1 |
tim |
2685 |
\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
2 |
|
|
|
3 |
tim |
2693 |
\section{\label{introSection:classicalMechanics}Classical |
4 |
|
|
Mechanics} |
5 |
tim |
2685 |
|
6 |
tim |
2692 |
Closely related to Classical Mechanics, Molecular Dynamics |
7 |
|
|
simulations are carried out by integrating the equations of motion |
8 |
|
|
for a given system of particles. There are three fundamental ideas |
9 |
|
|
behind classical mechanics. Firstly, One can determine the state of |
10 |
|
|
a mechanical system at any time of interest; Secondly, all the |
11 |
|
|
mechanical properties of the system at that time can be determined |
12 |
|
|
by combining the knowledge of the properties of the system with the |
13 |
|
|
specification of this state; Finally, the specification of the state |
14 |
|
|
when further combine with the laws of mechanics will also be |
15 |
|
|
sufficient to predict the future behavior of the system. |
16 |
tim |
2685 |
|
17 |
tim |
2693 |
\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
18 |
tim |
2694 |
The discovery of Newton's three laws of mechanics which govern the |
19 |
|
|
motion of particles is the foundation of the classical mechanics. |
20 |
|
|
Newton¡¯s first law defines a class of inertial frames. Inertial |
21 |
|
|
frames are reference frames where a particle not interacting with |
22 |
|
|
other bodies will move with constant speed in the same direction. |
23 |
|
|
With respect to inertial frames Newton¡¯s second law has the form |
24 |
|
|
\begin{equation} |
25 |
|
|
F = \frac {dp}{dt} = \frac {mv}{dt} |
26 |
|
|
\label{introEquation:newtonSecondLaw} |
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 $i$. |
32 |
|
|
Newton¡¯s third law states that |
33 |
|
|
\begin{equation} |
34 |
|
|
F_ij = -F_ji |
35 |
|
|
\label{introEquation:newtonThirdLaw} |
36 |
|
|
\end{equation} |
37 |
tim |
2692 |
|
38 |
tim |
2694 |
Conservation laws of Newtonian Mechanics play very important roles |
39 |
|
|
in solving mechanics problems. The linear momentum of a particle is |
40 |
|
|
conserved if it is free or it experiences no force. The second |
41 |
|
|
conservation theorem concerns the angular momentum of a particle. |
42 |
|
|
The angular momentum $L$ of a particle with respect to an origin |
43 |
|
|
from which $r$ is measured is defined to be |
44 |
|
|
\begin{equation} |
45 |
|
|
L \equiv r \times p \label{introEquation:angularMomentumDefinition} |
46 |
|
|
\end{equation} |
47 |
|
|
The torque $\tau$ with respect to the same origin is defined to be |
48 |
|
|
\begin{equation} |
49 |
|
|
N \equiv r \times F \label{introEquation:torqueDefinition} |
50 |
|
|
\end{equation} |
51 |
|
|
Differentiating Eq.~\ref{introEquation:angularMomentumDefinition}, |
52 |
|
|
\[ |
53 |
|
|
\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times |
54 |
|
|
\dot p) |
55 |
|
|
\] |
56 |
|
|
since |
57 |
|
|
\[ |
58 |
|
|
\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0 |
59 |
|
|
\] |
60 |
|
|
thus, |
61 |
|
|
\begin{equation} |
62 |
|
|
\dot L = r \times \dot p = N |
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 |
tim |
2696 |
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 |
tim |
2694 |
|
73 |
tim |
2693 |
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
74 |
tim |
2692 |
|
75 |
|
|
Newtonian Mechanics suffers from two important limitations: it |
76 |
|
|
describes their motion in special cartesian coordinate systems. |
77 |
|
|
Another limitation of Newtonian mechanics becomes obvious when we |
78 |
|
|
try to describe systems with large numbers of particles. It becomes |
79 |
|
|
very difficult to predict the properties of the system by carrying |
80 |
|
|
out calculations involving the each individual interaction between |
81 |
|
|
all the particles, even if we know all of the details of the |
82 |
|
|
interaction. In order to overcome some of the practical difficulties |
83 |
|
|
which arise in attempts to apply Newton's equation to complex |
84 |
|
|
system, alternative procedures may be developed. |
85 |
|
|
|
86 |
tim |
2694 |
\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
87 |
tim |
2692 |
Principle} |
88 |
|
|
|
89 |
|
|
Hamilton introduced the dynamical principle upon which it is |
90 |
|
|
possible to base all of mechanics and, indeed, most of classical |
91 |
|
|
physics. Hamilton's Principle may be stated as follow, |
92 |
|
|
|
93 |
|
|
The actual trajectory, along which a dynamical system may move from |
94 |
|
|
one point to another within a specified time, is derived by finding |
95 |
|
|
the path which minimizes the time integral of the difference between |
96 |
tim |
2694 |
the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
97 |
tim |
2692 |
\begin{equation} |
98 |
|
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
99 |
tim |
2693 |
\label{introEquation:halmitonianPrinciple1} |
100 |
tim |
2692 |
\end{equation} |
101 |
|
|
|
102 |
|
|
For simple mechanical systems, where the forces acting on the |
103 |
|
|
different part are derivable from a potential and the velocities are |
104 |
|
|
small compared with that of light, the Lagrangian function $L$ can |
105 |
|
|
be define as the difference between the kinetic energy of the system |
106 |
|
|
and its potential energy, |
107 |
|
|
\begin{equation} |
108 |
|
|
L \equiv K - U = L(q_i ,\dot q_i ) , |
109 |
|
|
\label{introEquation:lagrangianDef} |
110 |
|
|
\end{equation} |
111 |
|
|
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
112 |
|
|
\begin{equation} |
113 |
tim |
2693 |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
114 |
|
|
\label{introEquation:halmitonianPrinciple2} |
115 |
tim |
2692 |
\end{equation} |
116 |
|
|
|
117 |
tim |
2694 |
\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
118 |
tim |
2692 |
Equations of Motion in Lagrangian Mechanics} |
119 |
|
|
|
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 }} - |
124 |
|
|
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f |
125 |
tim |
2693 |
\label{introEquation:eqMotionLagrangian} |
126 |
tim |
2692 |
\end{equation} |
127 |
|
|
where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is |
128 |
|
|
generalized velocity. |
129 |
|
|
|
130 |
tim |
2693 |
\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} |
131 |
tim |
2692 |
|
132 |
|
|
Arising from Lagrangian Mechanics, Hamiltonian Mechanics was |
133 |
|
|
introduced by William Rowan Hamilton in 1833 as a re-formulation of |
134 |
|
|
classical mechanics. If the potential energy of a system is |
135 |
|
|
independent of generalized velocities, the generalized momenta can |
136 |
|
|
be defined as |
137 |
|
|
\begin{equation} |
138 |
|
|
p_i = \frac{\partial L}{\partial \dot q_i} |
139 |
|
|
\label{introEquation:generalizedMomenta} |
140 |
|
|
\end{equation} |
141 |
tim |
2693 |
The Lagrange equations of motion are then expressed by |
142 |
tim |
2692 |
\begin{equation} |
143 |
tim |
2693 |
p_i = \frac{{\partial L}}{{\partial q_i }} |
144 |
|
|
\label{introEquation:generalizedMomentaDot} |
145 |
|
|
\end{equation} |
146 |
|
|
|
147 |
|
|
With the help of the generalized momenta, we may now define a new |
148 |
|
|
quantity $H$ by the equation |
149 |
|
|
\begin{equation} |
150 |
|
|
H = \sum\limits_k {p_k \dot q_k } - L , |
151 |
tim |
2692 |
\label{introEquation:hamiltonianDefByLagrangian} |
152 |
|
|
\end{equation} |
153 |
|
|
where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
154 |
|
|
$L$ is the Lagrangian function for the system. |
155 |
|
|
|
156 |
tim |
2693 |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
157 |
|
|
one can obtain |
158 |
|
|
\begin{equation} |
159 |
|
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
160 |
|
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
161 |
|
|
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
162 |
|
|
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
163 |
|
|
\end{equation} |
164 |
|
|
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
165 |
|
|
second and fourth terms in the parentheses cancel. Therefore, |
166 |
|
|
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
167 |
|
|
\begin{equation} |
168 |
|
|
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
169 |
|
|
\right)} - \frac{{\partial L}}{{\partial t}}dt |
170 |
|
|
\label{introEquation:diffHamiltonian2} |
171 |
|
|
\end{equation} |
172 |
|
|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
173 |
|
|
find |
174 |
|
|
\begin{equation} |
175 |
|
|
\frac{{\partial H}}{{\partial p_k }} = q_k |
176 |
|
|
\label{introEquation:motionHamiltonianCoordinate} |
177 |
|
|
\end{equation} |
178 |
|
|
\begin{equation} |
179 |
|
|
\frac{{\partial H}}{{\partial q_k }} = - p_k |
180 |
|
|
\label{introEquation:motionHamiltonianMomentum} |
181 |
|
|
\end{equation} |
182 |
|
|
and |
183 |
|
|
\begin{equation} |
184 |
|
|
\frac{{\partial H}}{{\partial t}} = - \frac{{\partial L}}{{\partial |
185 |
|
|
t}} |
186 |
|
|
\label{introEquation:motionHamiltonianTime} |
187 |
|
|
\end{equation} |
188 |
|
|
|
189 |
|
|
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
190 |
|
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
191 |
|
|
equation of motion. Due to their symmetrical formula, they are also |
192 |
tim |
2694 |
known as the canonical equations of motions \cite{Goldstein01}. |
193 |
tim |
2693 |
|
194 |
tim |
2692 |
An important difference between Lagrangian approach and the |
195 |
|
|
Hamiltonian approach is that the Lagrangian is considered to be a |
196 |
|
|
function of the generalized velocities $\dot q_i$ and the |
197 |
|
|
generalized coordinates $q_i$, while the Hamiltonian is considered |
198 |
|
|
to be a function of the generalized momenta $p_i$ and the conjugate |
199 |
|
|
generalized coordinate $q_i$. Hamiltonian Mechanics is more |
200 |
|
|
appropriate for application to statistical mechanics and quantum |
201 |
|
|
mechanics, since it treats the coordinate and its time derivative as |
202 |
|
|
independent variables and it only works with 1st-order differential |
203 |
tim |
2694 |
equations\cite{Marion90}. |
204 |
tim |
2692 |
|
205 |
tim |
2696 |
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{{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 |
tim |
2698 |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
216 |
tim |
2696 |
\end{equation} |
217 |
|
|
|
218 |
tim |
2693 |
\section{\label{introSection:statisticalMechanics}Statistical |
219 |
|
|
Mechanics} |
220 |
tim |
2692 |
|
221 |
tim |
2694 |
The thermodynamic behaviors and properties of Molecular Dynamics |
222 |
tim |
2692 |
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. |
225 |
|
|
|
226 |
tim |
2696 |
\subsection{\label{introSection:ensemble}Ensemble and Phase Space} |
227 |
tim |
2692 |
|
228 |
tim |
2693 |
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
229 |
tim |
2692 |
|
230 |
tim |
2695 |
Various thermodynamic properties can be calculated from Molecular |
231 |
|
|
Dynamics simulation. By comparing experimental values with the |
232 |
|
|
calculated properties, one can determine the accuracy of the |
233 |
|
|
simulation and the quality of the underlying model. However, both of |
234 |
|
|
experiment and computer simulation are usually performed during a |
235 |
|
|
certain time interval and the measurements are averaged over a |
236 |
|
|
period of them which is different from the average behavior of |
237 |
|
|
many-body system in Statistical Mechanics. Fortunately, Ergodic |
238 |
|
|
Hypothesis is proposed to make a connection between time average and |
239 |
|
|
ensemble average. It states that time average and average over the |
240 |
|
|
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
241 |
|
|
\begin{equation} |
242 |
|
|
\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
243 |
|
|
\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma |
244 |
|
|
{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq |
245 |
|
|
\end{equation} |
246 |
|
|
where $\langle A \rangle_t$ is an equilibrium value of a physical |
247 |
|
|
quantity and $\rho (p(t), q(t))$ is the equilibrium distribution |
248 |
|
|
function. If an observation is averaged over a sufficiently long |
249 |
|
|
time (longer than relaxation time), all accessible microstates in |
250 |
|
|
phase space are assumed to be equally probed, giving a properly |
251 |
|
|
weighted statistical average. This allows the researcher freedom of |
252 |
|
|
choice when deciding how best to measure a given observable. In case |
253 |
|
|
an ensemble averaged approach sounds most reasonable, the Monte |
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 |
tim |
2696 |
will be the best choice\cite{Frenkel1996}. |
258 |
tim |
2694 |
|
259 |
tim |
2697 |
\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 |
tim |
2698 |
classical mechanics. |
306 |
tim |
2697 |
|
307 |
tim |
2698 |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
308 |
tim |
2697 |
|
309 |
tim |
2698 |
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 |
tim |
2697 |
|
334 |
tim |
2698 |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
335 |
|
|
\begin{equation} |
336 |
|
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
337 |
|
|
\end{equation} |
338 |
|
|
The most obvious change being that matrix $J$ now depends on $x$. |
339 |
|
|
The free rigid body is an example of Poisson system (actually a |
340 |
|
|
Lie-Poisson system) with Hamiltonian function of angular kinetic |
341 |
|
|
energy. |
342 |
|
|
\begin{equation} |
343 |
|
|
J(\pi ) = \left( {\begin{array}{*{20}c} |
344 |
|
|
0 & {\pi _3 } & { - \pi _2 } \\ |
345 |
|
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
346 |
|
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
347 |
|
|
\end{array}} \right) |
348 |
|
|
\end{equation} |
349 |
|
|
|
350 |
|
|
\begin{equation} |
351 |
|
|
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
352 |
|
|
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
353 |
|
|
\end{equation} |
354 |
|
|
|
355 |
|
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
356 |
|
|
Let $x(t)$ be the exact solution of the ODE system, |
357 |
|
|
\begin{equation} |
358 |
|
|
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
359 |
|
|
\end{equation} |
360 |
|
|
The exact flow(solution) $\varphi_\tau$ is defined by |
361 |
|
|
\[ |
362 |
|
|
x(t+\tau) =\varphi_\tau(x(t)) |
363 |
|
|
\] |
364 |
|
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
365 |
|
|
space to itself. In most cases, it is not easy to find the exact |
366 |
|
|
flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$, |
367 |
|
|
which is usually called integrator. The order of an integrator |
368 |
|
|
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
369 |
|
|
order $p$, |
370 |
|
|
\begin{equation} |
371 |
|
|
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
372 |
|
|
\end{equation} |
373 |
|
|
|
374 |
|
|
The hidden geometric properties of ODE and its flow play important |
375 |
|
|
roles in numerical studies. The flow of a Hamiltonian vector field |
376 |
|
|
on a symplectic manifold is a symplectomorphism. Let $\varphi$ be |
377 |
|
|
the flow of Hamiltonian vector field, $\varphi$ is a |
378 |
|
|
\emph{symplectic} flow if it satisfies, |
379 |
|
|
\begin{equation} |
380 |
|
|
d \varphi^T J d \varphi = J. |
381 |
|
|
\end{equation} |
382 |
|
|
According to Liouville's theorem, the symplectic volume is invariant |
383 |
|
|
under a Hamiltonian flow, which is the basis for classical |
384 |
|
|
statistical mechanics. As to the Poisson system, |
385 |
|
|
\begin{equation} |
386 |
|
|
d\varphi ^T Jd\varphi = J \circ \varphi |
387 |
|
|
\end{equation} |
388 |
|
|
is the property must be preserved by the integrator. It is possible |
389 |
|
|
to construct a \emph{volume-preserving} flow for a source free($ |
390 |
|
|
\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi = |
391 |
|
|
1$. Changing the variables $y = h(x)$ in a |
392 |
|
|
ODE\ref{introEquation:ODE} will result in a new system, |
393 |
|
|
\[ |
394 |
|
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
395 |
|
|
\] |
396 |
|
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
397 |
|
|
In other words, the flow of this vector field is reversible if and |
398 |
|
|
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. When |
399 |
|
|
designing any numerical methods, one should always try to preserve |
400 |
|
|
the structural properties of the original ODE and its flow. |
401 |
|
|
|
402 |
|
|
\subsection{\label{introSection:splittingAndComposition}Splitting and Composition Methods} |
403 |
|
|
|
404 |
tim |
2694 |
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
405 |
|
|
|
406 |
|
|
As a special discipline of molecular modeling, Molecular dynamics |
407 |
|
|
has proven to be a powerful tool for studying the functions of |
408 |
|
|
biological systems, providing structural, thermodynamic and |
409 |
|
|
dynamical information. |
410 |
|
|
|
411 |
|
|
\subsection{\label{introSec:mdInit}Initialization} |
412 |
|
|
|
413 |
|
|
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
414 |
|
|
|
415 |
tim |
2693 |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
416 |
tim |
2692 |
|
417 |
tim |
2694 |
A rigid body is a body in which the distance between any two given |
418 |
|
|
points of a rigid body remains constant regardless of external |
419 |
|
|
forces exerted on it. A rigid body therefore conserves its shape |
420 |
|
|
during its motion. |
421 |
|
|
|
422 |
|
|
Applications of dynamics of rigid bodies. |
423 |
|
|
|
424 |
tim |
2695 |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
425 |
tim |
2694 |
|
426 |
tim |
2695 |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
427 |
|
|
|
428 |
|
|
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
429 |
|
|
|
430 |
tim |
2694 |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
431 |
|
|
|
432 |
tim |
2693 |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
433 |
tim |
2692 |
|
434 |
tim |
2685 |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
435 |
|
|
|
436 |
tim |
2696 |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
437 |
|
|
|
438 |
tim |
2692 |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
439 |
tim |
2685 |
|
440 |
tim |
2696 |
\begin{equation} |
441 |
|
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
442 |
|
|
\label{introEquation:bathGLE} |
443 |
|
|
\end{equation} |
444 |
|
|
where $H_B$ is harmonic bath Hamiltonian, |
445 |
|
|
\[ |
446 |
|
|
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
447 |
|
|
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
448 |
|
|
\] |
449 |
|
|
and $\Delta U$ is bilinear system-bath coupling, |
450 |
|
|
\[ |
451 |
|
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
452 |
|
|
\] |
453 |
|
|
Completing the square, |
454 |
|
|
\[ |
455 |
|
|
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
456 |
|
|
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
457 |
|
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
458 |
|
|
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
459 |
|
|
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
460 |
|
|
\] |
461 |
|
|
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
462 |
|
|
\[ |
463 |
|
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
464 |
|
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
465 |
|
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
466 |
|
|
w_\alpha ^2 }}x} \right)^2 } \right\}} |
467 |
|
|
\] |
468 |
|
|
where |
469 |
|
|
\[ |
470 |
|
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
471 |
|
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
472 |
|
|
\] |
473 |
|
|
Since the first two terms of the new Hamiltonian depend only on the |
474 |
|
|
system coordinates, we can get the equations of motion for |
475 |
|
|
Generalized Langevin Dynamics by Hamilton's equations |
476 |
|
|
\ref{introEquation:motionHamiltonianCoordinate, |
477 |
|
|
introEquation:motionHamiltonianMomentum}, |
478 |
|
|
\begin{align} |
479 |
|
|
\dot p &= - \frac{{\partial H}}{{\partial x}} |
480 |
|
|
&= m\ddot x |
481 |
|
|
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
482 |
|
|
\label{introEq:Lp5} |
483 |
|
|
\end{align} |
484 |
|
|
, and |
485 |
|
|
\begin{align} |
486 |
|
|
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
487 |
|
|
&= m\ddot x_\alpha |
488 |
|
|
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
489 |
|
|
\end{align} |
490 |
|
|
|
491 |
|
|
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
492 |
|
|
|
493 |
|
|
\[ |
494 |
|
|
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
495 |
|
|
\] |
496 |
|
|
|
497 |
|
|
\[ |
498 |
|
|
L(x + y) = L(x) + L(y) |
499 |
|
|
\] |
500 |
|
|
|
501 |
|
|
\[ |
502 |
|
|
L(ax) = aL(x) |
503 |
|
|
\] |
504 |
|
|
|
505 |
|
|
\[ |
506 |
|
|
L(\dot x) = pL(x) - px(0) |
507 |
|
|
\] |
508 |
|
|
|
509 |
|
|
\[ |
510 |
|
|
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
511 |
|
|
\] |
512 |
|
|
|
513 |
|
|
\[ |
514 |
|
|
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
515 |
|
|
\] |
516 |
|
|
|
517 |
|
|
Some relatively important transformation, |
518 |
|
|
\[ |
519 |
|
|
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
520 |
|
|
\] |
521 |
|
|
|
522 |
|
|
\[ |
523 |
|
|
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
524 |
|
|
\] |
525 |
|
|
|
526 |
|
|
\[ |
527 |
|
|
L(1) = \frac{1}{p} |
528 |
|
|
\] |
529 |
|
|
|
530 |
|
|
First, the bath coordinates, |
531 |
|
|
\[ |
532 |
|
|
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
533 |
|
|
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
534 |
|
|
}}L(x) |
535 |
|
|
\] |
536 |
|
|
\[ |
537 |
|
|
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
538 |
|
|
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
539 |
|
|
\] |
540 |
|
|
Then, the system coordinates, |
541 |
|
|
\begin{align} |
542 |
|
|
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
543 |
|
|
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
544 |
|
|
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
545 |
|
|
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
546 |
|
|
}}\omega _\alpha ^2 L(x)} \right\}} |
547 |
|
|
% |
548 |
|
|
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
549 |
|
|
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
550 |
|
|
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
551 |
|
|
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
552 |
|
|
\end{align} |
553 |
|
|
Then, the inverse transform, |
554 |
|
|
|
555 |
|
|
\begin{align} |
556 |
|
|
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
557 |
|
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
558 |
|
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
559 |
|
|
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
560 |
|
|
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
561 |
|
|
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
562 |
|
|
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
563 |
|
|
% |
564 |
|
|
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
565 |
|
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
566 |
|
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
567 |
|
|
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
568 |
|
|
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
569 |
|
|
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
570 |
|
|
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
571 |
|
|
(\omega _\alpha t)} \right\}} |
572 |
|
|
\end{align} |
573 |
|
|
|
574 |
|
|
\begin{equation} |
575 |
|
|
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
576 |
|
|
(t)\dot x(t - \tau )d\tau } + R(t) |
577 |
|
|
\label{introEuqation:GeneralizedLangevinDynamics} |
578 |
|
|
\end{equation} |
579 |
|
|
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
580 |
|
|
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
581 |
|
|
\[ |
582 |
|
|
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
583 |
|
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
584 |
|
|
\] |
585 |
|
|
For an infinite harmonic bath, we can use the spectral density and |
586 |
|
|
an integral over frequencies. |
587 |
|
|
|
588 |
|
|
\[ |
589 |
|
|
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
590 |
|
|
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
591 |
|
|
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
592 |
|
|
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
593 |
|
|
\] |
594 |
|
|
The random forces depend only on initial conditions. |
595 |
|
|
|
596 |
|
|
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
597 |
|
|
So we can define a new set of coordinates, |
598 |
|
|
\[ |
599 |
|
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
600 |
|
|
^2 }}x(0) |
601 |
|
|
\] |
602 |
|
|
This makes |
603 |
|
|
\[ |
604 |
|
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
605 |
|
|
\] |
606 |
|
|
And since the $q$ coordinates are harmonic oscillators, |
607 |
|
|
\[ |
608 |
|
|
\begin{array}{l} |
609 |
|
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
610 |
|
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
611 |
|
|
\end{array} |
612 |
|
|
\] |
613 |
|
|
|
614 |
|
|
\begin{align} |
615 |
|
|
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
616 |
|
|
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
617 |
|
|
(t)q_\beta (0)} \right\rangle } } |
618 |
|
|
% |
619 |
|
|
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
620 |
|
|
\right\rangle \cos (\omega _\alpha t)} |
621 |
|
|
% |
622 |
|
|
&= kT\xi (t) |
623 |
|
|
\end{align} |
624 |
|
|
|
625 |
|
|
\begin{equation} |
626 |
|
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
627 |
|
|
\label{introEquation:secondFluctuationDissipation} |
628 |
|
|
\end{equation} |
629 |
|
|
|
630 |
|
|
\section{\label{introSection:hydroynamics}Hydrodynamics} |
631 |
|
|
|
632 |
|
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
633 |
|
|
\subsection{\label{introSection:analyticalApproach}Analytical |
634 |
|
|
Approach} |
635 |
|
|
|
636 |
|
|
\subsection{\label{introSection:approximationApproach}Approximation |
637 |
|
|
Approach} |
638 |
|
|
|
639 |
|
|
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
640 |
|
|
Body} |