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. |
260 |
– |
|
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 |
– |
|
240 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
241 |
|
Mechanics} |
242 |
|
|
277 |
|
system lends itself to a time averaging approach, the Molecular |
278 |
|
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
279 |
|
will be the best choice\cite{Frenkel1996}. |
280 |
+ |
|
281 |
+ |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
282 |
+ |
A variety of numerical integrators were proposed to simulate the |
283 |
+ |
motions. They usually begin with an initial conditionals and move |
284 |
+ |
the objects in the direction governed by the differential equations. |
285 |
+ |
However, most of them ignore the hidden physical law contained |
286 |
+ |
within the equations. Since 1990, geometric integrators, which |
287 |
+ |
preserve various phase-flow invariants such as symplectic structure, |
288 |
+ |
volume and time reversal symmetry, are developed to address this |
289 |
+ |
issue. The velocity verlet method, which happens to be a simple |
290 |
+ |
example of symplectic integrator, continues to gain its popularity |
291 |
+ |
in molecular dynamics community. This fact can be partly explained |
292 |
+ |
by its geometric nature. |
293 |
+ |
|
294 |
+ |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
295 |
+ |
A \emph{manifold} is an abstract mathematical space. It locally |
296 |
+ |
looks like Euclidean space, but when viewed globally, it may have |
297 |
+ |
more complicate structure. A good example of manifold is the surface |
298 |
+ |
of Earth. It seems to be flat locally, but it is round if viewed as |
299 |
+ |
a whole. A \emph{differentiable manifold} (also known as |
300 |
+ |
\emph{smooth manifold}) is a manifold with an open cover in which |
301 |
+ |
the covering neighborhoods are all smoothly isomorphic to one |
302 |
+ |
another. In other words,it is possible to apply calculus on |
303 |
+ |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
304 |
+ |
defined as a pair $(M, \omega)$ which consisting of a |
305 |
+ |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
306 |
+ |
bilinear symplectic form, $\omega$. A symplectic form on a vector |
307 |
+ |
space $V$ is a function $\omega(x, y)$ which satisfies |
308 |
+ |
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
309 |
+ |
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
310 |
+ |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
311 |
+ |
example of symplectic form. |
312 |
+ |
|
313 |
+ |
One of the motivations to study \emph{symplectic manifold} in |
314 |
+ |
Hamiltonian Mechanics is that a symplectic manifold can represent |
315 |
+ |
all possible configurations of the system and the phase space of the |
316 |
+ |
system can be described by it's cotangent bundle. Every symplectic |
317 |
+ |
manifold is even dimensional. For instance, in Hamilton equations, |
318 |
+ |
coordinate and momentum always appear in pairs. |
319 |
|
|
320 |
+ |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
321 |
+ |
\[ |
322 |
+ |
f : M \rightarrow N |
323 |
+ |
\] |
324 |
+ |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
325 |
+ |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
326 |
+ |
Canonical transformation is an example of symplectomorphism in |
327 |
+ |
classical mechanics. According to Liouville's theorem, the |
328 |
+ |
Hamiltonian \emph{flow} or \emph{symplectomorphism} generated by the |
329 |
+ |
Hamiltonian vector filed preserves the volume form on the phase |
330 |
+ |
space, which is the basis of classical statistical mechanics. |
331 |
+ |
|
332 |
+ |
\subsection{\label{introSection:exactFlow}The Exact Flow of ODE} |
333 |
+ |
|
334 |
+ |
\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting} |
335 |
+ |
|
336 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
337 |
|
|
338 |
|
As a special discipline of molecular modeling, Molecular dynamics |