63 |
|
\end{equation} |
64 |
|
If there are no external torques acting on a body, the angular |
65 |
|
momentum of it is conserved. The last conservation theorem state |
66 |
< |
that if all forces are conservative, Energy $E = T + V$ is |
67 |
< |
conserved. All of these conserved quantities are important factors |
68 |
< |
to determine the quality of numerical integration scheme for rigid |
69 |
< |
body \cite{Dullweber1997}. |
66 |
> |
that if all forces are conservative, Energy |
67 |
> |
\begin{equation}E = T + V \label{introEquation:energyConservation} |
68 |
> |
\end{equation} |
69 |
> |
is conserved. All of these conserved quantities are |
70 |
> |
important factors to determine the quality of numerical integration |
71 |
> |
scheme for rigid body \cite{Dullweber1997}. |
72 |
|
|
73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
74 |
|
|
202 |
|
independent variables and it only works with 1st-order differential |
203 |
|
equations\cite{Marion90}. |
204 |
|
|
205 |
+ |
In Newtonian Mechanics, a system described by conservative forces |
206 |
+ |
conserves the total energy \ref{introEquation:energyConservation}. |
207 |
+ |
It follows that Hamilton's equations of motion conserve the total |
208 |
+ |
Hamiltonian. |
209 |
+ |
\begin{equation} |
210 |
+ |
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
211 |
+ |
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
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} |
237 |
|
\label{introEquation:compactHamiltonian} |
238 |
|
\end{equation} |
239 |
|
|
224 |
– |
%\subsection{\label{introSection:canonicalTransformation}Canonical |
225 |
– |
%Transformation} |
226 |
– |
|
240 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
241 |
|
|
242 |
< |
\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods} |
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 |
|
|
275 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
280 |
|
The following section will give a brief introduction to some of the |
281 |
|
Statistical Mechanics concepts presented in this dissertation. |
282 |
|
|
283 |
< |
\subsection{\label{introSection::ensemble}Ensemble and Phase Space} |
283 |
> |
\subsection{\label{introSection:ensemble}Ensemble and Phase Space} |
284 |
|
|
285 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
286 |
|
|
311 |
|
Carlo techniques\cite{metropolis:1949} can be utilized. Or if the |
312 |
|
system lends itself to a time averaging approach, the Molecular |
313 |
|
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
314 |
< |
will be the best choice. |
314 |
> |
will be the best choice\cite{Frenkel1996}. |
315 |
|
|
316 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
317 |
|
|
345 |
|
|
346 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
347 |
|
|
348 |
+ |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
349 |
+ |
|
350 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
351 |
|
|
352 |
< |
\subsection{\label{introSection:hydroynamics}Hydrodynamics} |
352 |
> |
\begin{equation} |
353 |
> |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
354 |
> |
\label{introEquation:bathGLE} |
355 |
> |
\end{equation} |
356 |
> |
where $H_B$ is harmonic bath Hamiltonian, |
357 |
> |
\[ |
358 |
> |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
359 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
360 |
> |
\] |
361 |
> |
and $\Delta U$ is bilinear system-bath coupling, |
362 |
> |
\[ |
363 |
> |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
364 |
> |
\] |
365 |
> |
Completing the square, |
366 |
> |
\[ |
367 |
> |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
368 |
> |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
369 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
370 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
371 |
> |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
372 |
> |
\] |
373 |
> |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
374 |
> |
\[ |
375 |
> |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
376 |
> |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
377 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
378 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
379 |
> |
\] |
380 |
> |
where |
381 |
> |
\[ |
382 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
383 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
384 |
> |
\] |
385 |
> |
Since the first two terms of the new Hamiltonian depend only on the |
386 |
> |
system coordinates, we can get the equations of motion for |
387 |
> |
Generalized Langevin Dynamics by Hamilton's equations |
388 |
> |
\ref{introEquation:motionHamiltonianCoordinate, |
389 |
> |
introEquation:motionHamiltonianMomentum}, |
390 |
> |
\begin{align} |
391 |
> |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
392 |
> |
&= m\ddot x |
393 |
> |
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
394 |
> |
\label{introEq:Lp5} |
395 |
> |
\end{align} |
396 |
> |
, and |
397 |
> |
\begin{align} |
398 |
> |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
399 |
> |
&= m\ddot x_\alpha |
400 |
> |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
401 |
> |
\end{align} |
402 |
> |
|
403 |
> |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
404 |
> |
|
405 |
> |
\[ |
406 |
> |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
407 |
> |
\] |
408 |
> |
|
409 |
> |
\[ |
410 |
> |
L(x + y) = L(x) + L(y) |
411 |
> |
\] |
412 |
> |
|
413 |
> |
\[ |
414 |
> |
L(ax) = aL(x) |
415 |
> |
\] |
416 |
> |
|
417 |
> |
\[ |
418 |
> |
L(\dot x) = pL(x) - px(0) |
419 |
> |
\] |
420 |
> |
|
421 |
> |
\[ |
422 |
> |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
423 |
> |
\] |
424 |
> |
|
425 |
> |
\[ |
426 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
427 |
> |
\] |
428 |
> |
|
429 |
> |
Some relatively important transformation, |
430 |
> |
\[ |
431 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
432 |
> |
\] |
433 |
> |
|
434 |
> |
\[ |
435 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
436 |
> |
\] |
437 |
> |
|
438 |
> |
\[ |
439 |
> |
L(1) = \frac{1}{p} |
440 |
> |
\] |
441 |
> |
|
442 |
> |
First, the bath coordinates, |
443 |
> |
\[ |
444 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
445 |
> |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
446 |
> |
}}L(x) |
447 |
> |
\] |
448 |
> |
\[ |
449 |
> |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
450 |
> |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
451 |
> |
\] |
452 |
> |
Then, the system coordinates, |
453 |
> |
\begin{align} |
454 |
> |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
455 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
456 |
> |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
457 |
> |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
458 |
> |
}}\omega _\alpha ^2 L(x)} \right\}} |
459 |
> |
% |
460 |
> |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
461 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
462 |
> |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
463 |
> |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
464 |
> |
\end{align} |
465 |
> |
Then, the inverse transform, |
466 |
> |
|
467 |
> |
\begin{align} |
468 |
> |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
469 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
470 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
471 |
> |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
472 |
> |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
473 |
> |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
474 |
> |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
475 |
> |
% |
476 |
> |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
477 |
> |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
478 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
479 |
> |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
480 |
> |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
481 |
> |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
482 |
> |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
483 |
> |
(\omega _\alpha t)} \right\}} |
484 |
> |
\end{align} |
485 |
> |
|
486 |
> |
\begin{equation} |
487 |
> |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
488 |
> |
(t)\dot x(t - \tau )d\tau } + R(t) |
489 |
> |
\label{introEuqation:GeneralizedLangevinDynamics} |
490 |
> |
\end{equation} |
491 |
> |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
492 |
> |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
493 |
> |
\[ |
494 |
> |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
495 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
496 |
> |
\] |
497 |
> |
For an infinite harmonic bath, we can use the spectral density and |
498 |
> |
an integral over frequencies. |
499 |
> |
|
500 |
> |
\[ |
501 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
502 |
> |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
503 |
> |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
504 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
505 |
> |
\] |
506 |
> |
The random forces depend only on initial conditions. |
507 |
> |
|
508 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
509 |
> |
So we can define a new set of coordinates, |
510 |
> |
\[ |
511 |
> |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
512 |
> |
^2 }}x(0) |
513 |
> |
\] |
514 |
> |
This makes |
515 |
> |
\[ |
516 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
517 |
> |
\] |
518 |
> |
And since the $q$ coordinates are harmonic oscillators, |
519 |
> |
\[ |
520 |
> |
\begin{array}{l} |
521 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
522 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
523 |
> |
\end{array} |
524 |
> |
\] |
525 |
> |
|
526 |
> |
\begin{align} |
527 |
> |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
528 |
> |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
529 |
> |
(t)q_\beta (0)} \right\rangle } } |
530 |
> |
% |
531 |
> |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
532 |
> |
\right\rangle \cos (\omega _\alpha t)} |
533 |
> |
% |
534 |
> |
&= kT\xi (t) |
535 |
> |
\end{align} |
536 |
> |
|
537 |
> |
\begin{equation} |
538 |
> |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
539 |
> |
\label{introEquation:secondFluctuationDissipation} |
540 |
> |
\end{equation} |
541 |
> |
|
542 |
> |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
543 |
> |
|
544 |
> |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
545 |
> |
\subsection{\label{introSection:analyticalApproach}Analytical |
546 |
> |
Approach} |
547 |
> |
|
548 |
> |
\subsection{\label{introSection:approximationApproach}Approximation |
549 |
> |
Approach} |
550 |
> |
|
551 |
> |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
552 |
> |
Body} |