| 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 |
|
|
