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 |