| 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 |
