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 }} - |
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 |
+ |
Liouville's theorem can be expresses in a variety of different forms |
400 |
+ |
which are convenient within different contexts. For any two function |
401 |
+ |
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
402 |
+ |
bracket ${F, G}$ is defined as |
403 |
+ |
\begin{equation} |
404 |
+ |
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
405 |
+ |
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
406 |
+ |
\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial |
407 |
+ |
q_i }}} \right)}. |
408 |
+ |
\label{introEquation:poissonBracket} |
409 |
+ |
\end{equation} |
410 |
+ |
Substituting equations of motion in Hamiltonian formalism( |
411 |
+ |
\ref{introEquation:motionHamiltonianCoordinate} , |
412 |
+ |
\ref{introEquation:motionHamiltonianMomentum} ) into |
413 |
+ |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
414 |
+ |
theorem using Poisson bracket notion, |
415 |
+ |
\begin{equation} |
416 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
417 |
+ |
{\rho ,H} \right\}. |
418 |
+ |
\label{introEquation:liouvilleTheromInPoissin} |
419 |
+ |
\end{equation} |
420 |
+ |
Moreover, the Liouville operator is defined as |
421 |
+ |
\begin{equation} |
422 |
+ |
iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial |
423 |
+ |
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial |
424 |
+ |
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} |
425 |
+ |
\label{introEquation:liouvilleOperator} |
426 |
+ |
\end{equation} |
427 |
+ |
In terms of Liouville operator, Liouville's equation can also be |
428 |
+ |
expressed as |
429 |
+ |
\begin{equation} |
430 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
431 |
+ |
\label{introEquation:liouvilleTheoremInOperator} |
432 |
+ |
\end{equation} |
433 |
+ |
|
434 |
+ |
|
435 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
436 |
|
|
437 |
|
Various thermodynamic properties can be calculated from Molecular |
446 |
|
ensemble average. It states that time average and average over the |
447 |
|
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
448 |
|
\begin{equation} |
449 |
< |
\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
450 |
< |
\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma |
451 |
< |
{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq |
449 |
> |
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
450 |
> |
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
451 |
> |
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp |
452 |
|
\end{equation} |
453 |
< |
where $\langle A \rangle_t$ is an equilibrium value of a physical |
454 |
< |
quantity and $\rho (p(t), q(t))$ is the equilibrium distribution |
455 |
< |
function. If an observation is averaged over a sufficiently long |
456 |
< |
time (longer than relaxation time), all accessible microstates in |
457 |
< |
phase space are assumed to be equally probed, giving a properly |
458 |
< |
weighted statistical average. This allows the researcher freedom of |
459 |
< |
choice when deciding how best to measure a given observable. In case |
460 |
< |
an ensemble averaged approach sounds most reasonable, the Monte |
461 |
< |
Carlo techniques\cite{metropolis:1949} can be utilized. Or if the |
462 |
< |
system lends itself to a time averaging approach, the Molecular |
463 |
< |
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
464 |
< |
will be the best choice\cite{Frenkel1996}. |
453 |
> |
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
454 |
> |
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
455 |
> |
distribution function. If an observation is averaged over a |
456 |
> |
sufficiently long time (longer than relaxation time), all accessible |
457 |
> |
microstates in phase space are assumed to be equally probed, giving |
458 |
> |
a properly weighted statistical average. This allows the researcher |
459 |
> |
freedom of choice when deciding how best to measure a given |
460 |
> |
observable. In case an ensemble averaged approach sounds most |
461 |
> |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
462 |
> |
utilized. Or if the system lends itself to a time averaging |
463 |
> |
approach, the Molecular Dynamics techniques in |
464 |
> |
Sec.~\ref{introSection:molecularDynamics} will be the best |
465 |
> |
choice\cite{Frenkel1996}. |
466 |
|
|
467 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
468 |
|
A variety of numerical integrators were proposed to simulate the |
650 |
|
\label{introEquation:SymplecticFlowComposition} |
651 |
|
\end{equation} |
652 |
|
Suppose that a Hamiltonian system has a form with $H = T + V$ |
445 |
– |
|
446 |
– |
|
653 |
|
|
654 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
655 |
|
|