . |
| 795 |
|
\] |
| 796 |
|
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
| 797 |
< |
to the Sprang splitting, we can obtain |
| 797 |
> |
to the Strang splitting, we can obtain |
| 798 |
|
\begin{eqnarray*} |
| 799 |
|
\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 \\ |
| 800 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 802 |
|
). |
| 803 |
|
\end{eqnarray*} |
| 804 |
|
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
| 805 |
< |
error of Spring splitting is proportional to $h^3$. The same |
| 805 |
> |
error of Strang splitting is proportional to $h^3$. The same |
| 806 |
|
procedure can be applied to a general splitting of the form |
| 807 |
|
\begin{equation} |
| 808 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 838 |
|
dynamical information. The basic idea of molecular dynamics is that |
| 839 |
|
macroscopic properties are related to microscopic behavior and |
| 840 |
|
microscopic behavior can be calculated from the trajectories in |
| 841 |
< |
simulations. For instance, instantaneous temperature of an |
| 842 |
< |
Hamiltonian system of $N$ particle can be measured by |
| 841 |
> |
simulations. For instance, instantaneous temperature of a |
| 842 |
> |
Hamiltonian system of $N$ particles can be measured by |
| 843 |
|
\[ |
| 844 |
|
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
| 845 |
|
\] |
| 846 |
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
| 847 |
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
| 848 |
< |
the boltzman constant. |
| 848 |
> |
the Boltzman constant. |
| 849 |
|
|
| 850 |
|
A typical molecular dynamics run consists of three essential steps: |
| 851 |
|
\begin{enumerate} |
| 862 |
|
These three individual steps will be covered in the following |
| 863 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 864 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
| 865 |
< |
will discusse issues in production run. |
| 865 |
> |
will discuss issues of production runs. |
| 866 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 867 |
< |
trajectory analysis. |
| 867 |
> |
analysis of trajectories. |
| 868 |
|
|
| 869 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 870 |
|
|
| 876 |
|
thousands of crystal structures of molecules are discovered every |
| 877 |
|
year, many more remain unknown due to the difficulties of |
| 878 |
|
purification and crystallization. Even for molecules with known |
| 879 |
< |
structure, some important information is missing. For example, a |
| 879 |
> |
structures, some important information is missing. For example, a |
| 880 |
|
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 881 |
< |
be added. Moreover, in order to include electrostatic interaction, |
| 881 |
> |
be added. Moreover, in order to include electrostatic interactions, |
| 882 |
|
one may need to specify the partial charges for individual atoms. |
| 883 |
|
Under some circumstances, we may even need to prepare the system in |
| 884 |
|
a special configuration. For instance, when studying transport |
| 898 |
|
surface and to locate the local minimum. While converging slowly |
| 899 |
|
near the minimum, steepest descent method is extremely robust when |
| 900 |
|
systems are strongly anharmonic. Thus, it is often used to refine |
| 901 |
< |
structure from crystallographic data. Relied on the gradient or |
| 902 |
< |
hessian, advanced methods like Newton-Raphson converge rapidly to a |
| 903 |
< |
local minimum, but become unstable if the energy surface is far from |
| 901 |
> |
structures from crystallographic data. Relying on the Hessian, |
| 902 |
> |
advanced methods like Newton-Raphson converge rapidly to a local |
| 903 |
> |
minimum, but become unstable if the energy surface is far from |
| 904 |
|
quadratic. Another factor that must be taken into account, when |
| 905 |
|
choosing energy minimization method, is the size of the system. |
| 906 |
|
Steepest descent and conjugate gradient can deal with models of any |
| 907 |
|
size. Because of the limits on computer memory to store the hessian |
| 908 |
< |
matrix and the computing power needed to diagonalized these |
| 909 |
< |
matrices, most Newton-Raphson methods can not be used with very |
| 905 |
< |
large systems. |
| 908 |
> |
matrix and the computing power needed to diagonalize these matrices, |
| 909 |
> |
most Newton-Raphson methods can not be used with very large systems. |
| 910 |
|
|
| 911 |
|
\subsubsection{\textbf{Heating}} |
| 912 |
|
|
| 913 |
< |
Typically, Heating is performed by assigning random velocities |
| 913 |
> |
Typically, heating is performed by assigning random velocities |
| 914 |
|
according to a Maxwell-Boltzman distribution for a desired |
| 915 |
|
temperature. Beginning at a lower temperature and gradually |
| 916 |
|
increasing the temperature by assigning larger random velocities, we |
| 917 |
< |
end up with setting the temperature of the system to a final |
| 918 |
< |
temperature at which the simulation will be conducted. In heating |
| 919 |
< |
phase, we should also keep the system from drifting or rotating as a |
| 920 |
< |
whole. To do this, the net linear momentum and angular momentum of |
| 921 |
< |
the system is shifted to zero after each resampling from the Maxwell |
| 922 |
< |
-Boltzman distribution. |
| 917 |
> |
end up setting the temperature of the system to a final temperature |
| 918 |
> |
at which the simulation will be conducted. In heating phase, we |
| 919 |
> |
should also keep the system from drifting or rotating as a whole. To |
| 920 |
> |
do this, the net linear momentum and angular momentum of the system |
| 921 |
> |
is shifted to zero after each resampling from the Maxwell -Boltzman |
| 922 |
> |
distribution. |
| 923 |
|
|
| 924 |
|
\subsubsection{\textbf{Equilibration}} |
| 925 |
|
|
| 946 |
|
calculation of non-bonded forces, such as van der Waals force and |
| 947 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
| 948 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
| 949 |
< |
which making large simulations prohibitive in the absence of any |
| 949 |
> |
which makes large simulations prohibitive in the absence of any |
| 950 |
|
algorithmic tricks. A natural approach to avoid system size issues |
| 951 |
|
is to represent the bulk behavior by a finite number of the |
| 952 |
< |
particles. However, this approach will suffer from the surface |
| 953 |
< |
effect at the edges of the simulation. To offset this, |
| 954 |
< |
\textit{Periodic boundary conditions} (see Fig.~\ref{introFig:pbc}) |
| 955 |
< |
is developed to simulate bulk properties with a relatively small |
| 956 |
< |
number of particles. In this method, the simulation box is |
| 957 |
< |
replicated throughout space to form an infinite lattice. During the |
| 958 |
< |
simulation, when a particle moves in the primary cell, its image in |
| 959 |
< |
other cells move in exactly the same direction with exactly the same |
| 952 |
> |
particles. However, this approach will suffer from surface effects |
| 953 |
> |
at the edges of the simulation. To offset this, \textit{Periodic |
| 954 |
> |
boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to |
| 955 |
> |
simulate bulk properties with a relatively small number of |
| 956 |
> |
particles. In this method, the simulation box is replicated |
| 957 |
> |
throughout space to form an infinite lattice. During the simulation, |
| 958 |
> |
when a particle moves in the primary cell, its image in other cells |
| 959 |
> |
move in exactly the same direction with exactly the same |
| 960 |
|
orientation. Thus, as a particle leaves the primary cell, one of its |
| 961 |
|
images will enter through the opposite face. |
| 962 |
|
\begin{figure} |
| 970 |
|
|
| 971 |
|
%cutoff and minimum image convention |
| 972 |
|
Another important technique to improve the efficiency of force |
| 973 |
< |
evaluation is to apply spherical cutoff where particles farther than |
| 974 |
< |
a predetermined distance are not included in the calculation |
| 973 |
> |
evaluation is to apply spherical cutoffs where particles farther |
| 974 |
> |
than a predetermined distance are not included in the calculation |
| 975 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 976 |
|
discontinuity in the potential energy curve. Fortunately, one can |
| 977 |
< |
shift simple radial potential to ensure the potential curve go |
| 977 |
> |
shift a simple radial potential to ensure the potential curve go |
| 978 |
|
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 979 |
|
well for Lennard-Jones interaction because of its short range |
| 980 |
|
nature. However, simply truncating the electrostatic interaction |
| 1020 |
|
Recently, advanced visualization technique have become applied to |
| 1021 |
|
monitor the motions of molecules. Although the dynamics of the |
| 1022 |
|
system can be described qualitatively from animation, quantitative |
| 1023 |
< |
trajectory analysis are more useful. According to the principles of |
| 1023 |
> |
trajectory analysis is more useful. According to the principles of |
| 1024 |
|
Statistical Mechanics in |
| 1025 |
|
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 1026 |
|
thermodynamic properties, analyze fluctuations of structural |
| 1055 |
|
distribution functions. Among these functions,the \emph{pair |
| 1056 |
|
distribution function}, also known as \emph{radial distribution |
| 1057 |
|
function}, is of most fundamental importance to liquid theory. |
| 1058 |
< |
Experimentally, pair distribution function can be gathered by |
| 1058 |
> |
Experimentally, pair distribution functions can be gathered by |
| 1059 |
|
Fourier transforming raw data from a series of neutron diffraction |
| 1060 |
|
experiments and integrating over the surface factor |
| 1061 |
|
\cite{Powles1973}. The experimental results can serve as a criterion |
| 1062 |
|
to justify the correctness of a liquid model. Moreover, various |
| 1063 |
|
equilibrium thermodynamic and structural properties can also be |
| 1064 |
< |
expressed in terms of radial distribution function \cite{Allen1987}. |
| 1065 |
< |
The pair distribution functions $g(r)$ gives the probability that a |
| 1066 |
< |
particle $i$ will be located at a distance $r$ from a another |
| 1067 |
< |
particle $j$ in the system |
| 1064 |
> |
expressed in terms of the radial distribution function |
| 1065 |
> |
\cite{Allen1987}. The pair distribution functions $g(r)$ gives the |
| 1066 |
> |
probability that a particle $i$ will be located at a distance $r$ |
| 1067 |
> |
from a another particle $j$ in the system |
| 1068 |
|
\begin{equation} |
| 1069 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1070 |
|
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
| 1096 |
|
\right\rangle } dt |
| 1097 |
|
\] |
| 1098 |
|
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1099 |
< |
function, which is averaging over time origins and over all the |
| 1100 |
< |
atoms, the dipole autocorrelation functions are calculated for the |
| 1099 |
> |
function, which is averaged over time origins and over all the |
| 1100 |
> |
atoms, the dipole autocorrelation functions is calculated for the |
| 1101 |
|
entire system. The dipole autocorrelation function is given by: |
| 1102 |
|
\[ |
| 1103 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1108 |
|
\[ |
| 1109 |
|
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
| 1110 |
|
\] |
| 1111 |
< |
In principle, many time correlation functions can be related with |
| 1111 |
> |
In principle, many time correlation functions can be related to |
| 1112 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1113 |
|
scattering spectra of molecular liquids. In practice, one can |
| 1114 |
< |
extract the IR spectrum from the intensity of dipole fluctuation at |
| 1115 |
< |
each frequency using the following relationship: |
| 1114 |
> |
extract the IR spectrum from the intensity of the molecular dipole |
| 1115 |
> |
fluctuation at each frequency using the following relationship: |
| 1116 |
|
\[ |
| 1117 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1118 |
|
i2\pi vt} dt}. |
| 1122 |
|
|
| 1123 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1124 |
|
areas, from engineering, physics, to chemistry. For example, |
| 1125 |
< |
missiles and vehicle are usually modeled by rigid bodies. The |
| 1126 |
< |
movement of the objects in 3D gaming engine or other physics |
| 1127 |
< |
simulator is governed by rigid body dynamics. In molecular |
| 1125 |
> |
missiles and vehicles are usually modeled by rigid bodies. The |
| 1126 |
> |
movement of the objects in 3D gaming engines or other physics |
| 1127 |
> |
simulators is governed by rigid body dynamics. In molecular |
| 1128 |
|
simulations, rigid bodies are used to simplify protein-protein |
| 1129 |
|
docking studies\cite{Gray2003}. |
| 1130 |
|
|
| 1133 |
|
freedom. Euler angles are the natural choice to describe the |
| 1134 |
|
rotational degrees of freedom. However, due to $\frac {1}{sin |
| 1135 |
|
\theta}$ singularities, the numerical integration of corresponding |
| 1136 |
< |
equations of motion is very inefficient and inaccurate. Although an |
| 1137 |
< |
alternative integrator using multiple sets of Euler angles can |
| 1138 |
< |
overcome this difficulty\cite{Barojas1973}, the computational |
| 1139 |
< |
penalty and the loss of angular momentum conservation still remain. |
| 1140 |
< |
A singularity-free representation utilizing quaternions was |
| 1141 |
< |
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
| 1142 |
< |
approach uses a nonseparable Hamiltonian resulting from the |
| 1143 |
< |
quaternion representation, which prevents the symplectic algorithm |
| 1144 |
< |
to be utilized. Another different approach is to apply holonomic |
| 1145 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
| 1146 |
< |
moves independently under the normal forces deriving from potential |
| 1147 |
< |
energy and constraint forces which are used to guarantee the |
| 1148 |
< |
rigidness. However, due to their iterative nature, the SHAKE and |
| 1149 |
< |
Rattle algorithms also converge very slowly when the number of |
| 1150 |
< |
constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1136 |
> |
equations of these motion is very inefficient and inaccurate. |
| 1137 |
> |
Although an alternative integrator using multiple sets of Euler |
| 1138 |
> |
angles can overcome this difficulty\cite{Barojas1973}, the |
| 1139 |
> |
computational penalty and the loss of angular momentum conservation |
| 1140 |
> |
still remain. A singularity-free representation utilizing |
| 1141 |
> |
quaternions was developed by Evans in 1977\cite{Evans1977}. |
| 1142 |
> |
Unfortunately, this approach uses a nonseparable Hamiltonian |
| 1143 |
> |
resulting from the quaternion representation, which prevents the |
| 1144 |
> |
symplectic algorithm from being utilized. Another different approach |
| 1145 |
> |
is to apply holonomic constraints to the atoms belonging to the |
| 1146 |
> |
rigid body. Each atom moves independently under the normal forces |
| 1147 |
> |
deriving from potential energy and constraint forces which are used |
| 1148 |
> |
to guarantee the rigidness. However, due to their iterative nature, |
| 1149 |
> |
the SHAKE and Rattle algorithms also converge very slowly when the |
| 1150 |
> |
number of constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1151 |
|
|
| 1152 |
|
A break-through in geometric literature suggests that, in order to |
| 1153 |
|
develop a long-term integration scheme, one should preserve the |
| 1154 |
< |
symplectic structure of the flow. By introducing a conjugate |
| 1154 |
> |
symplectic structure of the propagator. By introducing a conjugate |
| 1155 |
|
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
| 1156 |
|
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
| 1157 |
|
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1159 |
|
An alternative method using the quaternion representation was |
| 1160 |
|
developed by Omelyan\cite{Omelyan1998}. However, both of these |
| 1161 |
|
methods are iterative and inefficient. In this section, we descibe a |
| 1162 |
< |
symplectic Lie-Poisson integrator for rigid body developed by |
| 1162 |
> |
symplectic Lie-Poisson integrator for rigid bodies developed by |
| 1163 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1164 |
|
|
| 1165 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1352 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1353 |
|
). |
| 1354 |
|
\] |
| 1355 |
< |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1355 |
> |
The propagator maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1356 |
|
manner. In order to construct a second-order symplectic method, we |
| 1357 |
|
split the angular kinetic Hamiltonian function into five terms |
| 1358 |
|
\[ |
