| 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} |
