1 |
|
\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
2 |
|
|
3 |
< |
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
3 |
> |
\section{\label{introSection:classicalMechanics}Classical |
4 |
> |
Mechanics} |
5 |
|
|
5 |
– |
As a special discipline of molecular modeling, Molecular dynamics |
6 |
– |
has proven to be a powerful tool for studying the functions of |
7 |
– |
biological systems, providing structural, thermodynamic and |
8 |
– |
dynamical information. |
9 |
– |
|
10 |
– |
\subsection{\label{introSection:classicalMechanics}Classical Mechanics} |
11 |
– |
|
6 |
|
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 |
14 |
|
when further combine with the laws of mechanics will also be |
15 |
|
sufficient to predict the future behavior of the system. |
16 |
|
|
17 |
< |
\subsubsection{\label{introSection:newtonian}Newtonian Mechanics} |
17 |
> |
\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
18 |
> |
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 |
|
|
38 |
< |
\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
38 |
> |
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 |
> |
that if all forces are conservative, Energy |
67 |
> |
\begin{equation}E = T + V \label{introEquation:energyConservation} |
68 |
> |
\end{equation} |
69 |
> |
is conserved. All of these conserved quantities are |
70 |
> |
important factors to determine the quality of numerical integration |
71 |
> |
scheme for rigid body \cite{Dullweber1997}. |
72 |
|
|
73 |
+ |
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
74 |
+ |
|
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 |
83 |
|
which arise in attempts to apply Newton's equation to complex |
84 |
|
system, alternative procedures may be developed. |
85 |
|
|
86 |
< |
\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
86 |
> |
\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
87 |
|
Principle} |
88 |
|
|
89 |
|
Hamilton introduced the dynamical principle upon which it is |
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 |
< |
the kinetic, $K$, and potential energies, $U$. |
96 |
> |
the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
97 |
|
\begin{equation} |
98 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
99 |
< |
\lable{introEquation:halmitonianPrinciple1} |
99 |
> |
\label{introEquation:halmitonianPrinciple1} |
100 |
|
\end{equation} |
101 |
|
|
102 |
|
For simple mechanical systems, where the forces acting on the |
110 |
|
\end{equation} |
111 |
|
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
112 |
|
\begin{equation} |
113 |
< |
\delta \int_{t_1 }^{t_2 } {K dt = 0} , |
114 |
< |
\lable{introEquation:halmitonianPrinciple2} |
113 |
> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
114 |
> |
\label{introEquation:halmitonianPrinciple2} |
115 |
|
\end{equation} |
116 |
|
|
117 |
< |
\subsubsubsection{\label{introSection:equationOfMotionLagrangian}The |
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 }} - |
124 |
|
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f |
125 |
< |
\lable{introEquation:eqMotionLagrangian} |
125 |
> |
\label{introEquation:eqMotionLagrangian} |
126 |
|
\end{equation} |
127 |
|
where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is |
128 |
|
generalized velocity. |
129 |
|
|
130 |
< |
\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} |
130 |
> |
\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} |
131 |
|
|
132 |
|
Arising from Lagrangian Mechanics, Hamiltonian Mechanics was |
133 |
|
introduced by William Rowan Hamilton in 1833 as a re-formulation of |
138 |
|
p_i = \frac{\partial L}{\partial \dot q_i} |
139 |
|
\label{introEquation:generalizedMomenta} |
140 |
|
\end{equation} |
141 |
< |
With the help of these momenta, we may now define a new quantity $H$ |
94 |
< |
by the equation |
141 |
> |
The Lagrange equations of motion are then expressed by |
142 |
|
\begin{equation} |
143 |
< |
H = p_1 \dot q_1 + \ldots + p_f \dot q_f - L, |
143 |
> |
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 |
|
\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 |
+ |
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 |
+ |
known as the canonical equations of motions \cite{Goldstein01}. |
193 |
+ |
|
194 |
|
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 |
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 |
< |
equations. |
203 |
> |
equations\cite{Marion90}. |
204 |
|
|
205 |
+ |
In Newtonian Mechanics, a system described by conservative forces |
206 |
+ |
conserves the total energy \ref{introEquation:energyConservation}. |
207 |
+ |
It follows that Hamilton's equations of motion conserve the total |
208 |
+ |
Hamiltonian. |
209 |
+ |
\begin{equation} |
210 |
+ |
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
211 |
+ |
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
212 |
+ |
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
213 |
+ |
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
214 |
+ |
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
215 |
+ |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
216 |
+ |
\end{equation} |
217 |
|
|
218 |
< |
\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation} |
218 |
> |
\section{\label{introSection:statisticalMechanics}Statistical |
219 |
> |
Mechanics} |
220 |
|
|
221 |
< |
\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics} |
117 |
< |
|
118 |
< |
The thermodynamic behaviors and properties of Molecular Dynamics |
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 |
< |
\subsubsection{\label{introSection::ensemble}Ensemble} |
227 |
> |
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
228 |
|
|
229 |
< |
\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
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 |
< |
\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
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 |
< |
\subsection{\label{introSection:correlationFunctions}Correlation Functions} |
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} \label{introEquation:NVEPartition}. |
319 |
> |
\end{equation} |
320 |
> |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
321 |
> |
can share its energy with a large heat reservoir. The distribution |
322 |
> |
of the total energy amongst the possible dynamical states is given |
323 |
> |
by the partition function, |
324 |
> |
\begin{equation} |
325 |
> |
\Omega (N,V,T) = e^{ - \beta A} |
326 |
> |
\label{introEquation:NVTPartition} |
327 |
> |
\end{equation} |
328 |
> |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
329 |
> |
TS$. Since most experiment are carried out under constant pressure |
330 |
> |
condition, isothermal-isobaric ensemble(NPT) play a very important |
331 |
> |
role in molecular simulation. The isothermal-isobaric ensemble allow |
332 |
> |
the system to exchange energy with a heat bath of temperature $T$ |
333 |
> |
and to change the volume as well. Its partition function is given as |
334 |
> |
\begin{equation} |
335 |
> |
\Delta (N,P,T) = - e^{\beta G}. |
336 |
> |
\label{introEquation:NPTPartition} |
337 |
> |
\end{equation} |
338 |
> |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
339 |
> |
|
340 |
> |
\subsection{\label{introSection:liouville}Liouville's theorem} |
341 |
> |
|
342 |
> |
The Liouville's theorem is the foundation on which statistical |
343 |
> |
mechanics rests. It describes the time evolution of phase space |
344 |
> |
distribution function. In order to calculate the rate of change of |
345 |
> |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
346 |
> |
consider the two faces perpendicular to the $q_1$ axis, which are |
347 |
> |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
348 |
> |
leaving the opposite face is given by the expression, |
349 |
> |
\begin{equation} |
350 |
> |
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
351 |
> |
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
352 |
> |
}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 |
353 |
> |
\ldots \delta p_f . |
354 |
> |
\end{equation} |
355 |
> |
Summing all over the phase space, we obtain |
356 |
> |
\begin{equation} |
357 |
> |
\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho |
358 |
> |
\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + |
359 |
> |
\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( |
360 |
> |
{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial |
361 |
> |
\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 |
362 |
> |
\ldots \delta q_f \delta p_1 \ldots \delta p_f . |
363 |
> |
\end{equation} |
364 |
> |
Differentiating the equations of motion in Hamiltonian formalism |
365 |
> |
(\ref{introEquation:motionHamiltonianCoordinate}, |
366 |
> |
\ref{introEquation:motionHamiltonianMomentum}), we can show, |
367 |
> |
\begin{equation} |
368 |
> |
\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} |
369 |
> |
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
370 |
> |
\end{equation} |
371 |
> |
which cancels the first terms of the right hand side. Furthermore, |
372 |
> |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
373 |
> |
p_f $ in both sides, we can write out Liouville's theorem in a |
374 |
> |
simple form, |
375 |
> |
\begin{equation} |
376 |
> |
\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f |
377 |
> |
{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + |
378 |
> |
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
379 |
> |
\label{introEquation:liouvilleTheorem} |
380 |
> |
\end{equation} |
381 |
> |
|
382 |
> |
Liouville's theorem states that the distribution function is |
383 |
> |
constant along any trajectory in phase space. In classical |
384 |
> |
statistical mechanics, since the number of particles in the system |
385 |
> |
is huge, we may be able to believe the system is stationary, |
386 |
> |
\begin{equation} |
387 |
> |
\frac{{\partial \rho }}{{\partial t}} = 0. |
388 |
> |
\label{introEquation:stationary} |
389 |
> |
\end{equation} |
390 |
> |
In such stationary system, the density of distribution $\rho$ can be |
391 |
> |
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
392 |
> |
distribution, |
393 |
> |
\begin{equation} |
394 |
> |
\rho \propto e^{ - \beta H} |
395 |
> |
\label{introEquation:densityAndHamiltonian} |
396 |
> |
\end{equation} |
397 |
|
|
398 |
+ |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
399 |
+ |
Lets consider a region in the phase space, |
400 |
+ |
\begin{equation} |
401 |
+ |
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
402 |
+ |
\end{equation} |
403 |
+ |
If this region is small enough, the density $\rho$ can be regarded |
404 |
+ |
as uniform over the whole phase space. Thus, the number of phase |
405 |
+ |
points inside this region is given by, |
406 |
+ |
\begin{equation} |
407 |
+ |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
408 |
+ |
dp_1 } ..dp_f. |
409 |
+ |
\end{equation} |
410 |
+ |
|
411 |
+ |
\begin{equation} |
412 |
+ |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
413 |
+ |
\frac{d}{{dt}}(\delta v) = 0. |
414 |
+ |
\end{equation} |
415 |
+ |
With the help of stationary assumption |
416 |
+ |
(\ref{introEquation:stationary}), we obtain the principle of the |
417 |
+ |
\emph{conservation of extension in phase space}, |
418 |
+ |
\begin{equation} |
419 |
+ |
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
420 |
+ |
...dq_f dp_1 } ..dp_f = 0. |
421 |
+ |
\label{introEquation:volumePreserving} |
422 |
+ |
\end{equation} |
423 |
+ |
|
424 |
+ |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
425 |
+ |
|
426 |
+ |
Liouville's theorem can be expresses in a variety of different forms |
427 |
+ |
which are convenient within different contexts. For any two function |
428 |
+ |
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
429 |
+ |
bracket ${F, G}$ is defined as |
430 |
+ |
\begin{equation} |
431 |
+ |
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
432 |
+ |
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
433 |
+ |
\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial |
434 |
+ |
q_i }}} \right)}. |
435 |
+ |
\label{introEquation:poissonBracket} |
436 |
+ |
\end{equation} |
437 |
+ |
Substituting equations of motion in Hamiltonian formalism( |
438 |
+ |
\ref{introEquation:motionHamiltonianCoordinate} , |
439 |
+ |
\ref{introEquation:motionHamiltonianMomentum} ) into |
440 |
+ |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
441 |
+ |
theorem using Poisson bracket notion, |
442 |
+ |
\begin{equation} |
443 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
444 |
+ |
{\rho ,H} \right\}. |
445 |
+ |
\label{introEquation:liouvilleTheromInPoissin} |
446 |
+ |
\end{equation} |
447 |
+ |
Moreover, the Liouville operator is defined as |
448 |
+ |
\begin{equation} |
449 |
+ |
iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial |
450 |
+ |
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial |
451 |
+ |
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} |
452 |
+ |
\label{introEquation:liouvilleOperator} |
453 |
+ |
\end{equation} |
454 |
+ |
In terms of Liouville operator, Liouville's equation can also be |
455 |
+ |
expressed as |
456 |
+ |
\begin{equation} |
457 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
458 |
+ |
\label{introEquation:liouvilleTheoremInOperator} |
459 |
+ |
\end{equation} |
460 |
+ |
|
461 |
+ |
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
462 |
+ |
|
463 |
+ |
Various thermodynamic properties can be calculated from Molecular |
464 |
+ |
Dynamics simulation. By comparing experimental values with the |
465 |
+ |
calculated properties, one can determine the accuracy of the |
466 |
+ |
simulation and the quality of the underlying model. However, both of |
467 |
+ |
experiment and computer simulation are usually performed during a |
468 |
+ |
certain time interval and the measurements are averaged over a |
469 |
+ |
period of them which is different from the average behavior of |
470 |
+ |
many-body system in Statistical Mechanics. Fortunately, Ergodic |
471 |
+ |
Hypothesis is proposed to make a connection between time average and |
472 |
+ |
ensemble average. It states that time average and average over the |
473 |
+ |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
474 |
+ |
\begin{equation} |
475 |
+ |
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
476 |
+ |
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
477 |
+ |
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp |
478 |
+ |
\end{equation} |
479 |
+ |
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
480 |
+ |
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
481 |
+ |
distribution function. If an observation is averaged over a |
482 |
+ |
sufficiently long time (longer than relaxation time), all accessible |
483 |
+ |
microstates in phase space are assumed to be equally probed, giving |
484 |
+ |
a properly weighted statistical average. This allows the researcher |
485 |
+ |
freedom of choice when deciding how best to measure a given |
486 |
+ |
observable. In case an ensemble averaged approach sounds most |
487 |
+ |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
488 |
+ |
utilized. Or if the system lends itself to a time averaging |
489 |
+ |
approach, the Molecular Dynamics techniques in |
490 |
+ |
Sec.~\ref{introSection:molecularDynamics} will be the best |
491 |
+ |
choice\cite{Frenkel1996}. |
492 |
+ |
|
493 |
+ |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
494 |
+ |
A variety of numerical integrators were proposed to simulate the |
495 |
+ |
motions. They usually begin with an initial conditionals and move |
496 |
+ |
the objects in the direction governed by the differential equations. |
497 |
+ |
However, most of them ignore the hidden physical law contained |
498 |
+ |
within the equations. Since 1990, geometric integrators, which |
499 |
+ |
preserve various phase-flow invariants such as symplectic structure, |
500 |
+ |
volume and time reversal symmetry, are developed to address this |
501 |
+ |
issue. The velocity verlet method, which happens to be a simple |
502 |
+ |
example of symplectic integrator, continues to gain its popularity |
503 |
+ |
in molecular dynamics community. This fact can be partly explained |
504 |
+ |
by its geometric nature. |
505 |
+ |
|
506 |
+ |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
507 |
+ |
A \emph{manifold} is an abstract mathematical space. It locally |
508 |
+ |
looks like Euclidean space, but when viewed globally, it may have |
509 |
+ |
more complicate structure. A good example of manifold is the surface |
510 |
+ |
of Earth. It seems to be flat locally, but it is round if viewed as |
511 |
+ |
a whole. A \emph{differentiable manifold} (also known as |
512 |
+ |
\emph{smooth manifold}) is a manifold with an open cover in which |
513 |
+ |
the covering neighborhoods are all smoothly isomorphic to one |
514 |
+ |
another. In other words,it is possible to apply calculus on |
515 |
+ |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
516 |
+ |
defined as a pair $(M, \omega)$ which consisting of a |
517 |
+ |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
518 |
+ |
bilinear symplectic form, $\omega$. A symplectic form on a vector |
519 |
+ |
space $V$ is a function $\omega(x, y)$ which satisfies |
520 |
+ |
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
521 |
+ |
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
522 |
+ |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
523 |
+ |
example of symplectic form. |
524 |
+ |
|
525 |
+ |
One of the motivations to study \emph{symplectic manifold} in |
526 |
+ |
Hamiltonian Mechanics is that a symplectic manifold can represent |
527 |
+ |
all possible configurations of the system and the phase space of the |
528 |
+ |
system can be described by it's cotangent bundle. Every symplectic |
529 |
+ |
manifold is even dimensional. For instance, in Hamilton equations, |
530 |
+ |
coordinate and momentum always appear in pairs. |
531 |
+ |
|
532 |
+ |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
533 |
+ |
\[ |
534 |
+ |
f : M \rightarrow N |
535 |
+ |
\] |
536 |
+ |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
537 |
+ |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
538 |
+ |
Canonical transformation is an example of symplectomorphism in |
539 |
+ |
classical mechanics. |
540 |
+ |
|
541 |
+ |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
542 |
+ |
|
543 |
+ |
For a ordinary differential system defined as |
544 |
+ |
\begin{equation} |
545 |
+ |
\dot x = f(x) |
546 |
+ |
\end{equation} |
547 |
+ |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
548 |
+ |
\begin{equation} |
549 |
+ |
f(r) = J\nabla _x H(r). |
550 |
+ |
\end{equation} |
551 |
+ |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
552 |
+ |
matrix |
553 |
+ |
\begin{equation} |
554 |
+ |
J = \left( {\begin{array}{*{20}c} |
555 |
+ |
0 & I \\ |
556 |
+ |
{ - I} & 0 \\ |
557 |
+ |
\end{array}} \right) |
558 |
+ |
\label{introEquation:canonicalMatrix} |
559 |
+ |
\end{equation} |
560 |
+ |
where $I$ is an identity matrix. Using this notation, Hamiltonian |
561 |
+ |
system can be rewritten as, |
562 |
+ |
\begin{equation} |
563 |
+ |
\frac{d}{{dt}}x = J\nabla _x H(x) |
564 |
+ |
\label{introEquation:compactHamiltonian} |
565 |
+ |
\end{equation}In this case, $f$ is |
566 |
+ |
called a \emph{Hamiltonian vector field}. |
567 |
+ |
|
568 |
+ |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
569 |
+ |
\begin{equation} |
570 |
+ |
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
571 |
+ |
\end{equation} |
572 |
+ |
The most obvious change being that matrix $J$ now depends on $x$. |
573 |
+ |
The free rigid body is an example of Poisson system (actually a |
574 |
+ |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
575 |
+ |
energy. |
576 |
+ |
\begin{equation} |
577 |
+ |
J(\pi ) = \left( {\begin{array}{*{20}c} |
578 |
+ |
0 & {\pi _3 } & { - \pi _2 } \\ |
579 |
+ |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
580 |
+ |
{\pi _2 } & { - \pi _1 } & 0 \\ |
581 |
+ |
\end{array}} \right) |
582 |
+ |
\end{equation} |
583 |
+ |
|
584 |
+ |
\begin{equation} |
585 |
+ |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
586 |
+ |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
587 |
+ |
\end{equation} |
588 |
+ |
|
589 |
+ |
\subsection{\label{introSection:exactFlow}Exact Flow} |
590 |
+ |
|
591 |
+ |
Let $x(t)$ be the exact solution of the ODE system, |
592 |
+ |
\begin{equation} |
593 |
+ |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
594 |
+ |
\end{equation} |
595 |
+ |
The exact flow(solution) $\varphi_\tau$ is defined by |
596 |
+ |
\[ |
597 |
+ |
x(t+\tau) =\varphi_\tau(x(t)) |
598 |
+ |
\] |
599 |
+ |
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
600 |
+ |
space to itself. The flow has the continuous group property, |
601 |
+ |
\begin{equation} |
602 |
+ |
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
603 |
+ |
+ \tau _2 } . |
604 |
+ |
\end{equation} |
605 |
+ |
In particular, |
606 |
+ |
\begin{equation} |
607 |
+ |
\varphi _\tau \circ \varphi _{ - \tau } = I |
608 |
+ |
\end{equation} |
609 |
+ |
Therefore, the exact flow is self-adjoint, |
610 |
+ |
\begin{equation} |
611 |
+ |
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
612 |
+ |
\end{equation} |
613 |
+ |
The exact flow can also be written in terms of the of an operator, |
614 |
+ |
\begin{equation} |
615 |
+ |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
616 |
+ |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
617 |
+ |
\label{introEquation:exponentialOperator} |
618 |
+ |
\end{equation} |
619 |
+ |
|
620 |
+ |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
621 |
+ |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
622 |
+ |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
623 |
+ |
the Taylor series of $\psi_\tau$ agree to order $p$, |
624 |
+ |
\begin{equation} |
625 |
+ |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
626 |
+ |
\end{equation} |
627 |
+ |
|
628 |
+ |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
629 |
+ |
|
630 |
+ |
The hidden geometric properties of ODE and its flow play important |
631 |
+ |
roles in numerical studies. Many of them can be found in systems |
632 |
+ |
which occur naturally in applications. |
633 |
+ |
|
634 |
+ |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
635 |
+ |
a \emph{symplectic} flow if it satisfies, |
636 |
+ |
\begin{equation} |
637 |
+ |
{\varphi '}^T J \varphi ' = J. |
638 |
+ |
\end{equation} |
639 |
+ |
According to Liouville's theorem, the symplectic volume is invariant |
640 |
+ |
under a Hamiltonian flow, which is the basis for classical |
641 |
+ |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
642 |
+ |
field on a symplectic manifold can be shown to be a |
643 |
+ |
symplectomorphism. As to the Poisson system, |
644 |
+ |
\begin{equation} |
645 |
+ |
{\varphi '}^T J \varphi ' = J \circ \varphi |
646 |
+ |
\end{equation} |
647 |
+ |
is the property must be preserved by the integrator. |
648 |
+ |
|
649 |
+ |
It is possible to construct a \emph{volume-preserving} flow for a |
650 |
+ |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
651 |
+ |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
652 |
+ |
be volume-preserving. |
653 |
+ |
|
654 |
+ |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
655 |
+ |
will result in a new system, |
656 |
+ |
\[ |
657 |
+ |
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
658 |
+ |
\] |
659 |
+ |
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
660 |
+ |
In other words, the flow of this vector field is reversible if and |
661 |
+ |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
662 |
+ |
|
663 |
+ |
A \emph{first integral}, or conserved quantity of a general |
664 |
+ |
differential function is a function $ G:R^{2d} \to R^d $ which is |
665 |
+ |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
666 |
+ |
\[ |
667 |
+ |
\frac{{dG(x(t))}}{{dt}} = 0. |
668 |
+ |
\] |
669 |
+ |
Using chain rule, one may obtain, |
670 |
+ |
\[ |
671 |
+ |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
672 |
+ |
\] |
673 |
+ |
which is the condition for conserving \emph{first integral}. For a |
674 |
+ |
canonical Hamiltonian system, the time evolution of an arbitrary |
675 |
+ |
smooth function $G$ is given by, |
676 |
+ |
\begin{equation} |
677 |
+ |
\begin{array}{c} |
678 |
+ |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
679 |
+ |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
680 |
+ |
\end{array} |
681 |
+ |
\label{introEquation:firstIntegral1} |
682 |
+ |
\end{equation} |
683 |
+ |
Using poisson bracket notion, Equation |
684 |
+ |
\ref{introEquation:firstIntegral1} can be rewritten as |
685 |
+ |
\[ |
686 |
+ |
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
687 |
+ |
\] |
688 |
+ |
Therefore, the sufficient condition for $G$ to be the \emph{first |
689 |
+ |
integral} of a Hamiltonian system is |
690 |
+ |
\[ |
691 |
+ |
\left\{ {G,H} \right\} = 0. |
692 |
+ |
\] |
693 |
+ |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
694 |
+ |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
695 |
+ |
0$. |
696 |
+ |
|
697 |
+ |
|
698 |
+ |
When designing any numerical methods, one should always try to |
699 |
+ |
preserve the structural properties of the original ODE and its flow. |
700 |
+ |
|
701 |
+ |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
702 |
+ |
A lot of well established and very effective numerical methods have |
703 |
+ |
been successful precisely because of their symplecticities even |
704 |
+ |
though this fact was not recognized when they were first |
705 |
+ |
constructed. The most famous example is leapfrog methods in |
706 |
+ |
molecular dynamics. In general, symplectic integrators can be |
707 |
+ |
constructed using one of four different methods. |
708 |
+ |
\begin{enumerate} |
709 |
+ |
\item Generating functions |
710 |
+ |
\item Variational methods |
711 |
+ |
\item Runge-Kutta methods |
712 |
+ |
\item Splitting methods |
713 |
+ |
\end{enumerate} |
714 |
+ |
|
715 |
+ |
Generating function tends to lead to methods which are cumbersome |
716 |
+ |
and difficult to use. In dissipative systems, variational methods |
717 |
+ |
can capture the decay of energy accurately. Since their |
718 |
+ |
geometrically unstable nature against non-Hamiltonian perturbations, |
719 |
+ |
ordinary implicit Runge-Kutta methods are not suitable for |
720 |
+ |
Hamiltonian system. Recently, various high-order explicit |
721 |
+ |
Runge--Kutta methods have been developed to overcome this |
722 |
+ |
instability. However, due to computational penalty involved in |
723 |
+ |
implementing the Runge-Kutta methods, they do not attract too much |
724 |
+ |
attention from Molecular Dynamics community. Instead, splitting have |
725 |
+ |
been widely accepted since they exploit natural decompositions of |
726 |
+ |
the system\cite{Tuckerman92}. |
727 |
+ |
|
728 |
+ |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
729 |
+ |
|
730 |
+ |
The main idea behind splitting methods is to decompose the discrete |
731 |
+ |
$\varphi_h$ as a composition of simpler flows, |
732 |
+ |
\begin{equation} |
733 |
+ |
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
734 |
+ |
\varphi _{h_n } |
735 |
+ |
\label{introEquation:FlowDecomposition} |
736 |
+ |
\end{equation} |
737 |
+ |
where each of the sub-flow is chosen such that each represent a |
738 |
+ |
simpler integration of the system. |
739 |
+ |
|
740 |
+ |
Suppose that a Hamiltonian system takes the form, |
741 |
+ |
\[ |
742 |
+ |
H = H_1 + H_2. |
743 |
+ |
\] |
744 |
+ |
Here, $H_1$ and $H_2$ may represent different physical processes of |
745 |
+ |
the system. For instance, they may relate to kinetic and potential |
746 |
+ |
energy respectively, which is a natural decomposition of the |
747 |
+ |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
748 |
+ |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
749 |
+ |
order is then given by the Lie-Trotter formula |
750 |
+ |
\begin{equation} |
751 |
+ |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
752 |
+ |
\label{introEquation:firstOrderSplitting} |
753 |
+ |
\end{equation} |
754 |
+ |
where $\varphi _h$ is the result of applying the corresponding |
755 |
+ |
continuous $\varphi _i$ over a time $h$. By definition, as |
756 |
+ |
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
757 |
+ |
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
758 |
+ |
It is easy to show that any composition of symplectic flows yields a |
759 |
+ |
symplectic map, |
760 |
+ |
\begin{equation} |
761 |
+ |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
762 |
+ |
'\phi ' = \phi '^T J\phi ' = J, |
763 |
+ |
\label{introEquation:SymplecticFlowComposition} |
764 |
+ |
\end{equation} |
765 |
+ |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
766 |
+ |
splitting in this context automatically generates a symplectic map. |
767 |
+ |
|
768 |
+ |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
769 |
+ |
introduces local errors proportional to $h^2$, while Strang |
770 |
+ |
splitting gives a second-order decomposition, |
771 |
+ |
\begin{equation} |
772 |
+ |
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
773 |
+ |
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
774 |
+ |
\end{equation} |
775 |
+ |
which has a local error proportional to $h^3$. Sprang splitting's |
776 |
+ |
popularity in molecular simulation community attribute to its |
777 |
+ |
symmetric property, |
778 |
+ |
\begin{equation} |
779 |
+ |
\varphi _h^{ - 1} = \varphi _{ - h}. |
780 |
+ |
\label{introEquation:timeReversible} |
781 |
+ |
\end{equation} |
782 |
+ |
|
783 |
+ |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
784 |
+ |
The classical equation for a system consisting of interacting |
785 |
+ |
particles can be written in Hamiltonian form, |
786 |
+ |
\[ |
787 |
+ |
H = T + V |
788 |
+ |
\] |
789 |
+ |
where $T$ is the kinetic energy and $V$ is the potential energy. |
790 |
+ |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
791 |
+ |
obtains the following: |
792 |
+ |
\begin{align} |
793 |
+ |
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
794 |
+ |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
795 |
+ |
\label{introEquation:Lp10a} \\% |
796 |
+ |
% |
797 |
+ |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
798 |
+ |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
799 |
+ |
\label{introEquation:Lp10b} |
800 |
+ |
\end{align} |
801 |
+ |
where $F(t)$ is the force at time $t$. This integration scheme is |
802 |
+ |
known as \emph{velocity verlet} which is |
803 |
+ |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
804 |
+ |
time-reversible(\ref{introEquation:timeReversible}) and |
805 |
+ |
volume-preserving (\ref{introEquation:volumePreserving}). These |
806 |
+ |
geometric properties attribute to its long-time stability and its |
807 |
+ |
popularity in the community. However, the most commonly used |
808 |
+ |
velocity verlet integration scheme is written as below, |
809 |
+ |
\begin{align} |
810 |
+ |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
811 |
+ |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
812 |
+ |
% |
813 |
+ |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
814 |
+ |
\label{introEquation:Lp9b}\\% |
815 |
+ |
% |
816 |
+ |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
817 |
+ |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
818 |
+ |
\end{align} |
819 |
+ |
From the preceding splitting, one can see that the integration of |
820 |
+ |
the equations of motion would follow: |
821 |
+ |
\begin{enumerate} |
822 |
+ |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
823 |
+ |
|
824 |
+ |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
825 |
+ |
|
826 |
+ |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
827 |
+ |
|
828 |
+ |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
829 |
+ |
\end{enumerate} |
830 |
+ |
|
831 |
+ |
Simply switching the order of splitting and composing, a new |
832 |
+ |
integrator, the \emph{position verlet} integrator, can be generated, |
833 |
+ |
\begin{align} |
834 |
+ |
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
835 |
+ |
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
836 |
+ |
\label{introEquation:positionVerlet1} \\% |
837 |
+ |
% |
838 |
+ |
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
839 |
+ |
q(\Delta t)} \right]. % |
840 |
+ |
\label{introEquation:positionVerlet1} |
841 |
+ |
\end{align} |
842 |
+ |
|
843 |
+ |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
844 |
+ |
|
845 |
+ |
Baker-Campbell-Hausdorff formula can be used to determine the local |
846 |
+ |
error of splitting method in terms of commutator of the |
847 |
+ |
operators(\ref{introEquation:exponentialOperator}) associated with |
848 |
+ |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
849 |
+ |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
850 |
+ |
\begin{equation} |
851 |
+ |
\exp (hX + hY) = \exp (hZ) |
852 |
+ |
\end{equation} |
853 |
+ |
where |
854 |
+ |
\begin{equation} |
855 |
+ |
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
856 |
+ |
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
857 |
+ |
\end{equation} |
858 |
+ |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
859 |
+ |
\[ |
860 |
+ |
[X,Y] = XY - YX . |
861 |
+ |
\] |
862 |
+ |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
863 |
+ |
can obtain |
864 |
+ |
\begin{eqnarray*} |
865 |
+ |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
866 |
+ |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
867 |
+ |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
868 |
+ |
\ldots ) |
869 |
+ |
\end{eqnarray*} |
870 |
+ |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
871 |
+ |
error of Spring splitting is proportional to $h^3$. The same |
872 |
+ |
procedure can be applied to general splitting, of the form |
873 |
+ |
\begin{equation} |
874 |
+ |
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
875 |
+ |
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
876 |
+ |
\end{equation} |
877 |
+ |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
878 |
+ |
order method. Yoshida proposed an elegant way to compose higher |
879 |
+ |
order methods based on symmetric splitting. Given a symmetric second |
880 |
+ |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
881 |
+ |
method can be constructed by composing, |
882 |
+ |
\[ |
883 |
+ |
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
884 |
+ |
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
885 |
+ |
\] |
886 |
+ |
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
887 |
+ |
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
888 |
+ |
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
889 |
+ |
\begin{equation} |
890 |
+ |
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
891 |
+ |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
892 |
+ |
\end{equation} |
893 |
+ |
, if the weights are chosen as |
894 |
+ |
\[ |
895 |
+ |
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
896 |
+ |
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
897 |
+ |
\] |
898 |
+ |
|
899 |
+ |
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
900 |
+ |
|
901 |
+ |
As a special discipline of molecular modeling, Molecular dynamics |
902 |
+ |
has proven to be a powerful tool for studying the functions of |
903 |
+ |
biological systems, providing structural, thermodynamic and |
904 |
+ |
dynamical information. |
905 |
+ |
|
906 |
+ |
\subsection{\label{introSec:mdInit}Initialization} |
907 |
+ |
|
908 |
+ |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
909 |
+ |
|
910 |
+ |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
911 |
+ |
|
912 |
+ |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
913 |
+ |
|
914 |
+ |
Rigid bodies are frequently involved in the modeling of different |
915 |
+ |
areas, from engineering, physics, to chemistry. For example, |
916 |
+ |
missiles and vehicle are usually modeled by rigid bodies. The |
917 |
+ |
movement of the objects in 3D gaming engine or other physics |
918 |
+ |
simulator is governed by the rigid body dynamics. In molecular |
919 |
+ |
simulation, rigid body is used to simplify the model in |
920 |
+ |
protein-protein docking study{\cite{Gray03}}. |
921 |
+ |
|
922 |
+ |
It is very important to develop stable and efficient methods to |
923 |
+ |
integrate the equations of motion of orientational degrees of |
924 |
+ |
freedom. Euler angles are the nature choice to describe the |
925 |
+ |
rotational degrees of freedom. However, due to its singularity, the |
926 |
+ |
numerical integration of corresponding equations of motion is very |
927 |
+ |
inefficient and inaccurate. Although an alternative integrator using |
928 |
+ |
different sets of Euler angles can overcome this difficulty\cite{}, |
929 |
+ |
the computational penalty and the lost of angular momentum |
930 |
+ |
conservation still remain. A singularity free representation |
931 |
+ |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
932 |
+ |
this approach suffer from the nonseparable Hamiltonian resulted from |
933 |
+ |
quaternion representation, which prevents the symplectic algorithm |
934 |
+ |
to be utilized. Another different approach is to apply holonomic |
935 |
+ |
constraints to the atoms belonging to the rigid body. Each atom |
936 |
+ |
moves independently under the normal forces deriving from potential |
937 |
+ |
energy and constraint forces which are used to guarantee the |
938 |
+ |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
939 |
+ |
algorithm converge very slowly when the number of constraint |
940 |
+ |
increases. |
941 |
+ |
|
942 |
+ |
The break through in geometric literature suggests that, in order to |
943 |
+ |
develop a long-term integration scheme, one should preserve the |
944 |
+ |
symplectic structure of the flow. Introducing conjugate momentum to |
945 |
+ |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
946 |
+ |
symplectic integrator, RSHAKE, was proposed to evolve the |
947 |
+ |
Hamiltonian system in a constraint manifold by iteratively |
948 |
+ |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
949 |
+ |
method using quaternion representation was developed by Omelyan. |
950 |
+ |
However, both of these methods are iterative and inefficient. In |
951 |
+ |
this section, we will present a symplectic Lie-Poisson integrator |
952 |
+ |
for rigid body developed by Dullweber and his |
953 |
+ |
coworkers\cite{Dullweber1997}. |
954 |
+ |
|
955 |
+ |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
956 |
+ |
|
957 |
+ |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
958 |
+ |
|
959 |
+ |
\begin{equation} |
960 |
+ |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
961 |
+ |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
962 |
+ |
\label{introEquation:RBHamiltonian} |
963 |
+ |
\end{equation} |
964 |
+ |
Here, $q$ and $Q$ are the position and rotation matrix for the |
965 |
+ |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
966 |
+ |
$J$, a diagonal matrix, is defined by |
967 |
+ |
\[ |
968 |
+ |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
969 |
+ |
\] |
970 |
+ |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
971 |
+ |
constrained Hamiltonian equation subjects to a holonomic constraint, |
972 |
+ |
\begin{equation} |
973 |
+ |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
974 |
+ |
\end{equation} |
975 |
+ |
which is used to ensure rotation matrix's orthogonality. |
976 |
+ |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
977 |
+ |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
978 |
+ |
\begin{equation} |
979 |
+ |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
980 |
+ |
\label{introEquation:RBFirstOrderConstraint} |
981 |
+ |
\end{equation} |
982 |
+ |
|
983 |
+ |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
984 |
+ |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
985 |
+ |
the equations of motion, |
986 |
+ |
\[ |
987 |
+ |
\begin{array}{c} |
988 |
+ |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
989 |
+ |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
990 |
+ |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
991 |
+ |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
992 |
+ |
\end{array} |
993 |
+ |
\] |
994 |
+ |
|
995 |
+ |
In general, there are two ways to satisfy the holonomic constraints. |
996 |
+ |
We can use constraint force provided by lagrange multiplier on the |
997 |
+ |
normal manifold to keep the motion on constraint space. Or we can |
998 |
+ |
simply evolve the system in constraint manifold. The two method are |
999 |
+ |
proved to be equivalent. The holonomic constraint and equations of |
1000 |
+ |
motions define a constraint manifold for rigid body |
1001 |
+ |
\[ |
1002 |
+ |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1003 |
+ |
\right\}. |
1004 |
+ |
\] |
1005 |
+ |
|
1006 |
+ |
Unfortunately, this constraint manifold is not the cotangent bundle |
1007 |
+ |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
1008 |
+ |
transformation, the cotangent space and the phase space are |
1009 |
+ |
diffeomorphic. Introducing |
1010 |
+ |
\[ |
1011 |
+ |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1012 |
+ |
\] |
1013 |
+ |
the mechanical system subject to a holonomic constraint manifold $M$ |
1014 |
+ |
can be re-formulated as a Hamiltonian system on the cotangent space |
1015 |
+ |
\[ |
1016 |
+ |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
1017 |
+ |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
1018 |
+ |
\] |
1019 |
+ |
|
1020 |
+ |
For a body fixed vector $X_i$ with respect to the center of mass of |
1021 |
+ |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
1022 |
+ |
given as |
1023 |
+ |
\begin{equation} |
1024 |
+ |
X_i^{lab} = Q X_i + q. |
1025 |
+ |
\end{equation} |
1026 |
+ |
Therefore, potential energy $V(q,Q)$ is defined by |
1027 |
+ |
\[ |
1028 |
+ |
V(q,Q) = V(Q X_0 + q). |
1029 |
+ |
\] |
1030 |
+ |
Hence, |
1031 |
+ |
\[ |
1032 |
+ |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)} |
1033 |
+ |
\] |
1034 |
+ |
|
1035 |
+ |
\[ |
1036 |
+ |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
1037 |
+ |
\] |
1038 |
+ |
|
1039 |
+ |
As a common choice to describe the rotation dynamics of the rigid |
1040 |
+ |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
1041 |
+ |
rewrite the equations of motion, |
1042 |
+ |
\begin{equation} |
1043 |
+ |
\begin{array}{l} |
1044 |
+ |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1045 |
+ |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
1046 |
+ |
\end{array} |
1047 |
+ |
\label{introEqaution:RBMotionPI} |
1048 |
+ |
\end{equation} |
1049 |
+ |
, as well as holonomic constraints, |
1050 |
+ |
\[ |
1051 |
+ |
\begin{array}{l} |
1052 |
+ |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
1053 |
+ |
Q^T Q = 1 \\ |
1054 |
+ |
\end{array} |
1055 |
+ |
\] |
1056 |
+ |
|
1057 |
+ |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1058 |
+ |
so(3)^ \star$, the hat-map isomorphism, |
1059 |
+ |
\begin{equation} |
1060 |
+ |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1061 |
+ |
{\begin{array}{*{20}c} |
1062 |
+ |
0 & { - v_3 } & {v_2 } \\ |
1063 |
+ |
{v_3 } & 0 & { - v_1 } \\ |
1064 |
+ |
{ - v_2 } & {v_1 } & 0 \\ |
1065 |
+ |
\end{array}} \right), |
1066 |
+ |
\label{introEquation:hatmapIsomorphism} |
1067 |
+ |
\end{equation} |
1068 |
+ |
will let us associate the matrix products with traditional vector |
1069 |
+ |
operations |
1070 |
+ |
\[ |
1071 |
+ |
\hat vu = v \times u |
1072 |
+ |
\] |
1073 |
+ |
|
1074 |
+ |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1075 |
+ |
matrix, |
1076 |
+ |
\begin{equation} |
1077 |
+ |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
1078 |
+ |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
1079 |
+ |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1080 |
+ |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
1081 |
+ |
\end{equation} |
1082 |
+ |
Since $\Lambda$ is symmetric, the last term of Equation |
1083 |
+ |
\ref{introEquation:skewMatrixPI}, which implies the Lagrange |
1084 |
+ |
multiplier $\Lambda$ is ignored in the integration. |
1085 |
+ |
|
1086 |
+ |
Hence, applying hat-map isomorphism, we obtain the equation of |
1087 |
+ |
motion for angular momentum on body frame |
1088 |
+ |
\[ |
1089 |
+ |
\dot \pi = \pi \times I^{ - 1} \pi + Q^T \sum\limits_i {F_i (r,Q) |
1090 |
+ |
\times X_i } |
1091 |
+ |
\] |
1092 |
+ |
In the same manner, the equation of motion for rotation matrix is |
1093 |
+ |
given by |
1094 |
+ |
\[ |
1095 |
+ |
\dot Q = Qskew(M^{ - 1} \pi ) |
1096 |
+ |
\] |
1097 |
+ |
|
1098 |
+ |
The free rigid body equation is an example of a non-canonical |
1099 |
+ |
Hamiltonian system. |
1100 |
+ |
|
1101 |
+ |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Integration of Euler Equations} |
1102 |
+ |
|
1103 |
+ |
\[ |
1104 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1105 |
+ |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V} |
1106 |
+ |
\] |
1107 |
+ |
|
1108 |
+ |
\[ |
1109 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,R} \circ \varphi |
1110 |
+ |
_{\Delta t,\pi } |
1111 |
+ |
\] |
1112 |
+ |
|
1113 |
+ |
\[ |
1114 |
+ |
\varphi _{\Delta t,\pi } = \varphi _{\Delta t/2,\pi _1 } \circ |
1115 |
+ |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1116 |
+ |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1117 |
+ |
_1 } |
1118 |
+ |
\] |
1119 |
+ |
|
1120 |
+ |
\[ |
1121 |
+ |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1122 |
+ |
_{\Delta t/2,\tau } |
1123 |
+ |
\] |
1124 |
+ |
|
1125 |
+ |
|
1126 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1127 |
|
|
1128 |
+ |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
1129 |
+ |
|
1130 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
1131 |
|
|
1132 |
< |
\subsection{\label{introSection:hydroynamics}Hydrodynamics} |
1132 |
> |
\begin{equation} |
1133 |
> |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1134 |
> |
\label{introEquation:bathGLE} |
1135 |
> |
\end{equation} |
1136 |
> |
where $H_B$ is harmonic bath Hamiltonian, |
1137 |
> |
\[ |
1138 |
> |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1139 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
1140 |
> |
\] |
1141 |
> |
and $\Delta U$ is bilinear system-bath coupling, |
1142 |
> |
\[ |
1143 |
> |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1144 |
> |
\] |
1145 |
> |
Completing the square, |
1146 |
> |
\[ |
1147 |
> |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
1148 |
> |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1149 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1150 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
1151 |
> |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1152 |
> |
\] |
1153 |
> |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
1154 |
> |
\[ |
1155 |
> |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1156 |
> |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1157 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1158 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1159 |
> |
\] |
1160 |
> |
where |
1161 |
> |
\[ |
1162 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1163 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1164 |
> |
\] |
1165 |
> |
Since the first two terms of the new Hamiltonian depend only on the |
1166 |
> |
system coordinates, we can get the equations of motion for |
1167 |
> |
Generalized Langevin Dynamics by Hamilton's equations |
1168 |
> |
\ref{introEquation:motionHamiltonianCoordinate, |
1169 |
> |
introEquation:motionHamiltonianMomentum}, |
1170 |
> |
\begin{align} |
1171 |
> |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
1172 |
> |
&= m\ddot x |
1173 |
> |
&= - \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)} |
1174 |
> |
\label{introEquation:Lp5} |
1175 |
> |
\end{align} |
1176 |
> |
, and |
1177 |
> |
\begin{align} |
1178 |
> |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
1179 |
> |
&= m\ddot x_\alpha |
1180 |
> |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
1181 |
> |
\end{align} |
1182 |
> |
|
1183 |
> |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
1184 |
> |
|
1185 |
> |
\[ |
1186 |
> |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
1187 |
> |
\] |
1188 |
> |
|
1189 |
> |
\[ |
1190 |
> |
L(x + y) = L(x) + L(y) |
1191 |
> |
\] |
1192 |
> |
|
1193 |
> |
\[ |
1194 |
> |
L(ax) = aL(x) |
1195 |
> |
\] |
1196 |
> |
|
1197 |
> |
\[ |
1198 |
> |
L(\dot x) = pL(x) - px(0) |
1199 |
> |
\] |
1200 |
> |
|
1201 |
> |
\[ |
1202 |
> |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
1203 |
> |
\] |
1204 |
> |
|
1205 |
> |
\[ |
1206 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
1207 |
> |
\] |
1208 |
> |
|
1209 |
> |
Some relatively important transformation, |
1210 |
> |
\[ |
1211 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
1212 |
> |
\] |
1213 |
> |
|
1214 |
> |
\[ |
1215 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
1216 |
> |
\] |
1217 |
> |
|
1218 |
> |
\[ |
1219 |
> |
L(1) = \frac{1}{p} |
1220 |
> |
\] |
1221 |
> |
|
1222 |
> |
First, the bath coordinates, |
1223 |
> |
\[ |
1224 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
1225 |
> |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
1226 |
> |
}}L(x) |
1227 |
> |
\] |
1228 |
> |
\[ |
1229 |
> |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
1230 |
> |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
1231 |
> |
\] |
1232 |
> |
Then, the system coordinates, |
1233 |
> |
\begin{align} |
1234 |
> |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1235 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
1236 |
> |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
1237 |
> |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
1238 |
> |
}}\omega _\alpha ^2 L(x)} \right\}} |
1239 |
> |
% |
1240 |
> |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1241 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
1242 |
> |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
1243 |
> |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
1244 |
> |
\end{align} |
1245 |
> |
Then, the inverse transform, |
1246 |
> |
|
1247 |
> |
\begin{align} |
1248 |
> |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
1249 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1250 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
1251 |
> |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
1252 |
> |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
1253 |
> |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
1254 |
> |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
1255 |
> |
% |
1256 |
> |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1257 |
> |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1258 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1259 |
> |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
1260 |
> |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
1261 |
> |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
1262 |
> |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
1263 |
> |
(\omega _\alpha t)} \right\}} |
1264 |
> |
\end{align} |
1265 |
> |
|
1266 |
> |
\begin{equation} |
1267 |
> |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
1268 |
> |
(t)\dot x(t - \tau )d\tau } + R(t) |
1269 |
> |
\label{introEuqation:GeneralizedLangevinDynamics} |
1270 |
> |
\end{equation} |
1271 |
> |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
1272 |
> |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
1273 |
> |
\[ |
1274 |
> |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1275 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
1276 |
> |
\] |
1277 |
> |
For an infinite harmonic bath, we can use the spectral density and |
1278 |
> |
an integral over frequencies. |
1279 |
> |
|
1280 |
> |
\[ |
1281 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
1282 |
> |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
1283 |
> |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
1284 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
1285 |
> |
\] |
1286 |
> |
The random forces depend only on initial conditions. |
1287 |
> |
|
1288 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
1289 |
> |
So we can define a new set of coordinates, |
1290 |
> |
\[ |
1291 |
> |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1292 |
> |
^2 }}x(0) |
1293 |
> |
\] |
1294 |
> |
This makes |
1295 |
> |
\[ |
1296 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
1297 |
> |
\] |
1298 |
> |
And since the $q$ coordinates are harmonic oscillators, |
1299 |
> |
\[ |
1300 |
> |
\begin{array}{l} |
1301 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1302 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1303 |
> |
\end{array} |
1304 |
> |
\] |
1305 |
> |
|
1306 |
> |
\begin{align} |
1307 |
> |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
1308 |
> |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
1309 |
> |
(t)q_\beta (0)} \right\rangle } } |
1310 |
> |
% |
1311 |
> |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
1312 |
> |
\right\rangle \cos (\omega _\alpha t)} |
1313 |
> |
% |
1314 |
> |
&= kT\xi (t) |
1315 |
> |
\end{align} |
1316 |
> |
|
1317 |
> |
\begin{equation} |
1318 |
> |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1319 |
> |
\label{introEquation:secondFluctuationDissipation} |
1320 |
> |
\end{equation} |
1321 |
> |
|
1322 |
> |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
1323 |
> |
|
1324 |
> |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1325 |
> |
\subsection{\label{introSection:analyticalApproach}Analytical |
1326 |
> |
Approach} |
1327 |
> |
|
1328 |
> |
\subsection{\label{introSection:approximationApproach}Approximation |
1329 |
> |
Approach} |
1330 |
> |
|
1331 |
> |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
1332 |
> |
Body} |
1333 |
> |
|
1334 |
> |
\section{\label{introSection:correlationFunctions}Correlation Functions} |