| 212 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
| 213 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
| 214 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
| 215 |
< |
q_i }}} \right) = 0} |
| 216 |
< |
\label{introEquation:conserveHalmitonian} |
| 215 |
> |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
| 216 |
|
\end{equation} |
| 218 |
– |
|
| 219 |
– |
When studying Hamiltonian system, it is more convenient to use |
| 220 |
– |
notation |
| 221 |
– |
\begin{equation} |
| 222 |
– |
r = r(q,p)^T |
| 223 |
– |
\end{equation} |
| 224 |
– |
and to introduce a $2n \times 2n$ canonical structure matrix $J$, |
| 225 |
– |
\begin{equation} |
| 226 |
– |
J = \left( {\begin{array}{*{20}c} |
| 227 |
– |
0 & I \\ |
| 228 |
– |
{ - I} & 0 \\ |
| 229 |
– |
\end{array}} \right) |
| 230 |
– |
\label{introEquation:canonicalMatrix} |
| 231 |
– |
\end{equation} |
| 232 |
– |
where $I$ is a $n \times n$ identity matrix and $J$ is a |
| 233 |
– |
skew-symmetric matrix ($ J^T = - J $). Thus, Hamiltonian system |
| 234 |
– |
can be rewritten as, |
| 235 |
– |
\begin{equation} |
| 236 |
– |
\frac{d}{{dt}}r = J\nabla _r H(r) |
| 237 |
– |
\label{introEquation:compactHamiltonian} |
| 238 |
– |
\end{equation} |
| 239 |
– |
|
| 240 |
– |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 241 |
– |
|
| 242 |
– |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 243 |
– |
A \emph{manifold} is an abstract mathematical space. It locally |
| 244 |
– |
looks like Euclidean space, but when viewed globally, it may have |
| 245 |
– |
more complicate structure. A good example of manifold is the surface |
| 246 |
– |
of Earth. It seems to be flat locally, but it is round if viewed as |
| 247 |
– |
a whole. A \emph{differentiable manifold} (also known as |
| 248 |
– |
\emph{smooth manifold}) is a manifold with an open cover in which |
| 249 |
– |
the covering neighborhoods are all smoothly isomorphic to one |
| 250 |
– |
another. In other words,it is possible to apply calculus on |
| 251 |
– |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
| 252 |
– |
defined as a pair $(M, \omega)$ consisting of a \emph{differentiable |
| 253 |
– |
manifold} $M$ and a close, non-degenerated, bilinear symplectic |
| 254 |
– |
form, $\omega$. One of the motivation to study \emph{symplectic |
| 255 |
– |
manifold} in Hamiltonian Mechanics is that a symplectic manifold can |
| 256 |
– |
represent all possible configurations of the system and the phase |
| 257 |
– |
space of the system can be described by it's cotangent bundle. Every |
| 258 |
– |
symplectic manifold is even dimensional. For instance, in Hamilton |
| 259 |
– |
equations, coordinate and momentum always appear in pairs. |
| 217 |
|
|
| 261 |
– |
A \emph{symplectomorphism} is also known as a \emph{canonical |
| 262 |
– |
transformation}. |
| 263 |
– |
|
| 264 |
– |
Any real-valued differentiable function H on a symplectic manifold |
| 265 |
– |
can serve as an energy function or Hamiltonian. Associated to any |
| 266 |
– |
Hamiltonian is a Hamiltonian vector field; the integral curves of |
| 267 |
– |
the Hamiltonian vector field are solutions to the Hamilton-Jacobi |
| 268 |
– |
equations. The Hamiltonian vector field defines a flow on the |
| 269 |
– |
symplectic manifold, called a Hamiltonian flow or symplectomorphism. |
| 270 |
– |
By Liouville's theorem, Hamiltonian flows preserve the volume form |
| 271 |
– |
on the phase space. |
| 272 |
– |
|
| 273 |
– |
\subsection{\label{Construction of Symplectic Methods}} |
| 274 |
– |
|
| 218 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
| 219 |
|
Mechanics} |
| 220 |
|
|
| 255 |
|
system lends itself to a time averaging approach, the Molecular |
| 256 |
|
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
| 257 |
|
will be the best choice\cite{Frenkel1996}. |
| 258 |
+ |
|
| 259 |
+ |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 260 |
+ |
A variety of numerical integrators were proposed to simulate the |
| 261 |
+ |
motions. They usually begin with an initial conditionals and move |
| 262 |
+ |
the objects in the direction governed by the differential equations. |
| 263 |
+ |
However, most of them ignore the hidden physical law contained |
| 264 |
+ |
within the equations. Since 1990, geometric integrators, which |
| 265 |
+ |
preserve various phase-flow invariants such as symplectic structure, |
| 266 |
+ |
volume and time reversal symmetry, are developed to address this |
| 267 |
+ |
issue. The velocity verlet method, which happens to be a simple |
| 268 |
+ |
example of symplectic integrator, continues to gain its popularity |
| 269 |
+ |
in molecular dynamics community. This fact can be partly explained |
| 270 |
+ |
by its geometric nature. |
| 271 |
+ |
|
| 272 |
+ |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 273 |
+ |
A \emph{manifold} is an abstract mathematical space. It locally |
| 274 |
+ |
looks like Euclidean space, but when viewed globally, it may have |
| 275 |
+ |
more complicate structure. A good example of manifold is the surface |
| 276 |
+ |
of Earth. It seems to be flat locally, but it is round if viewed as |
| 277 |
+ |
a whole. A \emph{differentiable manifold} (also known as |
| 278 |
+ |
\emph{smooth manifold}) is a manifold with an open cover in which |
| 279 |
+ |
the covering neighborhoods are all smoothly isomorphic to one |
| 280 |
+ |
another. In other words,it is possible to apply calculus on |
| 281 |
+ |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
| 282 |
+ |
defined as a pair $(M, \omega)$ which consisting of a |
| 283 |
+ |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 284 |
+ |
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 285 |
+ |
space $V$ is a function $\omega(x, y)$ which satisfies |
| 286 |
+ |
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 287 |
+ |
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 288 |
+ |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
| 289 |
+ |
example of symplectic form. |
| 290 |
+ |
|
| 291 |
+ |
One of the motivations to study \emph{symplectic manifold} in |
| 292 |
+ |
Hamiltonian Mechanics is that a symplectic manifold can represent |
| 293 |
+ |
all possible configurations of the system and the phase space of the |
| 294 |
+ |
system can be described by it's cotangent bundle. Every symplectic |
| 295 |
+ |
manifold is even dimensional. For instance, in Hamilton equations, |
| 296 |
+ |
coordinate and momentum always appear in pairs. |
| 297 |
+ |
|
| 298 |
+ |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
| 299 |
+ |
\[ |
| 300 |
+ |
f : M \rightarrow N |
| 301 |
+ |
\] |
| 302 |
+ |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
| 303 |
+ |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
| 304 |
+ |
Canonical transformation is an example of symplectomorphism in |
| 305 |
+ |
classical mechanics. |
| 306 |
+ |
|
| 307 |
+ |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 308 |
+ |
|
| 309 |
+ |
For a ordinary differential system defined as |
| 310 |
+ |
\begin{equation} |
| 311 |
+ |
\dot x = f(x) |
| 312 |
+ |
\end{equation} |
| 313 |
+ |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
| 314 |
+ |
\begin{equation} |
| 315 |
+ |
f(r) = J\nabla _x H(r). |
| 316 |
+ |
\end{equation} |
| 317 |
+ |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
| 318 |
+ |
matrix |
| 319 |
+ |
\begin{equation} |
| 320 |
+ |
J = \left( {\begin{array}{*{20}c} |
| 321 |
+ |
0 & I \\ |
| 322 |
+ |
{ - I} & 0 \\ |
| 323 |
+ |
\end{array}} \right) |
| 324 |
+ |
\label{introEquation:canonicalMatrix} |
| 325 |
+ |
\end{equation} |
| 326 |
+ |
where $I$ is an identity matrix. Using this notation, Hamiltonian |
| 327 |
+ |
system can be rewritten as, |
| 328 |
+ |
\begin{equation} |
| 329 |
+ |
\frac{d}{{dt}}x = J\nabla _x H(x) |
| 330 |
+ |
\label{introEquation:compactHamiltonian} |
| 331 |
+ |
\end{equation}In this case, $f$ is |
| 332 |
+ |
called a \emph{Hamiltonian vector field}. |
| 333 |
|
|
| 334 |
+ |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
| 335 |
+ |
\begin{equation} |
| 336 |
+ |
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 337 |
+ |
\end{equation} |
| 338 |
+ |
The most obvious change being that matrix $J$ now depends on $x$. |
| 339 |
+ |
The free rigid body is an example of Poisson system (actually a |
| 340 |
+ |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
| 341 |
+ |
energy. |
| 342 |
+ |
\begin{equation} |
| 343 |
+ |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 344 |
+ |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 345 |
+ |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 346 |
+ |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 347 |
+ |
\end{array}} \right) |
| 348 |
+ |
\end{equation} |
| 349 |
+ |
|
| 350 |
+ |
\begin{equation} |
| 351 |
+ |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
| 352 |
+ |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
| 353 |
+ |
\end{equation} |
| 354 |
+ |
|
| 355 |
+ |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 356 |
+ |
Let $x(t)$ be the exact solution of the ODE system, |
| 357 |
+ |
\begin{equation} |
| 358 |
+ |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
| 359 |
+ |
\end{equation} |
| 360 |
+ |
The exact flow(solution) $\varphi_\tau$ is defined by |
| 361 |
+ |
\[ |
| 362 |
+ |
x(t+\tau) =\varphi_\tau(x(t)) |
| 363 |
+ |
\] |
| 364 |
+ |
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 365 |
+ |
space to itself. In most cases, it is not easy to find the exact |
| 366 |
+ |
flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$, |
| 367 |
+ |
which is usually called integrator. The order of an integrator |
| 368 |
+ |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
| 369 |
+ |
order $p$, |
| 370 |
+ |
\begin{equation} |
| 371 |
+ |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 372 |
+ |
\end{equation} |
| 373 |
+ |
|
| 374 |
+ |
The hidden geometric properties of ODE and its flow play important |
| 375 |
+ |
roles in numerical studies. Let $\varphi$ be the flow of Hamiltonian |
| 376 |
+ |
vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies, |
| 377 |
+ |
\begin{equation} |
| 378 |
+ |
'\varphi^T J '\varphi = J. |
| 379 |
+ |
\end{equation} |
| 380 |
+ |
According to Liouville's theorem, the symplectic volume is invariant |
| 381 |
+ |
under a Hamiltonian flow, which is the basis for classical |
| 382 |
+ |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
| 383 |
+ |
field on a symplectic manifold can be shown to be a |
| 384 |
+ |
symplectomorphism. As to the Poisson system, |
| 385 |
+ |
\begin{equation} |
| 386 |
+ |
'\varphi ^T J '\varphi = J \circ \varphi |
| 387 |
+ |
\end{equation} |
| 388 |
+ |
is the property must be preserved by the integrator. It is possible |
| 389 |
+ |
to construct a \emph{volume-preserving} flow for a source free($ |
| 390 |
+ |
\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi = |
| 391 |
+ |
1$. Changing the variables $y = h(x)$ in a |
| 392 |
+ |
ODE\ref{introEquation:ODE} will result in a new system, |
| 393 |
+ |
\[ |
| 394 |
+ |
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 395 |
+ |
\] |
| 396 |
+ |
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 397 |
+ |
In other words, the flow of this vector field is reversible if and |
| 398 |
+ |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. When |
| 399 |
+ |
designing any numerical methods, one should always try to preserve |
| 400 |
+ |
the structural properties of the original ODE and its flow. |
| 401 |
+ |
|
| 402 |
+ |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 403 |
+ |
A lot of well established and very effective numerical methods have |
| 404 |
+ |
been successful precisely because of their symplecticities even |
| 405 |
+ |
though this fact was not recognized when they were first |
| 406 |
+ |
constructed. The most famous example is leapfrog methods in |
| 407 |
+ |
molecular dynamics. In general, symplectic integrators can be |
| 408 |
+ |
constructed using one of four different methods. |
| 409 |
+ |
\begin{enumerate} |
| 410 |
+ |
\item Generating functions |
| 411 |
+ |
\item Variational methods |
| 412 |
+ |
\item Runge-Kutta methods |
| 413 |
+ |
\item Splitting methods |
| 414 |
+ |
\end{enumerate} |
| 415 |
+ |
|
| 416 |
+ |
Generating function tends to lead to methods which are cumbersome |
| 417 |
+ |
and difficult to use\cite{}. In dissipative systems, variational |
| 418 |
+ |
methods can capture the decay of energy accurately\cite{}. Since |
| 419 |
+ |
their geometrically unstable nature against non-Hamiltonian |
| 420 |
+ |
perturbations, ordinary implicit Runge-Kutta methods are not |
| 421 |
+ |
suitable for Hamiltonian system. Recently, various high-order |
| 422 |
+ |
explicit Runge--Kutta methods have been developed to overcome this |
| 423 |
+ |
instability \cite{}. However, due to computational penalty involved |
| 424 |
+ |
in implementing the Runge-Kutta methods, they do not attract too |
| 425 |
+ |
much attention from Molecular Dynamics community. Instead, splitting |
| 426 |
+ |
have been widely accepted since they exploit natural decompositions |
| 427 |
+ |
of the system\cite{Tuckerman92}. The main idea behind splitting |
| 428 |
+ |
methods is to decompose the discrete $\varphi_h$ as a composition of |
| 429 |
+ |
simpler flows, |
| 430 |
+ |
\begin{equation} |
| 431 |
+ |
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 432 |
+ |
\varphi _{h_n } |
| 433 |
+ |
\label{introEquation:FlowDecomposition} |
| 434 |
+ |
\end{equation} |
| 435 |
+ |
where each of the sub-flow is chosen such that each represent a |
| 436 |
+ |
simpler integration of the system. Let $\phi$ and $\psi$ both be |
| 437 |
+ |
symplectic maps, it is easy to show that any composition of |
| 438 |
+ |
symplectic flows yields a symplectic map, |
| 439 |
+ |
\begin{equation} |
| 440 |
+ |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 441 |
+ |
'\phi ' = \phi '^T J\phi ' = J. |
| 442 |
+ |
\label{introEquation:SymplecticFlowComposition} |
| 443 |
+ |
\end{equation} |
| 444 |
+ |
Suppose that a Hamiltonian system has a form with $H = T + V$ |
| 445 |
+ |
|
| 446 |
+ |
|
| 447 |
+ |
|
| 448 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 449 |
|
|
| 450 |
|
As a special discipline of molecular modeling, Molecular dynamics |
