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 |
< |
When studying Hamiltonian system, it is more convenient to use |
206 |
< |
notation |
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 |
< |
r = r(q,p)^T |
211 |
< |
\end{equation} |
212 |
< |
and to introduce a $2n \times 2n$ canonical structure matrix $J$, |
213 |
< |
\begin{equation} |
214 |
< |
J = \left( {\begin{array}{*{20}c} |
215 |
< |
0 & I \\ |
212 |
< |
{ - I} & 0 \\ |
213 |
< |
\end{array}} \right) |
214 |
< |
\label{introEquation:canonicalMatrix} |
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} \label{introEquation:conserveHalmitonian} |
216 |
|
\end{equation} |
216 |
– |
where $I$ is a $n \times n$ identity matrix and $J$ is a |
217 |
– |
skew-symmetric matrix ($ J^T = - J $). Thus, Hamiltonian system |
218 |
– |
can be rewritten as, |
219 |
– |
\begin{equation} |
220 |
– |
\frac{d}{{dt}}r = J\nabla _r H(r) |
221 |
– |
\label{introEquation:compactHamiltonian} |
222 |
– |
\end{equation} |
217 |
|
|
224 |
– |
%\subsection{\label{introSection:canonicalTransformation}Canonical |
225 |
– |
%Transformation} |
226 |
– |
|
227 |
– |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
228 |
– |
|
229 |
– |
\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods} |
230 |
– |
|
231 |
– |
\subsection{\label{Construction of Symplectic Methods}} |
232 |
– |
|
218 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
219 |
|
Mechanics} |
220 |
|
|
223 |
|
The following section will give a brief introduction to some of the |
224 |
|
Statistical Mechanics concepts presented in this dissertation. |
225 |
|
|
226 |
< |
\subsection{\label{introSection::ensemble}Ensemble and Phase Space} |
226 |
> |
\subsection{\label{introSection:ensemble}Ensemble and Phase Space} |
227 |
|
|
228 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
229 |
|
|
254 |
|
Carlo techniques\cite{metropolis:1949} can be utilized. Or if the |
255 |
|
system lends itself to a time averaging approach, the Molecular |
256 |
|
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
257 |
< |
will be the best choice. |
257 |
> |
will be the best choice\cite{Frenkel1996}. |
258 |
> |
|
259 |
> |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
260 |
> |
A variety of numerical integrators were proposed to simulate the |
261 |
> |
motions. They usually begin with an initial conditionals and move |
262 |
> |
the objects in the direction governed by the differential equations. |
263 |
> |
However, most of them ignore the hidden physical law contained |
264 |
> |
within the equations. Since 1990, geometric integrators, which |
265 |
> |
preserve various phase-flow invariants such as symplectic structure, |
266 |
> |
volume and time reversal symmetry, are developed to address this |
267 |
> |
issue. The velocity verlet method, which happens to be a simple |
268 |
> |
example of symplectic integrator, continues to gain its popularity |
269 |
> |
in molecular dynamics community. This fact can be partly explained |
270 |
> |
by its geometric nature. |
271 |
> |
|
272 |
> |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
273 |
> |
A \emph{manifold} is an abstract mathematical space. It locally |
274 |
> |
looks like Euclidean space, but when viewed globally, it may have |
275 |
> |
more complicate structure. A good example of manifold is the surface |
276 |
> |
of Earth. It seems to be flat locally, but it is round if viewed as |
277 |
> |
a whole. A \emph{differentiable manifold} (also known as |
278 |
> |
\emph{smooth manifold}) is a manifold with an open cover in which |
279 |
> |
the covering neighborhoods are all smoothly isomorphic to one |
280 |
> |
another. In other words,it is possible to apply calculus on |
281 |
> |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
282 |
> |
defined as a pair $(M, \omega)$ which consisting of a |
283 |
> |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
284 |
> |
bilinear symplectic form, $\omega$. A symplectic form on a vector |
285 |
> |
space $V$ is a function $\omega(x, y)$ which satisfies |
286 |
> |
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
287 |
> |
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
288 |
> |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
289 |
> |
example of symplectic form. |
290 |
> |
|
291 |
> |
One of the motivations to study \emph{symplectic manifold} in |
292 |
> |
Hamiltonian Mechanics is that a symplectic manifold can represent |
293 |
> |
all possible configurations of the system and the phase space of the |
294 |
> |
system can be described by it's cotangent bundle. Every symplectic |
295 |
> |
manifold is even dimensional. For instance, in Hamilton equations, |
296 |
> |
coordinate and momentum always appear in pairs. |
297 |
> |
|
298 |
> |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
299 |
> |
\[ |
300 |
> |
f : M \rightarrow N |
301 |
> |
\] |
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. |
306 |
> |
|
307 |
> |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
308 |
|
|
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. The flow of a Hamiltonian vector field |
376 |
+ |
on a symplectic manifold is a symplectomorphism. Let $\varphi$ be |
377 |
+ |
the flow of Hamiltonian vector field, $\varphi$ is a |
378 |
+ |
\emph{symplectic} flow if it satisfies, |
379 |
+ |
\begin{equation} |
380 |
+ |
d \varphi^T J d \varphi = J. |
381 |
+ |
\end{equation} |
382 |
+ |
According to Liouville's theorem, the symplectic volume is invariant |
383 |
+ |
under a Hamiltonian flow, which is the basis for classical |
384 |
+ |
statistical mechanics. As to the Poisson system, |
385 |
+ |
\begin{equation} |
386 |
+ |
d\varphi ^T Jd\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:splittingAndComposition}Splitting and Composition Methods} |
403 |
+ |
|
404 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
405 |
|
|
406 |
|
As a special discipline of molecular modeling, Molecular dynamics |
433 |
|
|
434 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
435 |
|
|
436 |
+ |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
437 |
+ |
|
438 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
439 |
|
|
440 |
< |
\subsection{\label{introSection:hydroynamics}Hydrodynamics} |
440 |
> |
\begin{equation} |
441 |
> |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
442 |
> |
\label{introEquation:bathGLE} |
443 |
> |
\end{equation} |
444 |
> |
where $H_B$ is harmonic bath Hamiltonian, |
445 |
> |
\[ |
446 |
> |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
447 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
448 |
> |
\] |
449 |
> |
and $\Delta U$ is bilinear system-bath coupling, |
450 |
> |
\[ |
451 |
> |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
452 |
> |
\] |
453 |
> |
Completing the square, |
454 |
> |
\[ |
455 |
> |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
456 |
> |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
457 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
458 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
459 |
> |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
460 |
> |
\] |
461 |
> |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
462 |
> |
\[ |
463 |
> |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
464 |
> |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
465 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
466 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
467 |
> |
\] |
468 |
> |
where |
469 |
> |
\[ |
470 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
471 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
472 |
> |
\] |
473 |
> |
Since the first two terms of the new Hamiltonian depend only on the |
474 |
> |
system coordinates, we can get the equations of motion for |
475 |
> |
Generalized Langevin Dynamics by Hamilton's equations |
476 |
> |
\ref{introEquation:motionHamiltonianCoordinate, |
477 |
> |
introEquation:motionHamiltonianMomentum}, |
478 |
> |
\begin{align} |
479 |
> |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
480 |
> |
&= m\ddot x |
481 |
> |
&= - \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)} |
482 |
> |
\label{introEq:Lp5} |
483 |
> |
\end{align} |
484 |
> |
, and |
485 |
> |
\begin{align} |
486 |
> |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
487 |
> |
&= m\ddot x_\alpha |
488 |
> |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
489 |
> |
\end{align} |
490 |
> |
|
491 |
> |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
492 |
> |
|
493 |
> |
\[ |
494 |
> |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
495 |
> |
\] |
496 |
> |
|
497 |
> |
\[ |
498 |
> |
L(x + y) = L(x) + L(y) |
499 |
> |
\] |
500 |
> |
|
501 |
> |
\[ |
502 |
> |
L(ax) = aL(x) |
503 |
> |
\] |
504 |
> |
|
505 |
> |
\[ |
506 |
> |
L(\dot x) = pL(x) - px(0) |
507 |
> |
\] |
508 |
> |
|
509 |
> |
\[ |
510 |
> |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
511 |
> |
\] |
512 |
> |
|
513 |
> |
\[ |
514 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
515 |
> |
\] |
516 |
> |
|
517 |
> |
Some relatively important transformation, |
518 |
> |
\[ |
519 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
520 |
> |
\] |
521 |
> |
|
522 |
> |
\[ |
523 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
524 |
> |
\] |
525 |
> |
|
526 |
> |
\[ |
527 |
> |
L(1) = \frac{1}{p} |
528 |
> |
\] |
529 |
> |
|
530 |
> |
First, the bath coordinates, |
531 |
> |
\[ |
532 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
533 |
> |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
534 |
> |
}}L(x) |
535 |
> |
\] |
536 |
> |
\[ |
537 |
> |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
538 |
> |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
539 |
> |
\] |
540 |
> |
Then, the system coordinates, |
541 |
> |
\begin{align} |
542 |
> |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
543 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
544 |
> |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
545 |
> |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
546 |
> |
}}\omega _\alpha ^2 L(x)} \right\}} |
547 |
> |
% |
548 |
> |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
549 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
550 |
> |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
551 |
> |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
552 |
> |
\end{align} |
553 |
> |
Then, the inverse transform, |
554 |
> |
|
555 |
> |
\begin{align} |
556 |
> |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
557 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
558 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
559 |
> |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
560 |
> |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
561 |
> |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
562 |
> |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
563 |
> |
% |
564 |
> |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
565 |
> |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
566 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
567 |
> |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
568 |
> |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
569 |
> |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
570 |
> |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
571 |
> |
(\omega _\alpha t)} \right\}} |
572 |
> |
\end{align} |
573 |
> |
|
574 |
> |
\begin{equation} |
575 |
> |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
576 |
> |
(t)\dot x(t - \tau )d\tau } + R(t) |
577 |
> |
\label{introEuqation:GeneralizedLangevinDynamics} |
578 |
> |
\end{equation} |
579 |
> |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
580 |
> |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
581 |
> |
\[ |
582 |
> |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
583 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
584 |
> |
\] |
585 |
> |
For an infinite harmonic bath, we can use the spectral density and |
586 |
> |
an integral over frequencies. |
587 |
> |
|
588 |
> |
\[ |
589 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
590 |
> |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
591 |
> |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
592 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
593 |
> |
\] |
594 |
> |
The random forces depend only on initial conditions. |
595 |
> |
|
596 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
597 |
> |
So we can define a new set of coordinates, |
598 |
> |
\[ |
599 |
> |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
600 |
> |
^2 }}x(0) |
601 |
> |
\] |
602 |
> |
This makes |
603 |
> |
\[ |
604 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
605 |
> |
\] |
606 |
> |
And since the $q$ coordinates are harmonic oscillators, |
607 |
> |
\[ |
608 |
> |
\begin{array}{l} |
609 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
610 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
611 |
> |
\end{array} |
612 |
> |
\] |
613 |
> |
|
614 |
> |
\begin{align} |
615 |
> |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
616 |
> |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
617 |
> |
(t)q_\beta (0)} \right\rangle } } |
618 |
> |
% |
619 |
> |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
620 |
> |
\right\rangle \cos (\omega _\alpha t)} |
621 |
> |
% |
622 |
> |
&= kT\xi (t) |
623 |
> |
\end{align} |
624 |
> |
|
625 |
> |
\begin{equation} |
626 |
> |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
627 |
> |
\label{introEquation:secondFluctuationDissipation} |
628 |
> |
\end{equation} |
629 |
> |
|
630 |
> |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
631 |
> |
|
632 |
> |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
633 |
> |
\subsection{\label{introSection:analyticalApproach}Analytical |
634 |
> |
Approach} |
635 |
> |
|
636 |
> |
\subsection{\label{introSection:approximationApproach}Approximation |
637 |
> |
Approach} |
638 |
> |
|
639 |
> |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
640 |
> |
Body} |