| 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} |
| 217 |
< |
\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} |
| 215 |
> |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
| 216 |
|
\end{equation} |
| 217 |
|
|
| 218 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
| 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. 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. |
| 305 |
> |
classical mechanics. |
| 306 |
|
|
| 307 |
< |
\subsection{\label{introSection:exactFlow}The Exact Flow of ODE} |
| 307 |
> |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 308 |
|
|
| 309 |
< |
\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting} |
| 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 |
