| 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 |
| 207 |
< |
\begin{equation} |
| 208 |
< |
r = r(q,p)^T |
| 207 |
< |
\end{equation} |
| 208 |
< |
and to introduce a $2n \times 2n$ canonical structure matrix $J$, |
| 209 |
< |
\begin{equation} |
| 210 |
< |
J = \left( {\begin{array}{*{20}c} |
| 211 |
< |
0 & I \\ |
| 212 |
< |
{ - I} & 0 \\ |
| 213 |
< |
\end{array}} \right) |
| 214 |
< |
\label{introEquation:canonicalMatrix} |
| 215 |
< |
\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, |
| 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{d}{{dt}}r = J\nabla _r H(r) |
| 211 |
< |
\label{introEquation:compactHamiltonian} |
| 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} |
| 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. 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 |
| 477 |
|
|
| 478 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 479 |
|
|
| 480 |
+ |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 481 |
+ |
|
| 482 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 483 |
|
|
| 484 |
< |
\subsection{\label{introSection:hydroynamics}Hydrodynamics} |
| 484 |
> |
\begin{equation} |
| 485 |
> |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 486 |
> |
\label{introEquation:bathGLE} |
| 487 |
> |
\end{equation} |
| 488 |
> |
where $H_B$ is harmonic bath Hamiltonian, |
| 489 |
> |
\[ |
| 490 |
> |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 491 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
| 492 |
> |
\] |
| 493 |
> |
and $\Delta U$ is bilinear system-bath coupling, |
| 494 |
> |
\[ |
| 495 |
> |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 496 |
> |
\] |
| 497 |
> |
Completing the square, |
| 498 |
> |
\[ |
| 499 |
> |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
| 500 |
> |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 501 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 502 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
| 503 |
> |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 504 |
> |
\] |
| 505 |
> |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
| 506 |
> |
\[ |
| 507 |
> |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 508 |
> |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 509 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 510 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 511 |
> |
\] |
| 512 |
> |
where |
| 513 |
> |
\[ |
| 514 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 515 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 516 |
> |
\] |
| 517 |
> |
Since the first two terms of the new Hamiltonian depend only on the |
| 518 |
> |
system coordinates, we can get the equations of motion for |
| 519 |
> |
Generalized Langevin Dynamics by Hamilton's equations |
| 520 |
> |
\ref{introEquation:motionHamiltonianCoordinate, |
| 521 |
> |
introEquation:motionHamiltonianMomentum}, |
| 522 |
> |
\begin{align} |
| 523 |
> |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 524 |
> |
&= m\ddot x |
| 525 |
> |
&= - \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)} |
| 526 |
> |
\label{introEq:Lp5} |
| 527 |
> |
\end{align} |
| 528 |
> |
, and |
| 529 |
> |
\begin{align} |
| 530 |
> |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
| 531 |
> |
&= m\ddot x_\alpha |
| 532 |
> |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
| 533 |
> |
\end{align} |
| 534 |
> |
|
| 535 |
> |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
| 536 |
> |
|
| 537 |
> |
\[ |
| 538 |
> |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
| 539 |
> |
\] |
| 540 |
> |
|
| 541 |
> |
\[ |
| 542 |
> |
L(x + y) = L(x) + L(y) |
| 543 |
> |
\] |
| 544 |
> |
|
| 545 |
> |
\[ |
| 546 |
> |
L(ax) = aL(x) |
| 547 |
> |
\] |
| 548 |
> |
|
| 549 |
> |
\[ |
| 550 |
> |
L(\dot x) = pL(x) - px(0) |
| 551 |
> |
\] |
| 552 |
> |
|
| 553 |
> |
\[ |
| 554 |
> |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
| 555 |
> |
\] |
| 556 |
> |
|
| 557 |
> |
\[ |
| 558 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
| 559 |
> |
\] |
| 560 |
> |
|
| 561 |
> |
Some relatively important transformation, |
| 562 |
> |
\[ |
| 563 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
| 564 |
> |
\] |
| 565 |
> |
|
| 566 |
> |
\[ |
| 567 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
| 568 |
> |
\] |
| 569 |
> |
|
| 570 |
> |
\[ |
| 571 |
> |
L(1) = \frac{1}{p} |
| 572 |
> |
\] |
| 573 |
> |
|
| 574 |
> |
First, the bath coordinates, |
| 575 |
> |
\[ |
| 576 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
| 577 |
> |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
| 578 |
> |
}}L(x) |
| 579 |
> |
\] |
| 580 |
> |
\[ |
| 581 |
> |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
| 582 |
> |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
| 583 |
> |
\] |
| 584 |
> |
Then, the system coordinates, |
| 585 |
> |
\begin{align} |
| 586 |
> |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 587 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
| 588 |
> |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
| 589 |
> |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
| 590 |
> |
}}\omega _\alpha ^2 L(x)} \right\}} |
| 591 |
> |
% |
| 592 |
> |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 593 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
| 594 |
> |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
| 595 |
> |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
| 596 |
> |
\end{align} |
| 597 |
> |
Then, the inverse transform, |
| 598 |
> |
|
| 599 |
> |
\begin{align} |
| 600 |
> |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
| 601 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 602 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
| 603 |
> |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
| 604 |
> |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
| 605 |
> |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 606 |
> |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
| 607 |
> |
% |
| 608 |
> |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 609 |
> |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 610 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 611 |
> |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
| 612 |
> |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
| 613 |
> |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
| 614 |
> |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
| 615 |
> |
(\omega _\alpha t)} \right\}} |
| 616 |
> |
\end{align} |
| 617 |
> |
|
| 618 |
> |
\begin{equation} |
| 619 |
> |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
| 620 |
> |
(t)\dot x(t - \tau )d\tau } + R(t) |
| 621 |
> |
\label{introEuqation:GeneralizedLangevinDynamics} |
| 622 |
> |
\end{equation} |
| 623 |
> |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
| 624 |
> |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
| 625 |
> |
\[ |
| 626 |
> |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 627 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 628 |
> |
\] |
| 629 |
> |
For an infinite harmonic bath, we can use the spectral density and |
| 630 |
> |
an integral over frequencies. |
| 631 |
> |
|
| 632 |
> |
\[ |
| 633 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 634 |
> |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 635 |
> |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 636 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
| 637 |
> |
\] |
| 638 |
> |
The random forces depend only on initial conditions. |
| 639 |
> |
|
| 640 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 641 |
> |
So we can define a new set of coordinates, |
| 642 |
> |
\[ |
| 643 |
> |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 644 |
> |
^2 }}x(0) |
| 645 |
> |
\] |
| 646 |
> |
This makes |
| 647 |
> |
\[ |
| 648 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
| 649 |
> |
\] |
| 650 |
> |
And since the $q$ coordinates are harmonic oscillators, |
| 651 |
> |
\[ |
| 652 |
> |
\begin{array}{l} |
| 653 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 654 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 655 |
> |
\end{array} |
| 656 |
> |
\] |
| 657 |
> |
|
| 658 |
> |
\begin{align} |
| 659 |
> |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
| 660 |
> |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
| 661 |
> |
(t)q_\beta (0)} \right\rangle } } |
| 662 |
> |
% |
| 663 |
> |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
| 664 |
> |
\right\rangle \cos (\omega _\alpha t)} |
| 665 |
> |
% |
| 666 |
> |
&= kT\xi (t) |
| 667 |
> |
\end{align} |
| 668 |
> |
|
| 669 |
> |
\begin{equation} |
| 670 |
> |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 671 |
> |
\label{introEquation:secondFluctuationDissipation} |
| 672 |
> |
\end{equation} |
| 673 |
> |
|
| 674 |
> |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
| 675 |
> |
|
| 676 |
> |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 677 |
> |
\subsection{\label{introSection:analyticalApproach}Analytical |
| 678 |
> |
Approach} |
| 679 |
> |
|
| 680 |
> |
\subsection{\label{introSection:approximationApproach}Approximation |
| 681 |
> |
Approach} |
| 682 |
> |
|
| 683 |
> |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 684 |
> |
Body} |
