27 |
|
\end{equation} |
28 |
|
A point mass interacting with other bodies moves with the |
29 |
|
acceleration along the direction of the force acting on it. Let |
30 |
< |
$F_ij$ be the force that particle $i$ exerts on particle $j$, and |
31 |
< |
$F_ji$ be the force that particle $j$ exerts on particle $i$. |
30 |
> |
$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and |
31 |
> |
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
32 |
|
Newton¡¯s third law states that |
33 |
|
\begin{equation} |
34 |
< |
F_ij = -F_ji |
34 |
> |
F_{ij} = -F_{ji} |
35 |
|
\label{introEquation:newtonThirdLaw} |
36 |
|
\end{equation} |
37 |
|
|
117 |
|
\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
118 |
|
Equations of Motion in Lagrangian Mechanics} |
119 |
|
|
120 |
< |
for a holonomic system of $f$ degrees of freedom, the equations of |
120 |
> |
For a holonomic system of $f$ degrees of freedom, the equations of |
121 |
|
motion in the Lagrangian form is |
122 |
|
\begin{equation} |
123 |
|
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_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} |
215 |
> |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
216 |
|
\end{equation} |
217 |
|
|
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} |
238 |
– |
\end{equation} |
239 |
– |
|
218 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
219 |
|
Mechanics} |
220 |
|
|
221 |
|
The thermodynamic behaviors and properties of Molecular Dynamics |
222 |
|
simulation are governed by the principle of Statistical Mechanics. |
223 |
|
The following section will give a brief introduction to some of the |
224 |
< |
Statistical Mechanics concepts presented in this dissertation. |
224 |
> |
Statistical Mechanics concepts and theorem presented in this |
225 |
> |
dissertation. |
226 |
|
|
227 |
< |
\subsection{\label{introSection:ensemble}Ensemble and Phase Space} |
227 |
> |
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
228 |
> |
|
229 |
> |
Mathematically, phase space is the space which represents all |
230 |
> |
possible states. Each possible state of the system corresponds to |
231 |
> |
one unique point in the phase space. For mechanical systems, the |
232 |
> |
phase space usually consists of all possible values of position and |
233 |
> |
momentum variables. Consider a dynamic system in a cartesian space, |
234 |
> |
where each of the $6f$ coordinates and momenta is assigned to one of |
235 |
> |
$6f$ mutually orthogonal axes, the phase space of this system is a |
236 |
> |
$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , |
237 |
> |
\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and |
238 |
> |
momenta is a phase space vector. |
239 |
> |
|
240 |
> |
A microscopic state or microstate of a classical system is |
241 |
> |
specification of the complete phase space vector of a system at any |
242 |
> |
instant in time. An ensemble is defined as a collection of systems |
243 |
> |
sharing one or more macroscopic characteristics but each being in a |
244 |
> |
unique microstate. The complete ensemble is specified by giving all |
245 |
> |
systems or microstates consistent with the common macroscopic |
246 |
> |
characteristics of the ensemble. Although the state of each |
247 |
> |
individual system in the ensemble could be precisely described at |
248 |
> |
any instance in time by a suitable phase space vector, when using |
249 |
> |
ensembles for statistical purposes, there is no need to maintain |
250 |
> |
distinctions between individual systems, since the numbers of |
251 |
> |
systems at any time in the different states which correspond to |
252 |
> |
different regions of the phase space are more interesting. Moreover, |
253 |
> |
in the point of view of statistical mechanics, one would prefer to |
254 |
> |
use ensembles containing a large enough population of separate |
255 |
> |
members so that the numbers of systems in such different states can |
256 |
> |
be regarded as changing continuously as we traverse different |
257 |
> |
regions of the phase space. The condition of an ensemble at any time |
258 |
> |
can be regarded as appropriately specified by the density $\rho$ |
259 |
> |
with which representative points are distributed over the phase |
260 |
> |
space. The density of distribution for an ensemble with $f$ degrees |
261 |
> |
of freedom is defined as, |
262 |
> |
\begin{equation} |
263 |
> |
\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). |
264 |
> |
\label{introEquation:densityDistribution} |
265 |
> |
\end{equation} |
266 |
> |
Governed by the principles of mechanics, the phase points change |
267 |
> |
their value which would change the density at any time at phase |
268 |
> |
space. Hence, the density of distribution is also to be taken as a |
269 |
> |
function of the time. |
270 |
> |
|
271 |
> |
The number of systems $\delta N$ at time $t$ can be determined by, |
272 |
> |
\begin{equation} |
273 |
> |
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
274 |
> |
\label{introEquation:deltaN} |
275 |
> |
\end{equation} |
276 |
> |
Assuming a large enough population of systems are exploited, we can |
277 |
> |
sufficiently approximate $\delta N$ without introducing |
278 |
> |
discontinuity when we go from one region in the phase space to |
279 |
> |
another. By integrating over the whole phase space, |
280 |
> |
\begin{equation} |
281 |
> |
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
282 |
> |
\label{introEquation:totalNumberSystem} |
283 |
> |
\end{equation} |
284 |
> |
gives us an expression for the total number of the systems. Hence, |
285 |
> |
the probability per unit in the phase space can be obtained by, |
286 |
> |
\begin{equation} |
287 |
> |
\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int |
288 |
> |
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
289 |
> |
\label{introEquation:unitProbability} |
290 |
> |
\end{equation} |
291 |
> |
With the help of Equation(\ref{introEquation:unitProbability}) and |
292 |
> |
the knowledge of the system, it is possible to calculate the average |
293 |
> |
value of any desired quantity which depends on the coordinates and |
294 |
> |
momenta of the system. Even when the dynamics of the real system is |
295 |
> |
complex, or stochastic, or even discontinuous, the average |
296 |
> |
properties of the ensemble of possibilities as a whole may still |
297 |
> |
remain well defined. For a classical system in thermal equilibrium |
298 |
> |
with its environment, the ensemble average of a mechanical quantity, |
299 |
> |
$\langle A(q , p) \rangle_t$, takes the form of an integral over the |
300 |
> |
phase space of the system, |
301 |
> |
\begin{equation} |
302 |
> |
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
303 |
> |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
304 |
> |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
305 |
> |
\label{introEquation:ensembelAverage} |
306 |
> |
\end{equation} |
307 |
> |
|
308 |
> |
There are several different types of ensembles with different |
309 |
> |
statistical characteristics. As a function of macroscopic |
310 |
> |
parameters, such as temperature \textit{etc}, partition function can |
311 |
> |
be used to describe the statistical properties of a system in |
312 |
> |
thermodynamic equilibrium. |
313 |
> |
|
314 |
> |
As an ensemble of systems, each of which is known to be thermally |
315 |
> |
isolated and conserve energy, Microcanonical ensemble(NVE) has a |
316 |
> |
partition function like, |
317 |
> |
\begin{equation} |
318 |
> |
\Omega (N,V,E) = e^{\beta TS} |
319 |
> |
\label{introEqaution:NVEPartition}. |
320 |
> |
\end{equation} |
321 |
> |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
322 |
> |
can share its energy with a large heat reservoir. The distribution |
323 |
> |
of the total energy amongst the possible dynamical states is given |
324 |
> |
by the partition function, |
325 |
> |
\begin{equation} |
326 |
> |
\Omega (N,V,T) = e^{ - \beta A} |
327 |
> |
\label{introEquation:NVTPartition} |
328 |
> |
\end{equation} |
329 |
> |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
330 |
> |
TS$. Since most experiment are carried out under constant pressure |
331 |
> |
condition, isothermal-isobaric ensemble(NPT) play a very important |
332 |
> |
role in molecular simulation. The isothermal-isobaric ensemble allow |
333 |
> |
the system to exchange energy with a heat bath of temperature $T$ |
334 |
> |
and to change the volume as well. Its partition function is given as |
335 |
> |
\begin{equation} |
336 |
> |
\Delta (N,P,T) = - e^{\beta G}. |
337 |
> |
\label{introEquation:NPTPartition} |
338 |
> |
\end{equation} |
339 |
> |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
340 |
> |
|
341 |
> |
\subsection{\label{introSection:liouville}Liouville's theorem} |
342 |
> |
|
343 |
> |
The Liouville's theorem is the foundation on which statistical |
344 |
> |
mechanics rests. It describes the time evolution of phase space |
345 |
> |
distribution function. In order to calculate the rate of change of |
346 |
> |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
347 |
> |
consider the two faces perpendicular to the $q_1$ axis, which are |
348 |
> |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
349 |
> |
leaving the opposite face is given by the expression, |
350 |
> |
\begin{equation} |
351 |
> |
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
352 |
> |
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
353 |
> |
}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 |
354 |
> |
\ldots \delta p_f . |
355 |
> |
\end{equation} |
356 |
> |
Summing all over the phase space, we obtain |
357 |
> |
\begin{equation} |
358 |
> |
\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho |
359 |
> |
\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + |
360 |
> |
\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( |
361 |
> |
{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial |
362 |
> |
\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 |
363 |
> |
\ldots \delta q_f \delta p_1 \ldots \delta p_f . |
364 |
> |
\end{equation} |
365 |
> |
Differentiating the equations of motion in Hamiltonian formalism |
366 |
> |
(\ref{introEquation:motionHamiltonianCoordinate}, |
367 |
> |
\ref{introEquation:motionHamiltonianMomentum}), we can show, |
368 |
> |
\begin{equation} |
369 |
> |
\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} |
370 |
> |
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
371 |
> |
\end{equation} |
372 |
> |
which cancels the first terms of the right hand side. Furthermore, |
373 |
> |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
374 |
> |
p_f $ in both sides, we can write out Liouville's theorem in a |
375 |
> |
simple form, |
376 |
> |
\begin{equation} |
377 |
> |
\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f |
378 |
> |
{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + |
379 |
> |
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
380 |
> |
\label{introEquation:liouvilleTheorem} |
381 |
> |
\end{equation} |
382 |
|
|
383 |
+ |
Liouville's theorem states that the distribution function is |
384 |
+ |
constant along any trajectory in phase space. In classical |
385 |
+ |
statistical mechanics, since the number of particles in the system |
386 |
+ |
is huge, we may be able to believe the system is stationary, |
387 |
+ |
\begin{equation} |
388 |
+ |
\frac{{\partial \rho }}{{\partial t}} = 0. |
389 |
+ |
\label{introEquation:stationary} |
390 |
+ |
\end{equation} |
391 |
+ |
In such stationary system, the density of distribution $\rho$ can be |
392 |
+ |
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
393 |
+ |
distribution, |
394 |
+ |
\begin{equation} |
395 |
+ |
\rho \propto e^{ - \beta H} |
396 |
+ |
\label{introEquation:densityAndHamiltonian} |
397 |
+ |
\end{equation} |
398 |
+ |
|
399 |
+ |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
400 |
+ |
Lets consider a region in the phase space, |
401 |
+ |
\begin{equation} |
402 |
+ |
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
403 |
+ |
\end{equation} |
404 |
+ |
If this region is small enough, the density $\rho$ can be regarded |
405 |
+ |
as uniform over the whole phase space. Thus, the number of phase |
406 |
+ |
points inside this region is given by, |
407 |
+ |
\begin{equation} |
408 |
+ |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
409 |
+ |
dp_1 } ..dp_f. |
410 |
+ |
\end{equation} |
411 |
+ |
|
412 |
+ |
\begin{equation} |
413 |
+ |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
414 |
+ |
\frac{d}{{dt}}(\delta v) = 0. |
415 |
+ |
\end{equation} |
416 |
+ |
With the help of stationary assumption |
417 |
+ |
(\ref{introEquation:stationary}), we obtain the principle of the |
418 |
+ |
\emph{conservation of extension in phase space}, |
419 |
+ |
\begin{equation} |
420 |
+ |
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
421 |
+ |
...dq_f dp_1 } ..dp_f = 0. |
422 |
+ |
\label{introEquation:volumePreserving} |
423 |
+ |
\end{equation} |
424 |
+ |
|
425 |
+ |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
426 |
+ |
|
427 |
+ |
Liouville's theorem can be expresses in a variety of different forms |
428 |
+ |
which are convenient within different contexts. For any two function |
429 |
+ |
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
430 |
+ |
bracket ${F, G}$ is defined as |
431 |
+ |
\begin{equation} |
432 |
+ |
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
433 |
+ |
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
434 |
+ |
\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial |
435 |
+ |
q_i }}} \right)}. |
436 |
+ |
\label{introEquation:poissonBracket} |
437 |
+ |
\end{equation} |
438 |
+ |
Substituting equations of motion in Hamiltonian formalism( |
439 |
+ |
\ref{introEquation:motionHamiltonianCoordinate} , |
440 |
+ |
\ref{introEquation:motionHamiltonianMomentum} ) into |
441 |
+ |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
442 |
+ |
theorem using Poisson bracket notion, |
443 |
+ |
\begin{equation} |
444 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
445 |
+ |
{\rho ,H} \right\}. |
446 |
+ |
\label{introEquation:liouvilleTheromInPoissin} |
447 |
+ |
\end{equation} |
448 |
+ |
Moreover, the Liouville operator is defined as |
449 |
+ |
\begin{equation} |
450 |
+ |
iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial |
451 |
+ |
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial |
452 |
+ |
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} |
453 |
+ |
\label{introEquation:liouvilleOperator} |
454 |
+ |
\end{equation} |
455 |
+ |
In terms of Liouville operator, Liouville's equation can also be |
456 |
+ |
expressed as |
457 |
+ |
\begin{equation} |
458 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
459 |
+ |
\label{introEquation:liouvilleTheoremInOperator} |
460 |
+ |
\end{equation} |
461 |
+ |
|
462 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
463 |
|
|
464 |
|
Various thermodynamic properties can be calculated from Molecular |
473 |
|
ensemble average. It states that time average and average over the |
474 |
|
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
475 |
|
\begin{equation} |
476 |
< |
\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
477 |
< |
\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma |
478 |
< |
{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq |
476 |
> |
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
477 |
> |
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
478 |
> |
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp |
479 |
|
\end{equation} |
480 |
< |
where $\langle A \rangle_t$ is an equilibrium value of a physical |
481 |
< |
quantity and $\rho (p(t), q(t))$ is the equilibrium distribution |
482 |
< |
function. If an observation is averaged over a sufficiently long |
483 |
< |
time (longer than relaxation time), all accessible microstates in |
484 |
< |
phase space are assumed to be equally probed, giving a properly |
485 |
< |
weighted statistical average. This allows the researcher freedom of |
486 |
< |
choice when deciding how best to measure a given observable. In case |
487 |
< |
an ensemble averaged approach sounds most reasonable, the Monte |
488 |
< |
Carlo techniques\cite{metropolis:1949} can be utilized. Or if the |
489 |
< |
system lends itself to a time averaging approach, the Molecular |
490 |
< |
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
491 |
< |
will be the best choice\cite{Frenkel1996}. |
480 |
> |
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
481 |
> |
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
482 |
> |
distribution function. If an observation is averaged over a |
483 |
> |
sufficiently long time (longer than relaxation time), all accessible |
484 |
> |
microstates in phase space are assumed to be equally probed, giving |
485 |
> |
a properly weighted statistical average. This allows the researcher |
486 |
> |
freedom of choice when deciding how best to measure a given |
487 |
> |
observable. In case an ensemble averaged approach sounds most |
488 |
> |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
489 |
> |
utilized. Or if the system lends itself to a time averaging |
490 |
> |
approach, the Molecular Dynamics techniques in |
491 |
> |
Sec.~\ref{introSection:molecularDynamics} will be the best |
492 |
> |
choice\cite{Frenkel1996}. |
493 |
|
|
494 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
495 |
|
A variety of numerical integrators were proposed to simulate the |
537 |
|
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
538 |
|
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
539 |
|
Canonical transformation is an example of symplectomorphism in |
540 |
< |
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. |
540 |
> |
classical mechanics. |
541 |
|
|
542 |
< |
\subsection{\label{introSection:exactFlow}The Exact Flow of ODE} |
542 |
> |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
543 |
|
|
544 |
< |
\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting} |
544 |
> |
For a ordinary differential system defined as |
545 |
> |
\begin{equation} |
546 |
> |
\dot x = f(x) |
547 |
> |
\end{equation} |
548 |
> |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
549 |
> |
\begin{equation} |
550 |
> |
f(r) = J\nabla _x H(r). |
551 |
> |
\end{equation} |
552 |
> |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
553 |
> |
matrix |
554 |
> |
\begin{equation} |
555 |
> |
J = \left( {\begin{array}{*{20}c} |
556 |
> |
0 & I \\ |
557 |
> |
{ - I} & 0 \\ |
558 |
> |
\end{array}} \right) |
559 |
> |
\label{introEquation:canonicalMatrix} |
560 |
> |
\end{equation} |
561 |
> |
where $I$ is an identity matrix. Using this notation, Hamiltonian |
562 |
> |
system can be rewritten as, |
563 |
> |
\begin{equation} |
564 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x) |
565 |
> |
\label{introEquation:compactHamiltonian} |
566 |
> |
\end{equation}In this case, $f$ is |
567 |
> |
called a \emph{Hamiltonian vector field}. |
568 |
> |
|
569 |
> |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
570 |
> |
\begin{equation} |
571 |
> |
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
572 |
> |
\end{equation} |
573 |
> |
The most obvious change being that matrix $J$ now depends on $x$. |
574 |
> |
The free rigid body is an example of Poisson system (actually a |
575 |
> |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
576 |
> |
energy. |
577 |
> |
\begin{equation} |
578 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
579 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
580 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
581 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
582 |
> |
\end{array}} \right) |
583 |
> |
\end{equation} |
584 |
> |
|
585 |
> |
\begin{equation} |
586 |
> |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
587 |
> |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
588 |
> |
\end{equation} |
589 |
> |
|
590 |
> |
\subsection{\label{introSection:exactFlow}Exact Flow} |
591 |
> |
|
592 |
> |
Let $x(t)$ be the exact solution of the ODE system, |
593 |
> |
\begin{equation} |
594 |
> |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
595 |
> |
\end{equation} |
596 |
> |
The exact flow(solution) $\varphi_\tau$ is defined by |
597 |
> |
\[ |
598 |
> |
x(t+\tau) =\varphi_\tau(x(t)) |
599 |
> |
\] |
600 |
> |
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
601 |
> |
space to itself. The flow has the continuous group property, |
602 |
> |
\begin{equation} |
603 |
> |
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
604 |
> |
+ \tau _2 } . |
605 |
> |
\end{equation} |
606 |
> |
In particular, |
607 |
> |
\begin{equation} |
608 |
> |
\varphi _\tau \circ \varphi _{ - \tau } = I |
609 |
> |
\end{equation} |
610 |
> |
Therefore, the exact flow is self-adjoint, |
611 |
> |
\begin{equation} |
612 |
> |
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
613 |
> |
\end{equation} |
614 |
> |
The exact flow can also be written in terms of the of an operator, |
615 |
> |
\begin{equation} |
616 |
> |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
617 |
> |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
618 |
> |
\label{introEquation:exponentialOperator} |
619 |
> |
\end{equation} |
620 |
> |
|
621 |
> |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
622 |
> |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
623 |
> |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
624 |
> |
the Taylor series of $\psi_\tau$ agree to order $p$, |
625 |
> |
\begin{equation} |
626 |
> |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
627 |
> |
\end{equation} |
628 |
> |
|
629 |
> |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
630 |
> |
|
631 |
> |
The hidden geometric properties of ODE and its flow play important |
632 |
> |
roles in numerical studies. Many of them can be found in systems |
633 |
> |
which occur naturally in applications. |
634 |
> |
|
635 |
> |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
636 |
> |
a \emph{symplectic} flow if it satisfies, |
637 |
> |
\begin{equation} |
638 |
> |
'\varphi^T J '\varphi = J. |
639 |
> |
\end{equation} |
640 |
> |
According to Liouville's theorem, the symplectic volume is invariant |
641 |
> |
under a Hamiltonian flow, which is the basis for classical |
642 |
> |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
643 |
> |
field on a symplectic manifold can be shown to be a |
644 |
> |
symplectomorphism. As to the Poisson system, |
645 |
> |
\begin{equation} |
646 |
> |
'\varphi ^T J '\varphi = J \circ \varphi |
647 |
> |
\end{equation} |
648 |
> |
is the property must be preserved by the integrator. |
649 |
> |
|
650 |
> |
It is possible to construct a \emph{volume-preserving} flow for a |
651 |
> |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
652 |
> |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
653 |
> |
be volume-preserving. |
654 |
> |
|
655 |
> |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
656 |
> |
will result in a new system, |
657 |
> |
\[ |
658 |
> |
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
659 |
> |
\] |
660 |
> |
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
661 |
> |
In other words, the flow of this vector field is reversible if and |
662 |
> |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
663 |
> |
|
664 |
> |
When designing any numerical methods, one should always try to |
665 |
> |
preserve the structural properties of the original ODE and its flow. |
666 |
> |
|
667 |
> |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
668 |
> |
A lot of well established and very effective numerical methods have |
669 |
> |
been successful precisely because of their symplecticities even |
670 |
> |
though this fact was not recognized when they were first |
671 |
> |
constructed. The most famous example is leapfrog methods in |
672 |
> |
molecular dynamics. In general, symplectic integrators can be |
673 |
> |
constructed using one of four different methods. |
674 |
> |
\begin{enumerate} |
675 |
> |
\item Generating functions |
676 |
> |
\item Variational methods |
677 |
> |
\item Runge-Kutta methods |
678 |
> |
\item Splitting methods |
679 |
> |
\end{enumerate} |
680 |
> |
|
681 |
> |
Generating function tends to lead to methods which are cumbersome |
682 |
> |
and difficult to use. In dissipative systems, variational methods |
683 |
> |
can capture the decay of energy accurately. Since their |
684 |
> |
geometrically unstable nature against non-Hamiltonian perturbations, |
685 |
> |
ordinary implicit Runge-Kutta methods are not suitable for |
686 |
> |
Hamiltonian system. Recently, various high-order explicit |
687 |
> |
Runge--Kutta methods have been developed to overcome this |
688 |
> |
instability \cite{}. However, due to computational penalty involved |
689 |
> |
in implementing the Runge-Kutta methods, they do not attract too |
690 |
> |
much attention from Molecular Dynamics community. Instead, splitting |
691 |
> |
have been widely accepted since they exploit natural decompositions |
692 |
> |
of the system\cite{Tuckerman92}. |
693 |
> |
|
694 |
> |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
695 |
> |
|
696 |
> |
The main idea behind splitting methods is to decompose the discrete |
697 |
> |
$\varphi_h$ as a composition of simpler flows, |
698 |
> |
\begin{equation} |
699 |
> |
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
700 |
> |
\varphi _{h_n } |
701 |
> |
\label{introEquation:FlowDecomposition} |
702 |
> |
\end{equation} |
703 |
> |
where each of the sub-flow is chosen such that each represent a |
704 |
> |
simpler integration of the system. |
705 |
> |
|
706 |
> |
Suppose that a Hamiltonian system takes the form, |
707 |
> |
\[ |
708 |
> |
H = H_1 + H_2. |
709 |
> |
\] |
710 |
> |
Here, $H_1$ and $H_2$ may represent different physical processes of |
711 |
> |
the system. For instance, they may relate to kinetic and potential |
712 |
> |
energy respectively, which is a natural decomposition of the |
713 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
714 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
715 |
> |
order is then given by the Lie-Trotter formula |
716 |
> |
\begin{equation} |
717 |
> |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
718 |
> |
\label{introEquation:firstOrderSplitting} |
719 |
> |
\end{equation} |
720 |
> |
where $\varphi _h$ is the result of applying the corresponding |
721 |
> |
continuous $\varphi _i$ over a time $h$. By definition, as |
722 |
> |
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
723 |
> |
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
724 |
> |
It is easy to show that any composition of symplectic flows yields a |
725 |
> |
symplectic map, |
726 |
> |
\begin{equation} |
727 |
> |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
728 |
> |
'\phi ' = \phi '^T J\phi ' = J, |
729 |
> |
\label{introEquation:SymplecticFlowComposition} |
730 |
> |
\end{equation} |
731 |
> |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
732 |
> |
splitting in this context automatically generates a symplectic map. |
733 |
> |
|
734 |
> |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
735 |
> |
introduces local errors proportional to $h^2$, while Strang |
736 |
> |
splitting gives a second-order decomposition, |
737 |
> |
\begin{equation} |
738 |
> |
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
739 |
> |
_{1,h/2} , |
740 |
> |
\label{introEqaution:secondOrderSplitting} |
741 |
> |
\end{equation} |
742 |
> |
which has a local error proportional to $h^3$. Sprang splitting's |
743 |
> |
popularity in molecular simulation community attribute to its |
744 |
> |
symmetric property, |
745 |
> |
\begin{equation} |
746 |
> |
\varphi _h^{ - 1} = \varphi _{ - h}. |
747 |
> |
\lable{introEquation:timeReversible} |
748 |
> |
\end{equation} |
749 |
> |
|
750 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
751 |
> |
The classical equation for a system consisting of interacting |
752 |
> |
particles can be written in Hamiltonian form, |
753 |
> |
\[ |
754 |
> |
H = T + V |
755 |
> |
\] |
756 |
> |
where $T$ is the kinetic energy and $V$ is the potential energy. |
757 |
> |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
758 |
> |
obtains the following: |
759 |
> |
\begin{align} |
760 |
> |
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
761 |
> |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
762 |
> |
\label{introEquation:Lp10a} \\% |
763 |
> |
% |
764 |
> |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
765 |
> |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
766 |
> |
\label{introEquation:Lp10b} |
767 |
> |
\end{align} |
768 |
> |
where $F(t)$ is the force at time $t$. This integration scheme is |
769 |
> |
known as \emph{velocity verlet} which is |
770 |
> |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
771 |
> |
time-reversible(\ref{introEquation:timeReversible}) and |
772 |
> |
volume-preserving (\ref{introEquation:volumePreserving}). These |
773 |
> |
geometric properties attribute to its long-time stability and its |
774 |
> |
popularity in the community. However, the most commonly used |
775 |
> |
velocity verlet integration scheme is written as below, |
776 |
> |
\begin{align} |
777 |
> |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
778 |
> |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
779 |
> |
% |
780 |
> |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
781 |
> |
\label{introEquation:Lp9b}\\% |
782 |
> |
% |
783 |
> |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
784 |
> |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
785 |
> |
\end{align} |
786 |
> |
From the preceding splitting, one can see that the integration of |
787 |
> |
the equations of motion would follow: |
788 |
> |
\begin{enumerate} |
789 |
> |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
790 |
> |
|
791 |
> |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
792 |
|
|
793 |
+ |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
794 |
+ |
|
795 |
+ |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
796 |
+ |
\end{enumerate} |
797 |
+ |
|
798 |
+ |
Simply switching the order of splitting and composing, a new |
799 |
+ |
integrator, the \emph{position verlet} integrator, can be generated, |
800 |
+ |
\begin{align} |
801 |
+ |
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
802 |
+ |
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
803 |
+ |
\label{introEquation:positionVerlet1} \\% |
804 |
+ |
% |
805 |
+ |
q(\Delta t) = q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
806 |
+ |
q(\Delta t)} \right]. % |
807 |
+ |
\label{introEquation:positionVerlet1} |
808 |
+ |
\end{align} |
809 |
+ |
|
810 |
+ |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
811 |
+ |
|
812 |
+ |
Baker-Campbell-Hausdorff formula can be used to determine the local |
813 |
+ |
error of splitting method in terms of commutator of the |
814 |
+ |
operators(\ref{introEquation:exponentialOperator}) associated with |
815 |
+ |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
816 |
+ |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
817 |
+ |
\begin{equation} |
818 |
+ |
\exp (hX + hY) = \exp (hZ) |
819 |
+ |
\end{equation} |
820 |
+ |
where |
821 |
+ |
\begin{equation} |
822 |
+ |
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
823 |
+ |
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
824 |
+ |
\end{equation} |
825 |
+ |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
826 |
+ |
\[ |
827 |
+ |
[X,Y] = XY - YX . |
828 |
+ |
\] |
829 |
+ |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
830 |
+ |
can obtain |
831 |
+ |
\begin{eqnarray} |
832 |
+ |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
833 |
+ |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 + |
834 |
+ |
h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 + \ldots ) |
835 |
+ |
\end{eqnarray} |
836 |
+ |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
837 |
+ |
error of Spring splitting is proportional to $h^3$. The same |
838 |
+ |
procedure can be applied to general splitting, of the form |
839 |
+ |
\begin{equation} |
840 |
+ |
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
841 |
+ |
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
842 |
+ |
\end{equation} |
843 |
+ |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
844 |
+ |
order method. Yoshida proposed an elegant way to compose higher |
845 |
+ |
order methods based on symmetric splitting. Given a symmetric second |
846 |
+ |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
847 |
+ |
method can be constructed by composing, |
848 |
+ |
\[ |
849 |
+ |
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
850 |
+ |
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
851 |
+ |
\] |
852 |
+ |
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
853 |
+ |
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
854 |
+ |
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
855 |
+ |
\begin{equation} |
856 |
+ |
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
857 |
+ |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
858 |
+ |
\end{equation} |
859 |
+ |
, if the weights are chosen as |
860 |
+ |
\[ |
861 |
+ |
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
862 |
+ |
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
863 |
+ |
\] |
864 |
+ |
|
865 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
866 |
|
|
867 |
|
As a special discipline of molecular modeling, Molecular dynamics |
888 |
|
|
889 |
|
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
890 |
|
|
362 |
– |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
363 |
– |
|
891 |
|
\section{\label{introSection:correlationFunctions}Correlation Functions} |
892 |
|
|
893 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
938 |
|
\dot p &= - \frac{{\partial H}}{{\partial x}} |
939 |
|
&= m\ddot x |
940 |
|
&= - \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)} |
941 |
< |
\label{introEq:Lp5} |
941 |
> |
\label{introEquation:Lp5} |
942 |
|
\end{align} |
943 |
|
, and |
944 |
|
\begin{align} |