3 |
|
\section{\label{introSection:classicalMechanics}Classical |
4 |
|
Mechanics} |
5 |
|
|
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 |
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. |
6 |
> |
Using equations of motion derived from Classical Mechanics, |
7 |
> |
Molecular Dynamics simulations are carried out by integrating the |
8 |
> |
equations of motion for a given system of particles. There are three |
9 |
> |
fundamental ideas behind classical mechanics. Firstly, one can |
10 |
> |
determine the state of a mechanical system at any time of interest; |
11 |
> |
Secondly, all the mechanical properties of the system at that time |
12 |
> |
can be determined by combining the knowledge of the properties of |
13 |
> |
the system with the specification of this state; Finally, the |
14 |
> |
specification of the state when further combined with the laws of |
15 |
> |
mechanics will also be sufficient to predict the future behavior of |
16 |
> |
the system. |
17 |
|
|
18 |
|
\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
19 |
|
The discovery of Newton's three laws of mechanics which govern the |
20 |
|
motion of particles is the foundation of the classical mechanics. |
21 |
< |
Newton¡¯s first law defines a class of inertial frames. Inertial |
21 |
> |
Newton's first law defines a class of inertial frames. Inertial |
22 |
|
frames are reference frames where a particle not interacting with |
23 |
|
other bodies will move with constant speed in the same direction. |
24 |
< |
With respect to inertial frames Newton¡¯s second law has the form |
24 |
> |
With respect to inertial frames, Newton's second law has the form |
25 |
|
\begin{equation} |
26 |
< |
F = \frac {dp}{dt} = \frac {mv}{dt} |
26 |
> |
F = \frac {dp}{dt} = \frac {mdv}{dt} |
27 |
|
\label{introEquation:newtonSecondLaw} |
28 |
|
\end{equation} |
29 |
|
A point mass interacting with other bodies moves with the |
30 |
|
acceleration along the direction of the force acting on it. Let |
31 |
|
$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and |
32 |
|
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
33 |
< |
Newton¡¯s third law states that |
33 |
> |
Newton's third law states that |
34 |
|
\begin{equation} |
35 |
< |
F_{ij} = -F_{ji} |
35 |
> |
F_{ij} = -F_{ji}. |
36 |
|
\label{introEquation:newtonThirdLaw} |
37 |
|
\end{equation} |
37 |
– |
|
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 |
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} |
49 |
> |
\tau \equiv r \times F \label{introEquation:torqueDefinition} |
50 |
|
\end{equation} |
51 |
|
Differentiating Eq.~\ref{introEquation:angularMomentumDefinition}, |
52 |
|
\[ |
59 |
|
\] |
60 |
|
thus, |
61 |
|
\begin{equation} |
62 |
< |
\dot L = r \times \dot p = N |
62 |
> |
\dot L = r \times \dot p = \tau |
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} |
66 |
> |
that if all forces are conservative, energy is conserved, |
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}. |
69 |
> |
All of these conserved quantities are important factors to determine |
70 |
> |
the quality of numerical integration schemes for rigid bodies |
71 |
> |
\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 |
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. |
75 |
> |
Newtonian Mechanics suffers from an important limitation: motion can |
76 |
> |
only be described in cartesian coordinate systems which make it |
77 |
> |
impossible to predict analytically the properties of the system even |
78 |
> |
if we know all of the details of the interaction. In order to |
79 |
> |
overcome some of the practical difficulties which arise in attempts |
80 |
> |
to apply Newton's equation to complex systems, approximate numerical |
81 |
> |
procedures may be developed. |
82 |
|
|
83 |
< |
\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
84 |
< |
Principle} |
83 |
> |
\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
84 |
> |
Principle}} |
85 |
|
|
86 |
|
Hamilton introduced the dynamical principle upon which it is |
87 |
< |
possible to base all of mechanics and, indeed, most of classical |
88 |
< |
physics. Hamilton's Principle may be stated as follow, |
89 |
< |
|
90 |
< |
The actual trajectory, along which a dynamical system may move from |
91 |
< |
one point to another within a specified time, is derived by finding |
92 |
< |
the path which minimizes the time integral of the difference between |
96 |
< |
the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
87 |
> |
possible to base all of mechanics and most of classical physics. |
88 |
> |
Hamilton's Principle may be stated as follows: the trajectory, along |
89 |
> |
which a dynamical system may move from one point to another within a |
90 |
> |
specified time, is derived by finding the path which minimizes the |
91 |
> |
time integral of the difference between the kinetic $K$, and |
92 |
> |
potential energies $U$, |
93 |
|
\begin{equation} |
94 |
< |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
94 |
> |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}. |
95 |
|
\label{introEquation:halmitonianPrinciple1} |
96 |
|
\end{equation} |
101 |
– |
|
97 |
|
For simple mechanical systems, where the forces acting on the |
98 |
< |
different part are derivable from a potential and the velocities are |
99 |
< |
small compared with that of light, the Lagrangian function $L$ can |
100 |
< |
be define as the difference between the kinetic energy of the system |
106 |
< |
and its potential energy, |
98 |
> |
different parts are derivable from a potential, the Lagrangian |
99 |
> |
function $L$ can be defined as the difference between the kinetic |
100 |
> |
energy of the system and its potential energy, |
101 |
|
\begin{equation} |
102 |
< |
L \equiv K - U = L(q_i ,\dot q_i ) , |
102 |
> |
L \equiv K - U = L(q_i ,\dot q_i ). |
103 |
|
\label{introEquation:lagrangianDef} |
104 |
|
\end{equation} |
105 |
< |
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
105 |
> |
Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
106 |
|
\begin{equation} |
107 |
< |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
107 |
> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} . |
108 |
|
\label{introEquation:halmitonianPrinciple2} |
109 |
|
\end{equation} |
110 |
|
|
111 |
< |
\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
112 |
< |
Equations of Motion in Lagrangian Mechanics} |
111 |
> |
\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The |
112 |
> |
Equations of Motion in Lagrangian Mechanics}} |
113 |
|
|
114 |
< |
For a holonomic system of $f$ degrees of freedom, the equations of |
115 |
< |
motion in the Lagrangian form is |
114 |
> |
For a system of $f$ degrees of freedom, the equations of motion in |
115 |
> |
the Lagrangian form is |
116 |
|
\begin{equation} |
117 |
|
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - |
118 |
|
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f |
126 |
|
Arising from Lagrangian Mechanics, Hamiltonian Mechanics was |
127 |
|
introduced by William Rowan Hamilton in 1833 as a re-formulation of |
128 |
|
classical mechanics. If the potential energy of a system is |
129 |
< |
independent of generalized velocities, the generalized momenta can |
136 |
< |
be defined as |
129 |
> |
independent of velocities, the momenta can be defined as |
130 |
|
\begin{equation} |
131 |
|
p_i = \frac{\partial L}{\partial \dot q_i} |
132 |
|
\label{introEquation:generalizedMomenta} |
136 |
|
p_i = \frac{{\partial L}}{{\partial q_i }} |
137 |
|
\label{introEquation:generalizedMomentaDot} |
138 |
|
\end{equation} |
146 |
– |
|
139 |
|
With the help of the generalized momenta, we may now define a new |
140 |
|
quantity $H$ by the equation |
141 |
|
\begin{equation} |
143 |
|
\label{introEquation:hamiltonianDefByLagrangian} |
144 |
|
\end{equation} |
145 |
|
where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
146 |
< |
$L$ is the Lagrangian function for the system. |
147 |
< |
|
156 |
< |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
157 |
< |
one can obtain |
146 |
> |
$L$ is the Lagrangian function for the system. Differentiating |
147 |
> |
Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain |
148 |
|
\begin{equation} |
149 |
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
150 |
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
151 |
|
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
152 |
< |
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
152 |
> |
L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1} |
153 |
|
\end{equation} |
154 |
< |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
155 |
< |
second and fourth terms in the parentheses cancel. Therefore, |
154 |
> |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second |
155 |
> |
and fourth terms in the parentheses cancel. Therefore, |
156 |
|
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
157 |
|
\begin{equation} |
158 |
|
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
159 |
< |
\right)} - \frac{{\partial L}}{{\partial t}}dt |
159 |
> |
\right)} - \frac{{\partial L}}{{\partial t}}dt . |
160 |
|
\label{introEquation:diffHamiltonian2} |
161 |
|
\end{equation} |
162 |
|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
163 |
|
find |
164 |
|
\begin{equation} |
165 |
< |
\frac{{\partial H}}{{\partial p_k }} = q_k |
165 |
> |
\frac{{\partial H}}{{\partial p_k }} = \dot {q_k} |
166 |
|
\label{introEquation:motionHamiltonianCoordinate} |
167 |
|
\end{equation} |
168 |
|
\begin{equation} |
169 |
< |
\frac{{\partial H}}{{\partial q_k }} = - p_k |
169 |
> |
\frac{{\partial H}}{{\partial q_k }} = - \dot {p_k} |
170 |
|
\label{introEquation:motionHamiltonianMomentum} |
171 |
|
\end{equation} |
172 |
|
and |
175 |
|
t}} |
176 |
|
\label{introEquation:motionHamiltonianTime} |
177 |
|
\end{equation} |
178 |
< |
|
189 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
178 |
> |
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
179 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
180 |
|
equation of motion. Due to their symmetrical formula, they are also |
181 |
< |
known as the canonical equations of motions \cite{Goldstein01}. |
181 |
> |
known as the canonical equations of motions \cite{Goldstein2001}. |
182 |
|
|
183 |
|
An important difference between Lagrangian approach and the |
184 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
185 |
< |
function of the generalized velocities $\dot q_i$ and the |
186 |
< |
generalized coordinates $q_i$, while the Hamiltonian is considered |
187 |
< |
to be a function of the generalized momenta $p_i$ and the conjugate |
188 |
< |
generalized coordinate $q_i$. Hamiltonian Mechanics is more |
189 |
< |
appropriate for application to statistical mechanics and quantum |
190 |
< |
mechanics, since it treats the coordinate and its time derivative as |
191 |
< |
independent variables and it only works with 1st-order differential |
203 |
< |
equations\cite{Marion90}. |
204 |
< |
|
185 |
> |
function of the generalized velocities $\dot q_i$ and coordinates |
186 |
> |
$q_i$, while the Hamiltonian is considered to be a function of the |
187 |
> |
generalized momenta $p_i$ and the conjugate coordinates $q_i$. |
188 |
> |
Hamiltonian Mechanics is more appropriate for application to |
189 |
> |
statistical mechanics and quantum mechanics, since it treats the |
190 |
> |
coordinate and its time derivative as independent variables and it |
191 |
> |
only works with 1st-order differential equations\cite{Marion1990}. |
192 |
|
In Newtonian Mechanics, a system described by conservative forces |
193 |
< |
conserves the total energy \ref{introEquation:energyConservation}. |
194 |
< |
It follows that Hamilton's equations of motion conserve the total |
195 |
< |
Hamiltonian. |
193 |
> |
conserves the total energy |
194 |
> |
(Eq.~\ref{introEquation:energyConservation}). It follows that |
195 |
> |
Hamilton's equations of motion conserve the total Hamiltonian |
196 |
|
\begin{equation} |
197 |
|
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
198 |
|
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
199 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
200 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
201 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
202 |
< |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
202 |
> |
q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian} |
203 |
|
\end{equation} |
204 |
|
|
205 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
214 |
|
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
215 |
|
|
216 |
|
Mathematically, phase space is the space which represents all |
217 |
< |
possible states. Each possible state of the system corresponds to |
218 |
< |
one unique point in the phase space. For mechanical systems, the |
219 |
< |
phase space usually consists of all possible values of position and |
220 |
< |
momentum variables. Consider a dynamic system in a cartesian space, |
221 |
< |
where each of the $6f$ coordinates and momenta is assigned to one of |
222 |
< |
$6f$ mutually orthogonal axes, the phase space of this system is a |
223 |
< |
$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , |
224 |
< |
\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and |
217 |
> |
possible states of a system. Each possible state of the system |
218 |
> |
corresponds to one unique point in the phase space. For mechanical |
219 |
> |
systems, the phase space usually consists of all possible values of |
220 |
> |
position and momentum variables. Consider a dynamic system of $f$ |
221 |
> |
particles in a cartesian space, where each of the $6f$ coordinates |
222 |
> |
and momenta is assigned to one of $6f$ mutually orthogonal axes, the |
223 |
> |
phase space of this system is a $6f$ dimensional space. A point, $x |
224 |
> |
= |
225 |
> |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
226 |
> |
\over q} _1 , \ldots |
227 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
228 |
> |
\over q} _f |
229 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
230 |
> |
\over p} _1 \ldots |
231 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
232 |
> |
\over p} _f )$ , with a unique set of values of $6f$ coordinates and |
233 |
|
momenta is a phase space vector. |
234 |
+ |
%%%fix me |
235 |
|
|
236 |
< |
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 |
236 |
> |
In statistical mechanics, the condition of an ensemble at any time |
237 |
|
can be regarded as appropriately specified by the density $\rho$ |
238 |
|
with which representative points are distributed over the phase |
239 |
< |
space. The density of distribution for an ensemble with $f$ degrees |
240 |
< |
of freedom is defined as, |
239 |
> |
space. The density distribution for an ensemble with $f$ degrees of |
240 |
> |
freedom is defined as, |
241 |
|
\begin{equation} |
242 |
|
\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). |
243 |
|
\label{introEquation:densityDistribution} |
244 |
|
\end{equation} |
245 |
|
Governed by the principles of mechanics, the phase points change |
246 |
< |
their value which would change the density at any time at phase |
247 |
< |
space. Hence, the density of distribution is also to be taken as a |
248 |
< |
function of the time. |
249 |
< |
|
271 |
< |
The number of systems $\delta N$ at time $t$ can be determined by, |
246 |
> |
their locations which changes the density at any time at phase |
247 |
> |
space. Hence, the density distribution is also to be taken as a |
248 |
> |
function of the time. The number of systems $\delta N$ at time $t$ |
249 |
> |
can be determined by, |
250 |
|
\begin{equation} |
251 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
252 |
|
\label{introEquation:deltaN} |
253 |
|
\end{equation} |
254 |
< |
Assuming a large enough population of systems are exploited, we can |
255 |
< |
sufficiently approximate $\delta N$ without introducing |
256 |
< |
discontinuity when we go from one region in the phase space to |
257 |
< |
another. By integrating over the whole phase space, |
254 |
> |
Assuming enough copies of the systems, we can sufficiently |
255 |
> |
approximate $\delta N$ without introducing discontinuity when we go |
256 |
> |
from one region in the phase space to another. By integrating over |
257 |
> |
the whole phase space, |
258 |
|
\begin{equation} |
259 |
|
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
260 |
|
\label{introEquation:totalNumberSystem} |
261 |
|
\end{equation} |
262 |
< |
gives us an expression for the total number of the systems. Hence, |
263 |
< |
the probability per unit in the phase space can be obtained by, |
262 |
> |
gives us an expression for the total number of copies. Hence, the |
263 |
> |
probability per unit volume in the phase space can be obtained by, |
264 |
|
\begin{equation} |
265 |
|
\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int |
266 |
|
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
267 |
|
\label{introEquation:unitProbability} |
268 |
|
\end{equation} |
269 |
< |
With the help of Equation(\ref{introEquation:unitProbability}) and |
270 |
< |
the knowledge of the system, it is possible to calculate the average |
269 |
> |
With the help of Eq.~\ref{introEquation:unitProbability} and the |
270 |
> |
knowledge of the system, it is possible to calculate the average |
271 |
|
value of any desired quantity which depends on the coordinates and |
272 |
< |
momenta of the system. Even when the dynamics of the real system is |
272 |
> |
momenta of the system. Even when the dynamics of the real system are |
273 |
|
complex, or stochastic, or even discontinuous, the average |
274 |
< |
properties of the ensemble of possibilities as a whole may still |
275 |
< |
remain well defined. For a classical system in thermal equilibrium |
276 |
< |
with its environment, the ensemble average of a mechanical quantity, |
277 |
< |
$\langle A(q , p) \rangle_t$, takes the form of an integral over the |
278 |
< |
phase space of the system, |
274 |
> |
properties of the ensemble of possibilities as a whole remain well |
275 |
> |
defined. For a classical system in thermal equilibrium with its |
276 |
> |
environment, the ensemble average of a mechanical quantity, $\langle |
277 |
> |
A(q , p) \rangle_t$, takes the form of an integral over the phase |
278 |
> |
space of the system, |
279 |
|
\begin{equation} |
280 |
|
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
281 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
282 |
< |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
282 |
> |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
283 |
|
\label{introEquation:ensembelAverage} |
284 |
|
\end{equation} |
285 |
|
|
286 |
|
There are several different types of ensembles with different |
287 |
|
statistical characteristics. As a function of macroscopic |
288 |
< |
parameters, such as temperature \textit{etc}, partition function can |
289 |
< |
be used to describe the statistical properties of a system in |
290 |
< |
thermodynamic equilibrium. |
291 |
< |
|
292 |
< |
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, |
288 |
> |
parameters, such as temperature \textit{etc}, the partition function |
289 |
> |
can be used to describe the statistical properties of a system in |
290 |
> |
thermodynamic equilibrium. As an ensemble of systems, each of which |
291 |
> |
is known to be thermally isolated and conserve energy, the |
292 |
> |
Microcanonical ensemble (NVE) has a partition function like, |
293 |
|
\begin{equation} |
294 |
< |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
294 |
> |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
295 |
|
\end{equation} |
296 |
< |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
296 |
> |
A canonical ensemble (NVT) is an ensemble of systems, each of which |
297 |
|
can share its energy with a large heat reservoir. The distribution |
298 |
|
of the total energy amongst the possible dynamical states is given |
299 |
|
by the partition function, |
300 |
|
\begin{equation} |
301 |
< |
\Omega (N,V,T) = e^{ - \beta A} |
301 |
> |
\Omega (N,V,T) = e^{ - \beta A}. |
302 |
|
\label{introEquation:NVTPartition} |
303 |
|
\end{equation} |
304 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
305 |
< |
TS$. Since most experiment are carried out under constant pressure |
306 |
< |
condition, isothermal-isobaric ensemble(NPT) play a very important |
307 |
< |
role in molecular simulation. The isothermal-isobaric ensemble allow |
308 |
< |
the system to exchange energy with a heat bath of temperature $T$ |
309 |
< |
and to change the volume as well. Its partition function is given as |
305 |
> |
TS$. Since most experiments are carried out under constant pressure |
306 |
> |
condition, the isothermal-isobaric ensemble (NPT) plays a very |
307 |
> |
important role in molecular simulations. The isothermal-isobaric |
308 |
> |
ensemble allow the system to exchange energy with a heat bath of |
309 |
> |
temperature $T$ and to change the volume as well. Its partition |
310 |
> |
function is given as |
311 |
|
\begin{equation} |
312 |
|
\Delta (N,P,T) = - e^{\beta G}. |
313 |
|
\label{introEquation:NPTPartition} |
316 |
|
|
317 |
|
\subsection{\label{introSection:liouville}Liouville's theorem} |
318 |
|
|
319 |
< |
The Liouville's theorem is the foundation on which statistical |
320 |
< |
mechanics rests. It describes the time evolution of phase space |
319 |
> |
Liouville's theorem is the foundation on which statistical mechanics |
320 |
> |
rests. It describes the time evolution of the phase space |
321 |
|
distribution function. In order to calculate the rate of change of |
322 |
< |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
323 |
< |
consider the two faces perpendicular to the $q_1$ axis, which are |
324 |
< |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
325 |
< |
leaving the opposite face is given by the expression, |
322 |
> |
$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider |
323 |
> |
the two faces perpendicular to the $q_1$ axis, which are located at |
324 |
> |
$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the |
325 |
> |
opposite face is given by the expression, |
326 |
|
\begin{equation} |
327 |
|
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
328 |
|
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
346 |
|
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
347 |
|
\end{equation} |
348 |
|
which cancels the first terms of the right hand side. Furthermore, |
349 |
< |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
349 |
> |
dividing $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
350 |
|
p_f $ in both sides, we can write out Liouville's theorem in a |
351 |
|
simple form, |
352 |
|
\begin{equation} |
355 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
356 |
|
\label{introEquation:liouvilleTheorem} |
357 |
|
\end{equation} |
381 |
– |
|
358 |
|
Liouville's theorem states that the distribution function is |
359 |
|
constant along any trajectory in phase space. In classical |
360 |
< |
statistical mechanics, since the number of particles in the system |
361 |
< |
is huge, we may be able to believe the system is stationary, |
360 |
> |
statistical mechanics, since the number of system copies in an |
361 |
> |
ensemble is huge and constant, we can assume the local density has |
362 |
> |
no reason (other than classical mechanics) to change, |
363 |
|
\begin{equation} |
364 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
365 |
|
\label{introEquation:stationary} |
372 |
|
\label{introEquation:densityAndHamiltonian} |
373 |
|
\end{equation} |
374 |
|
|
375 |
< |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
375 |
> |
\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}} |
376 |
|
Lets consider a region in the phase space, |
377 |
|
\begin{equation} |
378 |
|
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
379 |
|
\end{equation} |
380 |
|
If this region is small enough, the density $\rho$ can be regarded |
381 |
< |
as uniform over the whole phase space. Thus, the number of phase |
382 |
< |
points inside this region is given by, |
381 |
> |
as uniform over the whole integral. Thus, the number of phase points |
382 |
> |
inside this region is given by, |
383 |
|
\begin{equation} |
384 |
|
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
385 |
|
dp_1 } ..dp_f. |
389 |
|
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
390 |
|
\frac{d}{{dt}}(\delta v) = 0. |
391 |
|
\end{equation} |
392 |
< |
With the help of stationary assumption |
393 |
< |
(\ref{introEquation:stationary}), we obtain the principle of the |
394 |
< |
\emph{conservation of extension in phase space}, |
392 |
> |
With the help of the stationary assumption |
393 |
> |
(Eq.~\ref{introEquation:stationary}), we obtain the principle of |
394 |
> |
\emph{conservation of volume in phase space}, |
395 |
|
\begin{equation} |
396 |
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
397 |
|
...dq_f dp_1 } ..dp_f = 0. |
398 |
|
\label{introEquation:volumePreserving} |
399 |
|
\end{equation} |
400 |
|
|
401 |
< |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
401 |
> |
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
402 |
|
|
403 |
< |
Liouville's theorem can be expresses in a variety of different forms |
403 |
> |
Liouville's theorem can be expressed in a variety of different forms |
404 |
|
which are convenient within different contexts. For any two function |
405 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
406 |
|
bracket ${F, G}$ is defined as |
411 |
|
q_i }}} \right)}. |
412 |
|
\label{introEquation:poissonBracket} |
413 |
|
\end{equation} |
414 |
< |
Substituting equations of motion in Hamiltonian formalism( |
415 |
< |
\ref{introEquation:motionHamiltonianCoordinate} , |
416 |
< |
\ref{introEquation:motionHamiltonianMomentum} ) into |
417 |
< |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
418 |
< |
theorem using Poisson bracket notion, |
414 |
> |
Substituting equations of motion in Hamiltonian formalism |
415 |
> |
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
416 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into |
417 |
> |
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
418 |
> |
Liouville's theorem using Poisson bracket notion, |
419 |
|
\begin{equation} |
420 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
421 |
|
{\rho ,H} \right\}. |
434 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
435 |
|
\label{introEquation:liouvilleTheoremInOperator} |
436 |
|
\end{equation} |
437 |
< |
|
437 |
> |
which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$. |
438 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
439 |
|
|
440 |
|
Various thermodynamic properties can be calculated from Molecular |
441 |
|
Dynamics simulation. By comparing experimental values with the |
442 |
|
calculated properties, one can determine the accuracy of the |
443 |
< |
simulation and the quality of the underlying model. However, both of |
444 |
< |
experiment and computer simulation are usually performed during a |
443 |
> |
simulation and the quality of the underlying model. However, both |
444 |
> |
experiments and computer simulations are usually performed during a |
445 |
|
certain time interval and the measurements are averaged over a |
446 |
< |
period of them which is different from the average behavior of |
447 |
< |
many-body system in Statistical Mechanics. Fortunately, Ergodic |
448 |
< |
Hypothesis is proposed to make a connection between time average and |
449 |
< |
ensemble average. It states that time average and average over the |
450 |
< |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
446 |
> |
period of time which is different from the average behavior of |
447 |
> |
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
448 |
> |
Hypothesis makes a connection between time average and the ensemble |
449 |
> |
average. It states that the time average and average over the |
450 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
451 |
|
\begin{equation} |
452 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
453 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
456 |
|
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
457 |
|
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
458 |
|
distribution function. If an observation is averaged over a |
459 |
< |
sufficiently long time (longer than relaxation time), all accessible |
460 |
< |
microstates in phase space are assumed to be equally probed, giving |
461 |
< |
a properly weighted statistical average. This allows the researcher |
462 |
< |
freedom of choice when deciding how best to measure a given |
463 |
< |
observable. In case an ensemble averaged approach sounds most |
464 |
< |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
459 |
> |
sufficiently long time (longer than the relaxation time), all |
460 |
> |
accessible microstates in phase space are assumed to be equally |
461 |
> |
probed, giving a properly weighted statistical average. This allows |
462 |
> |
the researcher freedom of choice when deciding how best to measure a |
463 |
> |
given observable. In case an ensemble averaged approach sounds most |
464 |
> |
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
465 |
|
utilized. Or if the system lends itself to a time averaging |
466 |
|
approach, the Molecular Dynamics techniques in |
467 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
468 |
|
choice\cite{Frenkel1996}. |
469 |
|
|
470 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
471 |
< |
A variety of numerical integrators were proposed to simulate the |
472 |
< |
motions. They usually begin with an initial conditionals and move |
473 |
< |
the objects in the direction governed by the differential equations. |
474 |
< |
However, most of them ignore the hidden physical law contained |
475 |
< |
within the equations. Since 1990, geometric integrators, which |
476 |
< |
preserve various phase-flow invariants such as symplectic structure, |
477 |
< |
volume and time reversal symmetry, are developed to address this |
478 |
< |
issue. The velocity verlet method, which happens to be a simple |
479 |
< |
example of symplectic integrator, continues to gain its popularity |
480 |
< |
in molecular dynamics community. This fact can be partly explained |
481 |
< |
by its geometric nature. |
471 |
> |
A variety of numerical integrators have been proposed to simulate |
472 |
> |
the motions of atoms in MD simulation. They usually begin with |
473 |
> |
initial conditionals and move the objects in the direction governed |
474 |
> |
by the differential equations. However, most of them ignore the |
475 |
> |
hidden physical laws contained within the equations. Since 1990, |
476 |
> |
geometric integrators, which preserve various phase-flow invariants |
477 |
> |
such as symplectic structure, volume and time reversal symmetry, |
478 |
> |
were developed to address this issue\cite{Dullweber1997, |
479 |
> |
McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which |
480 |
> |
happens to be a simple example of symplectic integrator, continues |
481 |
> |
to gain popularity in the molecular dynamics community. This fact |
482 |
> |
can be partly explained by its geometric nature. |
483 |
|
|
484 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
485 |
< |
A \emph{manifold} is an abstract mathematical space. It locally |
486 |
< |
looks like Euclidean space, but when viewed globally, it may have |
487 |
< |
more complicate structure. A good example of manifold is the surface |
488 |
< |
of Earth. It seems to be flat locally, but it is round if viewed as |
489 |
< |
a whole. A \emph{differentiable manifold} (also known as |
490 |
< |
\emph{smooth manifold}) is a manifold with an open cover in which |
491 |
< |
the covering neighborhoods are all smoothly isomorphic to one |
492 |
< |
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 |
484 |
> |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
485 |
> |
A \emph{manifold} is an abstract mathematical space. It looks |
486 |
> |
locally like Euclidean space, but when viewed globally, it may have |
487 |
> |
more complicated structure. A good example of manifold is the |
488 |
> |
surface of Earth. It seems to be flat locally, but it is round if |
489 |
> |
viewed as a whole. A \emph{differentiable manifold} (also known as |
490 |
> |
\emph{smooth manifold}) is a manifold on which it is possible to |
491 |
> |
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
492 |
> |
defined as a pair $(M, \omega)$ which consists of a |
493 |
|
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
494 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
495 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
496 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
497 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
498 |
< |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
499 |
< |
example of symplectic form. |
498 |
> |
$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in |
499 |
> |
vector field is an example of symplectic form. One of the |
500 |
> |
motivations to study \emph{symplectic manifolds} in Hamiltonian |
501 |
> |
Mechanics is that a symplectic manifold can represent all possible |
502 |
> |
configurations of the system and the phase space of the system can |
503 |
> |
be described by it's cotangent bundle\cite{Jost2002}. Every |
504 |
> |
symplectic manifold is even dimensional. For instance, in Hamilton |
505 |
> |
equations, coordinate and momentum always appear in pairs. |
506 |
|
|
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 |
– |
|
507 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
508 |
|
|
509 |
< |
For a ordinary differential system defined as |
509 |
> |
For an ordinary differential system defined as |
510 |
|
\begin{equation} |
511 |
|
\dot x = f(x) |
512 |
|
\end{equation} |
513 |
< |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
513 |
> |
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
514 |
> |
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian |
515 |
> |
function and $J$ is the skew-symmetric matrix |
516 |
|
\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} |
517 |
|
J = \left( {\begin{array}{*{20}c} |
518 |
|
0 & I \\ |
519 |
|
{ - I} & 0 \\ |
523 |
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
524 |
|
system can be rewritten as, |
525 |
|
\begin{equation} |
526 |
< |
\frac{d}{{dt}}x = J\nabla _x H(x) |
526 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x). |
527 |
|
\label{introEquation:compactHamiltonian} |
528 |
|
\end{equation}In this case, $f$ is |
529 |
< |
called a \emph{Hamiltonian vector field}. |
530 |
< |
|
568 |
< |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
529 |
> |
called a \emph{Hamiltonian vector field}. Another generalization of |
530 |
> |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
531 |
|
\begin{equation} |
532 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
533 |
|
\end{equation} |
534 |
|
The most obvious change being that matrix $J$ now depends on $x$. |
535 |
|
|
536 |
< |
\subsection{\label{introSection:exactFlow}Exact Flow} |
536 |
> |
\subsection{\label{introSection:exactFlow}Exact Propagator} |
537 |
|
|
538 |
< |
Let $x(t)$ be the exact solution of the ODE system, |
539 |
< |
\begin{equation} |
540 |
< |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
541 |
< |
\end{equation} |
542 |
< |
The exact flow(solution) $\varphi_\tau$ is defined by |
581 |
< |
\[ |
582 |
< |
x(t+\tau) =\varphi_\tau(x(t)) |
538 |
> |
Let $x(t)$ be the exact solution of the ODE |
539 |
> |
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
540 |
> |
define its exact propagator(solution) $\varphi_\tau$ |
541 |
> |
\[ x(t+\tau) |
542 |
> |
=\varphi_\tau(x(t)) |
543 |
|
\] |
544 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
545 |
< |
space to itself. The flow has the continuous group property, |
545 |
> |
space to itself. The propagator has the continuous group property, |
546 |
|
\begin{equation} |
547 |
|
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
548 |
|
+ \tau _2 } . |
551 |
|
\begin{equation} |
552 |
|
\varphi _\tau \circ \varphi _{ - \tau } = I |
553 |
|
\end{equation} |
554 |
< |
Therefore, the exact flow is self-adjoint, |
554 |
> |
Therefore, the exact propagator is self-adjoint, |
555 |
|
\begin{equation} |
556 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
557 |
|
\end{equation} |
558 |
< |
The exact flow can also be written in terms of the of an operator, |
558 |
> |
The exact propagator can also be written in terms of operator, |
559 |
|
\begin{equation} |
560 |
|
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
561 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
562 |
|
\label{introEquation:exponentialOperator} |
563 |
|
\end{equation} |
564 |
< |
|
565 |
< |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
566 |
< |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
567 |
< |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
568 |
< |
the Taylor series of $\psi_\tau$ agree to order $p$, |
564 |
> |
In most cases, it is not easy to find the exact propagator |
565 |
> |
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
566 |
> |
which is usually called an integrator. The order of an integrator |
567 |
> |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
568 |
> |
order $p$, |
569 |
|
\begin{equation} |
570 |
< |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
570 |
> |
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
571 |
|
\end{equation} |
572 |
|
|
573 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
574 |
|
|
575 |
< |
The hidden geometric properties of ODE and its flow play important |
576 |
< |
roles in numerical studies. Many of them can be found in systems |
577 |
< |
which occur naturally in applications. |
578 |
< |
|
579 |
< |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
620 |
< |
a \emph{symplectic} flow if it satisfies, |
575 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
576 |
> |
ODE and its propagator play important roles in numerical studies. |
577 |
> |
Many of them can be found in systems which occur naturally in |
578 |
> |
applications. Let $\varphi$ be the propagator of Hamiltonian vector |
579 |
> |
field, $\varphi$ is a \emph{symplectic} propagator if it satisfies, |
580 |
|
\begin{equation} |
581 |
|
{\varphi '}^T J \varphi ' = J. |
582 |
|
\end{equation} |
583 |
|
According to Liouville's theorem, the symplectic volume is invariant |
584 |
< |
under a Hamiltonian flow, which is the basis for classical |
585 |
< |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
586 |
< |
field on a symplectic manifold can be shown to be a |
584 |
> |
under a Hamiltonian propagator, which is the basis for classical |
585 |
> |
statistical mechanics. Furthermore, the propagator of a Hamiltonian |
586 |
> |
vector field on a symplectic manifold can be shown to be a |
587 |
|
symplectomorphism. As to the Poisson system, |
588 |
|
\begin{equation} |
589 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
590 |
|
\end{equation} |
591 |
< |
is the property must be preserved by the integrator. |
592 |
< |
|
593 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
594 |
< |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
595 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
596 |
< |
be volume-preserving. |
597 |
< |
|
639 |
< |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
640 |
< |
will result in a new system, |
591 |
> |
is the property that must be preserved by the integrator. It is |
592 |
> |
possible to construct a \emph{volume-preserving} propagator for a |
593 |
> |
source free ODE ($ \nabla \cdot f = 0 $), if the propagator |
594 |
> |
satisfies $ \det d\varphi = 1$. One can show easily that a |
595 |
> |
symplectic propagator will be volume-preserving. Changing the |
596 |
> |
variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will |
597 |
> |
result in a new system, |
598 |
|
\[ |
599 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
600 |
|
\] |
601 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
602 |
< |
In other words, the flow of this vector field is reversible if and |
603 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
604 |
< |
|
605 |
< |
A \emph{first integral}, or conserved quantity of a general |
606 |
< |
differential function is a function $ G:R^{2d} \to R^d $ which is |
650 |
< |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
602 |
> |
In other words, the propagator of this vector field is reversible if |
603 |
> |
and only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
604 |
> |
conserved quantity of a general differential function is a function |
605 |
> |
$ G:R^{2d} \to R^d $ which is constant for all solutions of the ODE |
606 |
> |
$\frac{{dx}}{{dt}} = f(x)$ , |
607 |
|
\[ |
608 |
|
\frac{{dG(x(t))}}{{dt}} = 0. |
609 |
|
\] |
610 |
< |
Using chain rule, one may obtain, |
610 |
> |
Using the chain rule, one may obtain, |
611 |
|
\[ |
612 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
612 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
613 |
|
\] |
614 |
< |
which is the condition for conserving \emph{first integral}. For a |
615 |
< |
canonical Hamiltonian system, the time evolution of an arbitrary |
616 |
< |
smooth function $G$ is given by, |
617 |
< |
\begin{equation} |
618 |
< |
\begin{array}{c} |
619 |
< |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
664 |
< |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
665 |
< |
\end{array} |
614 |
> |
which is the condition for conserved quantities. For a canonical |
615 |
> |
Hamiltonian system, the time evolution of an arbitrary smooth |
616 |
> |
function $G$ is given by, |
617 |
> |
\begin{eqnarray} |
618 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
619 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
620 |
|
\label{introEquation:firstIntegral1} |
621 |
< |
\end{equation} |
622 |
< |
Using poisson bracket notion, Equation |
623 |
< |
\ref{introEquation:firstIntegral1} can be rewritten as |
621 |
> |
\end{eqnarray} |
622 |
> |
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1} |
623 |
> |
can be rewritten as |
624 |
|
\[ |
625 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
626 |
|
\] |
627 |
< |
Therefore, the sufficient condition for $G$ to be the \emph{first |
628 |
< |
integral} of a Hamiltonian system is |
629 |
< |
\[ |
630 |
< |
\left\{ {G,H} \right\} = 0. |
631 |
< |
\] |
632 |
< |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
633 |
< |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
680 |
< |
0$. |
627 |
> |
Therefore, the sufficient condition for $G$ to be a conserved |
628 |
> |
quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As |
629 |
> |
is well known, the Hamiltonian (or energy) H of a Hamiltonian system |
630 |
> |
is a conserved quantity, which is due to the fact $\{ H,H\} = 0$. |
631 |
> |
When designing any numerical methods, one should always try to |
632 |
> |
preserve the structural properties of the original ODE and its |
633 |
> |
propagator. |
634 |
|
|
682 |
– |
|
683 |
– |
When designing any numerical methods, one should always try to |
684 |
– |
preserve the structural properties of the original ODE and its flow. |
685 |
– |
|
635 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
636 |
|
A lot of well established and very effective numerical methods have |
637 |
< |
been successful precisely because of their symplecticities even |
637 |
> |
been successful precisely because of their symplectic nature even |
638 |
|
though this fact was not recognized when they were first |
639 |
< |
constructed. The most famous example is leapfrog methods in |
640 |
< |
molecular dynamics. In general, symplectic integrators can be |
639 |
> |
constructed. The most famous example is the Verlet-leapfrog method |
640 |
> |
in molecular dynamics. In general, symplectic integrators can be |
641 |
|
constructed using one of four different methods. |
642 |
|
\begin{enumerate} |
643 |
|
\item Generating functions |
645 |
|
\item Runge-Kutta methods |
646 |
|
\item Splitting methods |
647 |
|
\end{enumerate} |
648 |
+ |
Generating functions\cite{Channell1990} tend to lead to methods |
649 |
+ |
which are cumbersome and difficult to use. In dissipative systems, |
650 |
+ |
variational methods can capture the decay of energy |
651 |
+ |
accurately\cite{Kane2000}. Since they are geometrically unstable |
652 |
+ |
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
653 |
+ |
methods are not suitable for Hamiltonian system. Recently, various |
654 |
+ |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
655 |
+ |
have been developed to overcome this instability. However, due to |
656 |
+ |
computational penalty involved in implementing the Runge-Kutta |
657 |
+ |
methods, they have not attracted much attention from the Molecular |
658 |
+ |
Dynamics community. Instead, splitting methods have been widely |
659 |
+ |
accepted since they exploit natural decompositions of the |
660 |
+ |
system\cite{Tuckerman1992, McLachlan1998}. |
661 |
|
|
662 |
< |
Generating function tends to lead to methods which are cumbersome |
701 |
< |
and difficult to use. In dissipative systems, variational methods |
702 |
< |
can capture the decay of energy accurately. Since their |
703 |
< |
geometrically unstable nature against non-Hamiltonian perturbations, |
704 |
< |
ordinary implicit Runge-Kutta methods are not suitable for |
705 |
< |
Hamiltonian system. Recently, various high-order explicit |
706 |
< |
Runge--Kutta methods have been developed to overcome this |
707 |
< |
instability. However, due to computational penalty involved in |
708 |
< |
implementing the Runge-Kutta methods, they do not attract too much |
709 |
< |
attention from Molecular Dynamics community. Instead, splitting have |
710 |
< |
been widely accepted since they exploit natural decompositions of |
711 |
< |
the system\cite{Tuckerman92}. |
662 |
> |
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
663 |
|
|
713 |
– |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
714 |
– |
|
664 |
|
The main idea behind splitting methods is to decompose the discrete |
665 |
< |
$\varphi_h$ as a composition of simpler flows, |
665 |
> |
$\varphi_h$ as a composition of simpler propagators, |
666 |
|
\begin{equation} |
667 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
668 |
|
\varphi _{h_n } |
669 |
|
\label{introEquation:FlowDecomposition} |
670 |
|
\end{equation} |
671 |
< |
where each of the sub-flow is chosen such that each represent a |
672 |
< |
simpler integration of the system. |
673 |
< |
|
725 |
< |
Suppose that a Hamiltonian system takes the form, |
671 |
> |
where each of the sub-propagator is chosen such that each represent |
672 |
> |
a simpler integration of the system. Suppose that a Hamiltonian |
673 |
> |
system takes the form, |
674 |
|
\[ |
675 |
|
H = H_1 + H_2. |
676 |
|
\] |
677 |
|
Here, $H_1$ and $H_2$ may represent different physical processes of |
678 |
|
the system. For instance, they may relate to kinetic and potential |
679 |
|
energy respectively, which is a natural decomposition of the |
680 |
< |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
681 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
682 |
< |
order is then given by the Lie-Trotter formula |
680 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact |
681 |
> |
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
682 |
> |
simple first order expression is then given by the Lie-Trotter |
683 |
> |
formula |
684 |
|
\begin{equation} |
685 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
686 |
|
\label{introEquation:firstOrderSplitting} |
689 |
|
continuous $\varphi _i$ over a time $h$. By definition, as |
690 |
|
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
691 |
|
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
692 |
< |
It is easy to show that any composition of symplectic flows yields a |
693 |
< |
symplectic map, |
692 |
> |
It is easy to show that any composition of symplectic propagators |
693 |
> |
yields a symplectic map, |
694 |
|
\begin{equation} |
695 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
696 |
|
'\phi ' = \phi '^T J\phi ' = J, |
698 |
|
\end{equation} |
699 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
700 |
|
splitting in this context automatically generates a symplectic map. |
701 |
< |
|
702 |
< |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
703 |
< |
introduces local errors proportional to $h^2$, while Strang |
704 |
< |
splitting gives a second-order decomposition, |
701 |
> |
The Lie-Trotter |
702 |
> |
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
703 |
> |
local errors proportional to $h^2$, while the Strang splitting gives |
704 |
> |
a second-order decomposition, |
705 |
|
\begin{equation} |
706 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
707 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
708 |
|
\end{equation} |
709 |
< |
which has a local error proportional to $h^3$. Sprang splitting's |
710 |
< |
popularity in molecular simulation community attribute to its |
711 |
< |
symmetric property, |
709 |
> |
which has a local error proportional to $h^3$. The Strang |
710 |
> |
splitting's popularity in molecular simulation community attribute |
711 |
> |
to its symmetric property, |
712 |
|
\begin{equation} |
713 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
714 |
|
\label{introEquation:timeReversible} |
715 |
|
\end{equation} |
716 |
|
|
717 |
< |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
717 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
718 |
|
The classical equation for a system consisting of interacting |
719 |
|
particles can be written in Hamiltonian form, |
720 |
|
\[ |
721 |
|
H = T + V |
722 |
|
\] |
723 |
|
where $T$ is the kinetic energy and $V$ is the potential energy. |
724 |
< |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
724 |
> |
Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one |
725 |
|
obtains the following: |
726 |
|
\begin{align} |
727 |
|
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
748 |
|
\label{introEquation:Lp9b}\\% |
749 |
|
% |
750 |
|
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
751 |
< |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
751 |
> |
\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c} |
752 |
|
\end{align} |
753 |
|
From the preceding splitting, one can see that the integration of |
754 |
|
the equations of motion would follow: |
757 |
|
|
758 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
759 |
|
|
760 |
< |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
760 |
> |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
761 |
|
|
762 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
763 |
|
\end{enumerate} |
764 |
< |
|
765 |
< |
Simply switching the order of splitting and composing, a new |
766 |
< |
integrator, the \emph{position verlet} integrator, can be generated, |
764 |
> |
By simply switching the order of the propagators in the splitting |
765 |
> |
and composing a new integrator, the \emph{position verlet} |
766 |
> |
integrator, can be generated, |
767 |
|
\begin{align} |
768 |
|
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
769 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
774 |
|
\label{introEquation:positionVerlet2} |
775 |
|
\end{align} |
776 |
|
|
777 |
< |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
777 |
> |
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
778 |
|
|
779 |
< |
Baker-Campbell-Hausdorff formula can be used to determine the local |
780 |
< |
error of splitting method in terms of commutator of the |
779 |
> |
The Baker-Campbell-Hausdorff formula can be used to determine the |
780 |
> |
local error of a splitting method in terms of the commutator of the |
781 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
782 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
783 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
782 |
> |
the sub-propagator. For operators $hX$ and $hY$ which are associated |
783 |
> |
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
784 |
|
\begin{equation} |
785 |
|
\exp (hX + hY) = \exp (hZ) |
786 |
|
\end{equation} |
789 |
|
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
790 |
|
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
791 |
|
\end{equation} |
792 |
< |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
792 |
> |
Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by |
793 |
|
\[ |
794 |
|
[X,Y] = XY - YX . |
795 |
|
\] |
796 |
< |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
797 |
< |
can obtain |
796 |
> |
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
797 |
> |
to the Strang splitting, we can obtain |
798 |
> |
\begin{eqnarray*} |
799 |
> |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
800 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
801 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots |
802 |
> |
). |
803 |
> |
\end{eqnarray*} |
804 |
> |
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
805 |
> |
error of Strang splitting is proportional to $h^3$. The same |
806 |
> |
procedure can be applied to a general splitting of the form |
807 |
|
\begin{equation} |
850 |
– |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
851 |
– |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
852 |
– |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
853 |
– |
\ldots ) |
854 |
– |
\end{equation} |
855 |
– |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
856 |
– |
error of Spring splitting is proportional to $h^3$. The same |
857 |
– |
procedure can be applied to general splitting, of the form |
858 |
– |
\begin{equation} |
808 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
809 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
810 |
|
\end{equation} |
811 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
812 |
< |
order method. Yoshida proposed an elegant way to compose higher |
813 |
< |
order methods based on symmetric splitting. Given a symmetric second |
814 |
< |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
815 |
< |
method can be constructed by composing, |
811 |
> |
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
812 |
> |
order methods. Yoshida proposed an elegant way to compose higher |
813 |
> |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
814 |
> |
a symmetric second order base method $ \varphi _h^{(2)} $, a |
815 |
> |
fourth-order symmetric method can be constructed by composing, |
816 |
|
\[ |
817 |
|
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
818 |
|
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
822 |
|
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
823 |
|
\begin{equation} |
824 |
|
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
825 |
< |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
825 |
> |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)}, |
826 |
|
\end{equation} |
827 |
< |
, if the weights are chosen as |
827 |
> |
if the weights are chosen as |
828 |
|
\[ |
829 |
|
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
830 |
|
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
838 |
|
dynamical information. The basic idea of molecular dynamics is that |
839 |
|
macroscopic properties are related to microscopic behavior and |
840 |
|
microscopic behavior can be calculated from the trajectories in |
841 |
< |
simulations. For instance, instantaneous temperature of an |
842 |
< |
Hamiltonian system of $N$ particle can be measured by |
841 |
> |
simulations. For instance, instantaneous temperature of a |
842 |
> |
Hamiltonian system of $N$ particles can be measured by |
843 |
|
\[ |
844 |
|
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
845 |
|
\] |
846 |
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
847 |
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
848 |
< |
the boltzman constant. |
848 |
> |
the Boltzman constant. |
849 |
|
|
850 |
|
A typical molecular dynamics run consists of three essential steps: |
851 |
|
\begin{enumerate} |
861 |
|
\end{enumerate} |
862 |
|
These three individual steps will be covered in the following |
863 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
864 |
< |
initialization of a simulation. Sec.~\ref{introSec:production} will |
865 |
< |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
866 |
< |
provides the theoretical tools for trajectory analysis. |
864 |
> |
initialization of a simulation. Sec.~\ref{introSection:production} |
865 |
> |
will discuss issues of production runs. |
866 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
867 |
> |
analysis of trajectories. |
868 |
|
|
869 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
870 |
|
|
871 |
< |
\subsubsection{Preliminary preparation} |
871 |
> |
\subsubsection{\textbf{Preliminary preparation}} |
872 |
|
|
873 |
|
When selecting the starting structure of a molecule for molecular |
874 |
|
simulation, one may retrieve its Cartesian coordinates from public |
875 |
|
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
876 |
|
thousands of crystal structures of molecules are discovered every |
877 |
|
year, many more remain unknown due to the difficulties of |
878 |
< |
purification and crystallization. Even for the molecule with known |
879 |
< |
structure, some important information is missing. For example, the |
878 |
> |
purification and crystallization. Even for molecules with known |
879 |
> |
structures, some important information is missing. For example, a |
880 |
|
missing hydrogen atom which acts as donor in hydrogen bonding must |
881 |
< |
be added. Moreover, in order to include electrostatic interaction, |
881 |
> |
be added. Moreover, in order to include electrostatic interactions, |
882 |
|
one may need to specify the partial charges for individual atoms. |
883 |
|
Under some circumstances, we may even need to prepare the system in |
884 |
< |
a special setup. For instance, when studying transport phenomenon in |
885 |
< |
membrane system, we may prepare the lipids in bilayer structure |
886 |
< |
instead of placing lipids randomly in solvent, since we are not |
887 |
< |
interested in self-aggregation and it takes a long time to happen. |
884 |
> |
a special configuration. For instance, when studying transport |
885 |
> |
phenomenon in membrane systems, we may prepare the lipids in a |
886 |
> |
bilayer structure instead of placing lipids randomly in solvent, |
887 |
> |
since we are not interested in the slow self-aggregation process. |
888 |
|
|
889 |
< |
\subsubsection{Minimization} |
889 |
> |
\subsubsection{\textbf{Minimization}} |
890 |
|
|
891 |
|
It is quite possible that some of molecules in the system from |
892 |
< |
preliminary preparation may be overlapped with each other. This |
893 |
< |
close proximity leads to high potential energy which consequently |
894 |
< |
jeopardizes any molecular dynamics simulations. To remove these |
895 |
< |
steric overlaps, one typically performs energy minimization to find |
896 |
< |
a more reasonable conformation. Several energy minimization methods |
897 |
< |
have been developed to exploit the energy surface and to locate the |
898 |
< |
local minimum. While converging slowly near the minimum, steepest |
899 |
< |
descent method is extremely robust when systems are far from |
900 |
< |
harmonic. Thus, it is often used to refine structure from |
901 |
< |
crystallographic data. Relied on the gradient or hessian, advanced |
902 |
< |
methods like conjugate gradient and Newton-Raphson converge rapidly |
903 |
< |
to a local minimum, while become unstable if the energy surface is |
904 |
< |
far from quadratic. Another factor must be taken into account, when |
892 |
> |
preliminary preparation may be overlapping with each other. This |
893 |
> |
close proximity leads to high initial potential energy which |
894 |
> |
consequently jeopardizes any molecular dynamics simulations. To |
895 |
> |
remove these steric overlaps, one typically performs energy |
896 |
> |
minimization to find a more reasonable conformation. Several energy |
897 |
> |
minimization methods have been developed to exploit the energy |
898 |
> |
surface and to locate the local minimum. While converging slowly |
899 |
> |
near the minimum, steepest descent method is extremely robust when |
900 |
> |
systems are strongly anharmonic. Thus, it is often used to refine |
901 |
> |
structures from crystallographic data. Relying on the Hessian, |
902 |
> |
advanced methods like Newton-Raphson converge rapidly to a local |
903 |
> |
minimum, but become unstable if the energy surface is far from |
904 |
> |
quadratic. Another factor that must be taken into account, when |
905 |
|
choosing energy minimization method, is the size of the system. |
906 |
|
Steepest descent and conjugate gradient can deal with models of any |
907 |
< |
size. Because of the limit of computation power to calculate hessian |
908 |
< |
matrix and insufficient storage capacity to store them, most |
909 |
< |
Newton-Raphson methods can not be used with very large models. |
907 |
> |
size. Because of the limits on computer memory to store the hessian |
908 |
> |
matrix and the computing power needed to diagonalize these matrices, |
909 |
> |
most Newton-Raphson methods can not be used with very large systems. |
910 |
|
|
911 |
< |
\subsubsection{Heating} |
911 |
> |
\subsubsection{\textbf{Heating}} |
912 |
|
|
913 |
< |
Typically, Heating is performed by assigning random velocities |
914 |
< |
according to a Gaussian distribution for a temperature. Beginning at |
915 |
< |
a lower temperature and gradually increasing the temperature by |
916 |
< |
assigning greater random velocities, we end up with setting the |
917 |
< |
temperature of the system to a final temperature at which the |
918 |
< |
simulation will be conducted. In heating phase, we should also keep |
919 |
< |
the system from drifting or rotating as a whole. Equivalently, the |
920 |
< |
net linear momentum and angular momentum of the system should be |
921 |
< |
shifted to zero. |
913 |
> |
Typically, heating is performed by assigning random velocities |
914 |
> |
according to a Maxwell-Boltzman distribution for a desired |
915 |
> |
temperature. Beginning at a lower temperature and gradually |
916 |
> |
increasing the temperature by assigning larger random velocities, we |
917 |
> |
end up setting the temperature of the system to a final temperature |
918 |
> |
at which the simulation will be conducted. In heating phase, we |
919 |
> |
should also keep the system from drifting or rotating as a whole. To |
920 |
> |
do this, the net linear momentum and angular momentum of the system |
921 |
> |
is shifted to zero after each resampling from the Maxwell -Boltzman |
922 |
> |
distribution. |
923 |
|
|
924 |
< |
\subsubsection{Equilibration} |
924 |
> |
\subsubsection{\textbf{Equilibration}} |
925 |
|
|
926 |
|
The purpose of equilibration is to allow the system to evolve |
927 |
|
spontaneously for a period of time and reach equilibrium. The |
935 |
|
|
936 |
|
\subsection{\label{introSection:production}Production} |
937 |
|
|
938 |
< |
Production run is the most important steps of the simulation, in |
938 |
> |
The production run is the most important step of the simulation, in |
939 |
|
which the equilibrated structure is used as a starting point and the |
940 |
|
motions of the molecules are collected for later analysis. In order |
941 |
|
to capture the macroscopic properties of the system, the molecular |
942 |
< |
dynamics simulation must be performed in correct and efficient way. |
942 |
> |
dynamics simulation must be performed by sampling correctly and |
943 |
> |
efficiently from the relevant thermodynamic ensemble. |
944 |
|
|
945 |
|
The most expensive part of a molecular dynamics simulation is the |
946 |
|
calculation of non-bonded forces, such as van der Waals force and |
947 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
948 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
949 |
< |
which making large simulations prohibitive in the absence of any |
950 |
< |
computation saving techniques. |
949 |
> |
which makes large simulations prohibitive in the absence of any |
950 |
> |
algorithmic tricks. A natural approach to avoid system size issues |
951 |
> |
is to represent the bulk behavior by a finite number of the |
952 |
> |
particles. However, this approach will suffer from surface effects |
953 |
> |
at the edges of the simulation. To offset this, \textit{Periodic |
954 |
> |
boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to |
955 |
> |
simulate bulk properties with a relatively small number of |
956 |
> |
particles. In this method, the simulation box is replicated |
957 |
> |
throughout space to form an infinite lattice. During the simulation, |
958 |
> |
when a particle moves in the primary cell, its image in other cells |
959 |
> |
move in exactly the same direction with exactly the same |
960 |
> |
orientation. Thus, as a particle leaves the primary cell, one of its |
961 |
> |
images will enter through the opposite face. |
962 |
> |
\begin{figure} |
963 |
> |
\centering |
964 |
> |
\includegraphics[width=\linewidth]{pbc.eps} |
965 |
> |
\caption[An illustration of periodic boundary conditions]{A 2-D |
966 |
> |
illustration of periodic boundary conditions. As one particle leaves |
967 |
> |
the left of the simulation box, an image of it enters the right.} |
968 |
> |
\label{introFig:pbc} |
969 |
> |
\end{figure} |
970 |
|
|
1000 |
– |
A natural approach to avoid system size issue is to represent the |
1001 |
– |
bulk behavior by a finite number of the particles. However, this |
1002 |
– |
approach will suffer from the surface effect. To offset this, |
1003 |
– |
\textit{Periodic boundary condition} is developed to simulate bulk |
1004 |
– |
properties with a relatively small number of particles. In this |
1005 |
– |
method, the simulation box is replicated throughout space to form an |
1006 |
– |
infinite lattice. During the simulation, when a particle moves in |
1007 |
– |
the primary cell, its image in other cells move in exactly the same |
1008 |
– |
direction with exactly the same orientation. Thus, as a particle |
1009 |
– |
leaves the primary cell, one of its images will enter through the |
1010 |
– |
opposite face. |
1011 |
– |
%\begin{figure} |
1012 |
– |
%\centering |
1013 |
– |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1014 |
– |
%\caption[An illustration of periodic boundary conditions]{A 2-D |
1015 |
– |
%illustration of periodic boundary conditions. As one particle leaves |
1016 |
– |
%the right of the simulation box, an image of it enters the left.} |
1017 |
– |
%\label{introFig:pbc} |
1018 |
– |
%\end{figure} |
1019 |
– |
|
971 |
|
%cutoff and minimum image convention |
972 |
|
Another important technique to improve the efficiency of force |
973 |
< |
evaluation is to apply cutoff where particles farther than a |
974 |
< |
predetermined distance, are not included in the calculation |
973 |
> |
evaluation is to apply spherical cutoffs where particles farther |
974 |
> |
than a predetermined distance are not included in the calculation |
975 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
976 |
< |
discontinuity in the potential energy curve |
977 |
< |
(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the |
978 |
< |
potential to ensure the potential curve go smoothly to zero at the |
979 |
< |
cutoff radius. Cutoff strategy works pretty well for Lennard-Jones |
980 |
< |
interaction because of its short range nature. However, simply |
981 |
< |
truncating the electrostatic interaction with the use of cutoff has |
982 |
< |
been shown to lead to severe artifacts in simulations. Ewald |
983 |
< |
summation, in which the slowly conditionally convergent Coulomb |
984 |
< |
potential is transformed into direct and reciprocal sums with rapid |
985 |
< |
and absolute convergence, has proved to minimize the periodicity |
986 |
< |
artifacts in liquid simulations. Taking the advantages of the fast |
987 |
< |
Fourier transform (FFT) for calculating discrete Fourier transforms, |
988 |
< |
the particle mesh-based methods are accelerated from $O(N^{3/2})$ to |
989 |
< |
$O(N logN)$. An alternative approach is \emph{fast multipole |
990 |
< |
method}, which treats Coulombic interaction exactly at short range, |
991 |
< |
and approximate the potential at long range through multipolar |
992 |
< |
expansion. In spite of their wide acceptances at the molecular |
993 |
< |
simulation community, these two methods are hard to be implemented |
976 |
> |
discontinuity in the potential energy curve. Fortunately, one can |
977 |
> |
shift a simple radial potential to ensure the potential curve go |
978 |
> |
smoothly to zero at the cutoff radius. The cutoff strategy works |
979 |
> |
well for Lennard-Jones interaction because of its short range |
980 |
> |
nature. However, simply truncating the electrostatic interaction |
981 |
> |
with the use of cutoffs has been shown to lead to severe artifacts |
982 |
> |
in simulations. The Ewald summation, in which the slowly decaying |
983 |
> |
Coulomb potential is transformed into direct and reciprocal sums |
984 |
> |
with rapid and absolute convergence, has proved to minimize the |
985 |
> |
periodicity artifacts in liquid simulations. Taking the advantages |
986 |
> |
of the fast Fourier transform (FFT) for calculating discrete Fourier |
987 |
> |
transforms, the particle mesh-based |
988 |
> |
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
989 |
> |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
990 |
> |
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
991 |
> |
which treats Coulombic interactions exactly at short range, and |
992 |
> |
approximate the potential at long range through multipolar |
993 |
> |
expansion. In spite of their wide acceptance at the molecular |
994 |
> |
simulation community, these two methods are difficult to implement |
995 |
|
correctly and efficiently. Instead, we use a damped and |
996 |
|
charge-neutralized Coulomb potential method developed by Wolf and |
997 |
< |
his coworkers. The shifted Coulomb potential for particle $i$ and |
998 |
< |
particle $j$ at distance $r_{rj}$ is given by: |
997 |
> |
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
998 |
> |
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
999 |
|
\begin{equation} |
1000 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1001 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1005 |
|
where $\alpha$ is the convergence parameter. Due to the lack of |
1006 |
|
inherent periodicity and rapid convergence,this method is extremely |
1007 |
|
efficient and easy to implement. |
1008 |
< |
%\begin{figure} |
1009 |
< |
%\centering |
1010 |
< |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1011 |
< |
%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} |
1012 |
< |
%\label{introFigure:shiftedCoulomb} |
1013 |
< |
%\end{figure} |
1008 |
> |
\begin{figure} |
1009 |
> |
\centering |
1010 |
> |
\includegraphics[width=\linewidth]{shifted_coulomb.eps} |
1011 |
> |
\caption[An illustration of shifted Coulomb potential]{An |
1012 |
> |
illustration of shifted Coulomb potential.} |
1013 |
> |
\label{introFigure:shiftedCoulomb} |
1014 |
> |
\end{figure} |
1015 |
|
|
1016 |
|
%multiple time step |
1017 |
|
|
1018 |
|
\subsection{\label{introSection:Analysis} Analysis} |
1019 |
|
|
1020 |
< |
Recently, advanced visualization technique are widely applied to |
1020 |
> |
Recently, advanced visualization technique have become applied to |
1021 |
|
monitor the motions of molecules. Although the dynamics of the |
1022 |
|
system can be described qualitatively from animation, quantitative |
1023 |
< |
trajectory analysis are more appreciable. According to the |
1024 |
< |
principles of Statistical Mechanics, |
1023 |
> |
trajectory analysis is more useful. According to the principles of |
1024 |
> |
Statistical Mechanics in |
1025 |
|
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1026 |
< |
thermodynamics properties, analyze fluctuations of structural |
1026 |
> |
thermodynamic properties, analyze fluctuations of structural |
1027 |
|
parameters, and investigate time-dependent processes of the molecule |
1028 |
|
from the trajectories. |
1029 |
|
|
1030 |
< |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
1030 |
> |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
1031 |
|
|
1032 |
< |
Thermodynamics properties, which can be expressed in terms of some |
1032 |
> |
Thermodynamic properties, which can be expressed in terms of some |
1033 |
|
function of the coordinates and momenta of all particles in the |
1034 |
|
system, can be directly computed from molecular dynamics. The usual |
1035 |
|
way to measure the pressure is based on virial theorem of Clausius |
1049 |
|
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1050 |
|
\end{equation} |
1051 |
|
|
1052 |
< |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
1052 |
> |
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
1053 |
|
|
1054 |
|
Structural Properties of a simple fluid can be described by a set of |
1055 |
< |
distribution functions. Among these functions,\emph{pair |
1055 |
> |
distribution functions. Among these functions,the \emph{pair |
1056 |
|
distribution function}, also known as \emph{radial distribution |
1057 |
< |
function}, is of most fundamental importance to liquid-state theory. |
1058 |
< |
Pair distribution function can be gathered by Fourier transforming |
1059 |
< |
raw data from a series of neutron diffraction experiments and |
1060 |
< |
integrating over the surface factor \cite{Powles73}. The experiment |
1061 |
< |
result can serve as a criterion to justify the correctness of the |
1062 |
< |
theory. Moreover, various equilibrium thermodynamic and structural |
1063 |
< |
properties can also be expressed in terms of radial distribution |
1064 |
< |
function \cite{allen87:csl}. |
1065 |
< |
|
1066 |
< |
A pair distribution functions $g(r)$ gives the probability that a |
1067 |
< |
particle $i$ will be located at a distance $r$ from a another |
1068 |
< |
particle $j$ in the system |
1116 |
< |
\[ |
1057 |
> |
function}, is of most fundamental importance to liquid theory. |
1058 |
> |
Experimentally, pair distribution functions can be gathered by |
1059 |
> |
Fourier transforming raw data from a series of neutron diffraction |
1060 |
> |
experiments and integrating over the surface factor |
1061 |
> |
\cite{Powles1973}. The experimental results can serve as a criterion |
1062 |
> |
to justify the correctness of a liquid model. Moreover, various |
1063 |
> |
equilibrium thermodynamic and structural properties can also be |
1064 |
> |
expressed in terms of the radial distribution function |
1065 |
> |
\cite{Allen1987}. The pair distribution functions $g(r)$ gives the |
1066 |
> |
probability that a particle $i$ will be located at a distance $r$ |
1067 |
> |
from a another particle $j$ in the system |
1068 |
> |
\begin{equation} |
1069 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1070 |
< |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1071 |
< |
\] |
1070 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
1071 |
> |
(r)}{\rho}. |
1072 |
> |
\end{equation} |
1073 |
|
Note that the delta function can be replaced by a histogram in |
1074 |
< |
computer simulation. Figure |
1075 |
< |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1076 |
< |
distribution function for the liquid argon system. The occurrence of |
1124 |
< |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1125 |
< |
to find particles at certain radial values than at others. This is a |
1126 |
< |
result of the attractive interaction at such distances. Because of |
1127 |
< |
the strong repulsive forces at short distance, the probability of |
1128 |
< |
locating particles at distances less than about 2.5{\AA} from each |
1129 |
< |
other is essentially zero. |
1074 |
> |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
1075 |
> |
the height of these peaks gradually decreases to 1 as the liquid of |
1076 |
> |
large distance approaches the bulk density. |
1077 |
|
|
1131 |
– |
%\begin{figure} |
1132 |
– |
%\centering |
1133 |
– |
%\includegraphics[width=\linewidth]{pdf.eps} |
1134 |
– |
%\caption[Pair distribution function for the liquid argon |
1135 |
– |
%]{Pair distribution function for the liquid argon} |
1136 |
– |
%\label{introFigure:pairDistributionFunction} |
1137 |
– |
%\end{figure} |
1078 |
|
|
1079 |
< |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
1080 |
< |
Properties} |
1079 |
> |
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
1080 |
> |
Properties}} |
1081 |
|
|
1082 |
|
Time-dependent properties are usually calculated using \emph{time |
1083 |
< |
correlation function}, which correlates random variables $A$ and $B$ |
1084 |
< |
at two different time |
1083 |
> |
correlation functions}, which correlate random variables $A$ and $B$ |
1084 |
> |
at two different times, |
1085 |
|
\begin{equation} |
1086 |
|
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1087 |
|
\label{introEquation:timeCorrelationFunction} |
1088 |
|
\end{equation} |
1089 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
1090 |
< |
function is called \emph{auto correlation function}. One example of |
1091 |
< |
auto correlation function is velocity auto-correlation function |
1092 |
< |
which is directly related to transport properties of molecular |
1093 |
< |
liquids: |
1090 |
> |
function is called an \emph{autocorrelation function}. One example |
1091 |
> |
of an auto correlation function is the velocity auto-correlation |
1092 |
> |
function which is directly related to transport properties of |
1093 |
> |
molecular liquids: |
1094 |
|
\[ |
1095 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1096 |
|
\right\rangle } dt |
1097 |
|
\] |
1098 |
< |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1099 |
< |
function which is averaging over time origins and over all the |
1100 |
< |
atoms, dipole autocorrelation are calculated for the entire system. |
1101 |
< |
The dipole autocorrelation function is given by: |
1098 |
> |
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
1099 |
> |
function, which is averaged over time origins and over all the |
1100 |
> |
atoms, the dipole autocorrelation functions is calculated for the |
1101 |
> |
entire system. The dipole autocorrelation function is given by: |
1102 |
|
\[ |
1103 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1104 |
|
\right\rangle |
1106 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
1107 |
|
by |
1108 |
|
\[ |
1109 |
< |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1109 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
1110 |
|
\] |
1111 |
< |
In principle, many time correlation functions can be related with |
1111 |
> |
In principle, many time correlation functions can be related to |
1112 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
1113 |
|
scattering spectra of molecular liquids. In practice, one can |
1114 |
< |
extract the IR spectrum from the intensity of dipole fluctuation at |
1115 |
< |
each frequency using the following relationship: |
1114 |
> |
extract the IR spectrum from the intensity of the molecular dipole |
1115 |
> |
fluctuation at each frequency using the following relationship: |
1116 |
|
\[ |
1117 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1118 |
< |
i2\pi vt} dt} |
1118 |
> |
i2\pi vt} dt}. |
1119 |
|
\] |
1120 |
|
|
1121 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1122 |
|
|
1123 |
|
Rigid bodies are frequently involved in the modeling of different |
1124 |
|
areas, from engineering, physics, to chemistry. For example, |
1125 |
< |
missiles and vehicle are usually modeled by rigid bodies. The |
1126 |
< |
movement of the objects in 3D gaming engine or other physics |
1127 |
< |
simulator is governed by the rigid body dynamics. In molecular |
1128 |
< |
simulation, rigid body is used to simplify the model in |
1129 |
< |
protein-protein docking study{\cite{Gray03}}. |
1125 |
> |
missiles and vehicles are usually modeled by rigid bodies. The |
1126 |
> |
movement of the objects in 3D gaming engines or other physics |
1127 |
> |
simulators is governed by rigid body dynamics. In molecular |
1128 |
> |
simulations, rigid bodies are used to simplify protein-protein |
1129 |
> |
docking studies\cite{Gray2003}. |
1130 |
|
|
1131 |
|
It is very important to develop stable and efficient methods to |
1132 |
< |
integrate the equations of motion of orientational degrees of |
1133 |
< |
freedom. Euler angles are the nature choice to describe the |
1134 |
< |
rotational degrees of freedom. However, due to its singularity, the |
1135 |
< |
numerical integration of corresponding equations of motion is very |
1136 |
< |
inefficient and inaccurate. Although an alternative integrator using |
1137 |
< |
different sets of Euler angles can overcome this difficulty\cite{}, |
1138 |
< |
the computational penalty and the lost of angular momentum |
1139 |
< |
conservation still remain. A singularity free representation |
1140 |
< |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
1141 |
< |
this approach suffer from the nonseparable Hamiltonian resulted from |
1142 |
< |
quaternion representation, which prevents the symplectic algorithm |
1143 |
< |
to be utilized. Another different approach is to apply holonomic |
1144 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
1145 |
< |
moves independently under the normal forces deriving from potential |
1146 |
< |
energy and constraint forces which are used to guarantee the |
1147 |
< |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
1148 |
< |
algorithm converge very slowly when the number of constraint |
1149 |
< |
increases. |
1132 |
> |
integrate the equations of motion for orientational degrees of |
1133 |
> |
freedom. Euler angles are the natural choice to describe the |
1134 |
> |
rotational degrees of freedom. However, due to $\frac {1}{sin |
1135 |
> |
\theta}$ singularities, the numerical integration of corresponding |
1136 |
> |
equations of these motion is very inefficient and inaccurate. |
1137 |
> |
Although an alternative integrator using multiple sets of Euler |
1138 |
> |
angles can overcome this difficulty\cite{Barojas1973}, the |
1139 |
> |
computational penalty and the loss of angular momentum conservation |
1140 |
> |
still remain. A singularity-free representation utilizing |
1141 |
> |
quaternions was developed by Evans in 1977\cite{Evans1977}. |
1142 |
> |
Unfortunately, this approach uses a nonseparable Hamiltonian |
1143 |
> |
resulting from the quaternion representation, which prevents the |
1144 |
> |
symplectic algorithm from being utilized. Another different approach |
1145 |
> |
is to apply holonomic constraints to the atoms belonging to the |
1146 |
> |
rigid body. Each atom moves independently under the normal forces |
1147 |
> |
deriving from potential energy and constraint forces which are used |
1148 |
> |
to guarantee the rigidness. However, due to their iterative nature, |
1149 |
> |
the SHAKE and Rattle algorithms also converge very slowly when the |
1150 |
> |
number of constraints increases\cite{Ryckaert1977, Andersen1983}. |
1151 |
|
|
1152 |
< |
The break through in geometric literature suggests that, in order to |
1152 |
> |
A break-through in geometric literature suggests that, in order to |
1153 |
|
develop a long-term integration scheme, one should preserve the |
1154 |
< |
symplectic structure of the flow. Introducing conjugate momentum to |
1155 |
< |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
1156 |
< |
symplectic integrator, RSHAKE, was proposed to evolve the |
1157 |
< |
Hamiltonian system in a constraint manifold by iteratively |
1158 |
< |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
1159 |
< |
method using quaternion representation was developed by Omelyan. |
1160 |
< |
However, both of these methods are iterative and inefficient. In |
1161 |
< |
this section, we will present a symplectic Lie-Poisson integrator |
1162 |
< |
for rigid body developed by Dullweber and his |
1163 |
< |
coworkers\cite{Dullweber1997} in depth. |
1154 |
> |
symplectic structure of the propagator. By introducing a conjugate |
1155 |
> |
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
1156 |
> |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
1157 |
> |
proposed to evolve the Hamiltonian system in a constraint manifold |
1158 |
> |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
1159 |
> |
An alternative method using the quaternion representation was |
1160 |
> |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
1161 |
> |
methods are iterative and inefficient. In this section, we descibe a |
1162 |
> |
symplectic Lie-Poisson integrator for rigid bodies developed by |
1163 |
> |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1164 |
|
|
1165 |
< |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
1166 |
< |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
1165 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
1166 |
> |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
1167 |
|
function |
1168 |
|
\begin{equation} |
1169 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
1177 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
1178 |
|
\] |
1179 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1180 |
< |
constrained Hamiltonian equation subjects to a holonomic constraint, |
1180 |
> |
constrained Hamiltonian equation is subjected to a holonomic |
1181 |
> |
constraint, |
1182 |
|
\begin{equation} |
1183 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1184 |
|
\end{equation} |
1185 |
< |
which is used to ensure rotation matrix's orthogonality. |
1186 |
< |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1187 |
< |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
1185 |
> |
which is used to ensure rotation matrix's unitarity. Differentiating |
1186 |
> |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
1187 |
> |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
1188 |
|
\begin{equation} |
1189 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1190 |
|
\label{introEquation:RBFirstOrderConstraint} |
1191 |
|
\end{equation} |
1250 |
– |
|
1192 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1193 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1194 |
|
the equations of motion, |
1195 |
< |
\[ |
1196 |
< |
\begin{array}{c} |
1197 |
< |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1198 |
< |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1199 |
< |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1200 |
< |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
1260 |
< |
\end{array} |
1261 |
< |
\] |
1262 |
< |
|
1195 |
> |
\begin{eqnarray} |
1196 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
1197 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\ |
1198 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
1199 |
> |
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1200 |
> |
\end{eqnarray} |
1201 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1202 |
< |
We can use constraint force provided by lagrange multiplier on the |
1203 |
< |
normal manifold to keep the motion on constraint space. Or we can |
1204 |
< |
simply evolve the system in constraint manifold. The two method are |
1205 |
< |
proved to be equivalent. The holonomic constraint and equations of |
1206 |
< |
motions define a constraint manifold for rigid body |
1202 |
> |
We can use a constraint force provided by a Lagrange multiplier on |
1203 |
> |
the normal manifold to keep the motion on constraint space. Or we |
1204 |
> |
can simply evolve the system on the constraint manifold. These two |
1205 |
> |
methods have been proved to be equivalent. The holonomic constraint |
1206 |
> |
and equations of motions define a constraint manifold for rigid |
1207 |
> |
bodies |
1208 |
|
\[ |
1209 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1210 |
|
\right\}. |
1211 |
|
\] |
1273 |
– |
|
1212 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
1213 |
< |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
1213 |
> |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
1214 |
> |
rotation group $SO(3)$. However, it turns out that under symplectic |
1215 |
|
transformation, the cotangent space and the phase space are |
1216 |
< |
diffeomorphic. Introducing |
1216 |
> |
diffeomorphic. By introducing |
1217 |
|
\[ |
1218 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1219 |
|
\] |
1223 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
1224 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
1225 |
|
\] |
1287 |
– |
|
1226 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
1227 |
|
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
1228 |
|
given as |
1241 |
|
\[ |
1242 |
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
1243 |
|
\] |
1244 |
< |
respectively. |
1245 |
< |
|
1246 |
< |
As a common choice to describe the rotation dynamics of the rigid |
1309 |
< |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
1310 |
< |
rewrite the equations of motion, |
1244 |
> |
respectively. As a common choice to describe the rotation dynamics |
1245 |
> |
of the rigid body, the angular momentum on the body fixed frame $\Pi |
1246 |
> |
= Q^t P$ is introduced to rewrite the equations of motion, |
1247 |
|
\begin{equation} |
1248 |
|
\begin{array}{l} |
1249 |
< |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1250 |
< |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
1249 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda, \\ |
1250 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1}, \\ |
1251 |
|
\end{array} |
1252 |
|
\label{introEqaution:RBMotionPI} |
1253 |
|
\end{equation} |
1254 |
< |
, as well as holonomic constraints, |
1255 |
< |
\[ |
1256 |
< |
\begin{array}{l} |
1321 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
1322 |
< |
Q^T Q = 1 \\ |
1323 |
< |
\end{array} |
1324 |
< |
\] |
1325 |
< |
|
1326 |
< |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1327 |
< |
so(3)^ \star$, the hat-map isomorphism, |
1254 |
> |
as well as holonomic constraints $\Pi J^{ - 1} + J^{ - 1} \Pi ^t = |
1255 |
> |
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a |
1256 |
> |
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism, |
1257 |
|
\begin{equation} |
1258 |
|
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1259 |
|
{\begin{array}{*{20}c} |
1266 |
|
will let us associate the matrix products with traditional vector |
1267 |
|
operations |
1268 |
|
\[ |
1269 |
< |
\hat vu = v \times u |
1269 |
> |
\hat vu = v \times u. |
1270 |
|
\] |
1271 |
< |
|
1343 |
< |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1271 |
> |
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
1272 |
|
matrix, |
1273 |
+ |
\begin{eqnarray} |
1274 |
+ |
(\dot \Pi - \dot \Pi ^T )&= &(\Pi - \Pi ^T )(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
1275 |
+ |
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1276 |
+ |
(\Lambda - \Lambda ^T ). \label{introEquation:skewMatrixPI} |
1277 |
+ |
\end{eqnarray} |
1278 |
+ |
Since $\Lambda$ is symmetric, the last term of |
1279 |
+ |
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
1280 |
+ |
Lagrange multiplier $\Lambda$ is absent from the equations of |
1281 |
+ |
motion. This unique property eliminates the requirement of |
1282 |
+ |
iterations which can not be avoided in other methods\cite{Kol1997, |
1283 |
+ |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
1284 |
+ |
equation of motion for angular momentum on body frame |
1285 |
|
\begin{equation} |
1346 |
– |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
1347 |
– |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
1348 |
– |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1349 |
– |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
1350 |
– |
\end{equation} |
1351 |
– |
Since $\Lambda$ is symmetric, the last term of Equation |
1352 |
– |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1353 |
– |
multiplier $\Lambda$ is absent from the equations of motion. This |
1354 |
– |
unique property eliminate the requirement of iterations which can |
1355 |
– |
not be avoided in other methods\cite{}. |
1356 |
– |
|
1357 |
– |
Applying hat-map isomorphism, we obtain the equation of motion for |
1358 |
– |
angular momentum on body frame |
1359 |
– |
\begin{equation} |
1286 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
1287 |
|
F_i (r,Q)} \right) \times X_i }. |
1288 |
|
\label{introEquation:bodyAngularMotion} |
1290 |
|
In the same manner, the equation of motion for rotation matrix is |
1291 |
|
given by |
1292 |
|
\[ |
1293 |
< |
\dot Q = Qskew(I^{ - 1} \pi ) |
1293 |
> |
\dot Q = Qskew(I^{ - 1} \pi ). |
1294 |
|
\] |
1295 |
|
|
1296 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
1297 |
|
Lie-Poisson Integrator for Free Rigid Body} |
1298 |
|
|
1299 |
< |
If there is not external forces exerted on the rigid body, the only |
1300 |
< |
contribution to the rotational is from the kinetic potential (the |
1301 |
< |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
1302 |
< |
rigid body is an example of Lie-Poisson system with Hamiltonian |
1299 |
> |
If there are no external forces exerted on the rigid body, the only |
1300 |
> |
contribution to the rotational motion is from the kinetic energy |
1301 |
> |
(the first term of \ref{introEquation:bodyAngularMotion}). The free |
1302 |
> |
rigid body is an example of a Lie-Poisson system with Hamiltonian |
1303 |
|
function |
1304 |
|
\begin{equation} |
1305 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1312 |
|
0 & {\pi _3 } & { - \pi _2 } \\ |
1313 |
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
1314 |
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
1315 |
< |
\end{array}} \right) |
1315 |
> |
\end{array}} \right). |
1316 |
|
\end{equation} |
1317 |
|
Thus, the dynamics of free rigid body is governed by |
1318 |
|
\begin{equation} |
1319 |
< |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
1319 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
1320 |
|
\end{equation} |
1321 |
< |
|
1322 |
< |
One may notice that each $T_i^r$ in Equation |
1323 |
< |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1398 |
< |
instance, the equations of motion due to $T_1^r$ are given by |
1321 |
> |
One may notice that each $T_i^r$ in |
1322 |
> |
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly. |
1323 |
> |
For instance, the equations of motion due to $T_1^r$ are given by |
1324 |
|
\begin{equation} |
1325 |
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
1326 |
|
\label{introEqaution:RBMotionSingleTerm} |
1327 |
|
\end{equation} |
1328 |
< |
where |
1328 |
> |
with |
1329 |
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
1330 |
|
0 & 0 & 0 \\ |
1331 |
|
0 & 0 & {\pi _1 } \\ |
1332 |
|
0 & { - \pi _1 } & 0 \\ |
1333 |
|
\end{array}} \right). |
1334 |
|
\] |
1335 |
< |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1335 |
> |
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is |
1336 |
|
\[ |
1337 |
|
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
1338 |
|
Q(0)e^{\Delta tR_1 } |
1346 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1347 |
|
\] |
1348 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
1349 |
< |
tR_1 }$, we can use Cayley transformation, |
1349 |
> |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
1350 |
> |
propagator, |
1351 |
|
\[ |
1352 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1353 |
< |
) |
1353 |
> |
). |
1354 |
|
\] |
1355 |
< |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1356 |
< |
manner. |
1357 |
< |
|
1432 |
< |
In order to construct a second-order symplectic method, we split the |
1433 |
< |
angular kinetic Hamiltonian function can into five terms |
1355 |
> |
The propagator maps for $T_2^r$ and $T_3^r$ can be found in the same |
1356 |
> |
manner. In order to construct a second-order symplectic method, we |
1357 |
> |
split the angular kinetic Hamiltonian function into five terms |
1358 |
|
\[ |
1359 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1360 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1361 |
< |
(\pi _1 ) |
1362 |
< |
\]. |
1363 |
< |
Concatenating flows corresponding to these five terms, we can obtain |
1364 |
< |
an symplectic integrator, |
1361 |
> |
(\pi _1 ). |
1362 |
> |
\] |
1363 |
> |
By concatenating the propagators corresponding to these five terms, |
1364 |
> |
we can obtain an symplectic integrator, |
1365 |
|
\[ |
1366 |
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
1367 |
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1368 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1369 |
|
_1 }. |
1370 |
|
\] |
1447 |
– |
|
1371 |
|
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
1372 |
|
$F(\pi )$ and $G(\pi )$ is defined by |
1373 |
|
\[ |
1374 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
1375 |
< |
) |
1375 |
> |
). |
1376 |
|
\] |
1377 |
|
If the Poisson bracket of a function $F$ with an arbitrary smooth |
1378 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
1383 |
|
then by the chain rule |
1384 |
|
\[ |
1385 |
|
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
1386 |
< |
}}{2})\pi |
1387 |
< |
\] |
1388 |
< |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
1386 |
> |
}}{2})\pi. |
1387 |
> |
\] |
1388 |
> |
Thus, $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel |
1389 |
> |
\pi |
1390 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
1391 |
< |
Lie-Poisson integrator is found to be extremely efficient and stable |
1392 |
< |
which can be explained by the fact the small angle approximation is |
1393 |
< |
used and the norm of the angular momentum is conserved. |
1391 |
> |
Lie-Poisson integrator is found to be both extremely efficient and |
1392 |
> |
stable. These properties can be explained by the fact the small |
1393 |
> |
angle approximation is used and the norm of the angular momentum is |
1394 |
> |
conserved. |
1395 |
|
|
1396 |
|
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
1397 |
|
Splitting for Rigid Body} |
1398 |
|
|
1399 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
1400 |
< |
energy and potential energy, |
1401 |
< |
\[ |
1402 |
< |
H = T(p,\pi ) + V(q,Q) |
1403 |
< |
\] |
1404 |
< |
The equations of motion corresponding to potential energy and |
1480 |
< |
kinetic energy are listed in the below table, |
1400 |
> |
energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations |
1401 |
> |
of motion corresponding to potential energy and kinetic energy are |
1402 |
> |
listed in the below table, |
1403 |
> |
\begin{table} |
1404 |
> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
1405 |
|
\begin{center} |
1406 |
|
\begin{tabular}{|l|l|} |
1407 |
|
\hline |
1414 |
|
\hline |
1415 |
|
\end{tabular} |
1416 |
|
\end{center} |
1417 |
+ |
\end{table} |
1418 |
|
A second-order symplectic method is now obtained by the composition |
1419 |
< |
of the flow maps, |
1419 |
> |
of the position and velocity propagators, |
1420 |
|
\[ |
1421 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1422 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1423 |
|
\] |
1424 |
|
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
1425 |
< |
sub-flows which corresponding to force and torque respectively, |
1425 |
> |
sub-propagators which corresponding to force and torque |
1426 |
> |
respectively, |
1427 |
|
\[ |
1428 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1429 |
|
_{\Delta t/2,\tau }. |
1430 |
|
\] |
1431 |
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
1432 |
< |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
1433 |
< |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
1434 |
< |
|
1435 |
< |
Furthermore, kinetic potential can be separated to translational |
1510 |
< |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
1432 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order |
1433 |
> |
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the |
1434 |
> |
kinetic energy can be separated to translational kinetic term, $T^t |
1435 |
> |
(p)$, and rotational kinetic term, $T^r (\pi )$, |
1436 |
|
\begin{equation} |
1437 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
1438 |
|
\end{equation} |
1439 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
1440 |
< |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
1441 |
< |
corresponding flow maps are given by |
1440 |
> |
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore, |
1441 |
> |
the corresponding propagators are given by |
1442 |
|
\[ |
1443 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
1444 |
|
_{\Delta t,T^r }. |
1445 |
|
\] |
1446 |
< |
Finally, we obtain the overall symplectic flow maps for free moving |
1447 |
< |
rigid body |
1448 |
< |
\begin{equation} |
1449 |
< |
\begin{array}{c} |
1450 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
1451 |
< |
\circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \\ |
1527 |
< |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1528 |
< |
\end{array} |
1446 |
> |
Finally, we obtain the overall symplectic propagators for freely |
1447 |
> |
moving rigid bodies |
1448 |
> |
\begin{eqnarray} |
1449 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
1450 |
> |
& & \circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \notag\\ |
1451 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1452 |
|
\label{introEquation:overallRBFlowMaps} |
1453 |
< |
\end{equation} |
1453 |
> |
\end{eqnarray} |
1454 |
|
|
1455 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1456 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
1457 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
1458 |
|
has been applied in a variety of studies. This section will review |
1459 |
< |
the theory of Langevin dynamics simulation. A brief derivation of |
1460 |
< |
generalized Langevin equation will be given first. Follow that, we |
1461 |
< |
will discuss the physical meaning of the terms appearing in the |
1462 |
< |
equation as well as the calculation of friction tensor from |
1463 |
< |
hydrodynamics theory. |
1459 |
> |
the theory of Langevin dynamics. A brief derivation of generalized |
1460 |
> |
Langevin equation will be given first. Following that, we will |
1461 |
> |
discuss the physical meaning of the terms appearing in the equation |
1462 |
> |
as well as the calculation of friction tensor from hydrodynamics |
1463 |
> |
theory. |
1464 |
|
|
1465 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
1466 |
|
|
1467 |
< |
Harmonic bath model, in which an effective set of harmonic |
1467 |
> |
A harmonic bath model, in which an effective set of harmonic |
1468 |
|
oscillators are used to mimic the effect of a linearly responding |
1469 |
|
environment, has been widely used in quantum chemistry and |
1470 |
|
statistical mechanics. One of the successful applications of |
1471 |
< |
Harmonic bath model is the derivation of Deriving Generalized |
1472 |
< |
Langevin Dynamics. Lets consider a system, in which the degree of |
1471 |
> |
Harmonic bath model is the derivation of the Generalized Langevin |
1472 |
> |
Dynamics (GLE). Lets consider a system, in which the degree of |
1473 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
1474 |
|
Hamiltonian of the form |
1475 |
|
\begin{equation} |
1476 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1477 |
|
\label{introEquation:bathGLE}. |
1478 |
|
\end{equation} |
1479 |
< |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
1480 |
< |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
1479 |
> |
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated |
1480 |
> |
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
1481 |
|
\[ |
1482 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1483 |
|
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
1485 |
|
\] |
1486 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
1487 |
|
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
1488 |
< |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
1488 |
> |
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath |
1489 |
|
coupling, |
1490 |
|
\[ |
1491 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1492 |
|
\] |
1493 |
< |
where $g_\alpha$ are the coupling constants between the bath and the |
1494 |
< |
coordinate $x$. Introducing |
1493 |
> |
where $g_\alpha$ are the coupling constants between the bath |
1494 |
> |
coordinates ($x_ \alpha$) and the system coordinate ($x$). |
1495 |
> |
Introducing |
1496 |
|
\[ |
1497 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1498 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1499 |
< |
\] and combining the last two terms in Equation |
1500 |
< |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
1577 |
< |
Hamiltonian as |
1499 |
> |
\] |
1500 |
> |
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as |
1501 |
|
\[ |
1502 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1503 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1504 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1505 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1505 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}}. |
1506 |
|
\] |
1507 |
|
Since the first two terms of the new Hamiltonian depend only on the |
1508 |
|
system coordinates, we can get the equations of motion for |
1509 |
< |
Generalized Langevin Dynamics by Hamilton's equations |
1587 |
< |
\ref{introEquation:motionHamiltonianCoordinate, |
1588 |
< |
introEquation:motionHamiltonianMomentum}, |
1509 |
> |
Generalized Langevin Dynamics by Hamilton's equations, |
1510 |
|
\begin{equation} |
1511 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
1512 |
|
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
1519 |
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
1520 |
|
\label{introEquation:bathMotionGLE} |
1521 |
|
\end{equation} |
1601 |
– |
|
1522 |
|
In order to derive an equation for $x$, the dynamics of the bath |
1523 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
1524 |
|
transform which is particularly useful in solving linear ordinary |
1525 |
< |
differential equations, Laplace transform is the appropriate tool to |
1526 |
< |
solve this problem. The basic idea is to transform the difficult |
1525 |
> |
differential equations,the Laplace transform is the appropriate tool |
1526 |
> |
to solve this problem. The basic idea is to transform the difficult |
1527 |
|
differential equations into simple algebra problems which can be |
1528 |
< |
solved easily. Then applying inverse Laplace transform, also known |
1529 |
< |
as the Bromwich integral, we can retrieve the solutions of the |
1530 |
< |
original problems. |
1531 |
< |
|
1612 |
< |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
1613 |
< |
transform of f(t) is a new function defined as |
1528 |
> |
solved easily. Then, by applying the inverse Laplace transform, also |
1529 |
> |
known as the Bromwich integral, we can retrieve the solutions of the |
1530 |
> |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
1531 |
> |
$, the Laplace transform of $f(t)$ is a new function defined as |
1532 |
|
\[ |
1533 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1534 |
|
\] |
1535 |
|
where $p$ is real and $L$ is called the Laplace Transform |
1536 |
|
Operator. Below are some important properties of Laplace transform |
1537 |
< |
\begin{equation} |
1538 |
< |
\begin{array}{c} |
1539 |
< |
L(x + y) = L(x) + L(y) \\ |
1540 |
< |
L(ax) = aL(x) \\ |
1541 |
< |
L(\dot x) = pL(x) - px(0) \\ |
1542 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
1543 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
1544 |
< |
\end{array} |
1545 |
< |
\end{equation} |
1546 |
< |
|
1547 |
< |
Applying Laplace transform to the bath coordinates, we obtain |
1548 |
< |
\[ |
1631 |
< |
\begin{array}{c} |
1632 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1633 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1634 |
< |
\end{array} |
1635 |
< |
\] |
1537 |
> |
\begin{eqnarray*} |
1538 |
> |
L(x + y) & = & L(x) + L(y) \\ |
1539 |
> |
L(ax) & = & aL(x) \\ |
1540 |
> |
L(\dot x) & = & pL(x) - px(0) \\ |
1541 |
> |
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
1542 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
1543 |
> |
\end{eqnarray*} |
1544 |
> |
Applying the Laplace transform to the bath coordinates, we obtain |
1545 |
> |
\begin{eqnarray*} |
1546 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x), \\ |
1547 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
1548 |
> |
\end{eqnarray*} |
1549 |
|
By the same way, the system coordinates become |
1550 |
< |
\[ |
1551 |
< |
\begin{array}{c} |
1639 |
< |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1550 |
> |
\begin{eqnarray*} |
1551 |
> |
mL(\ddot x) & = & |
1552 |
|
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1553 |
< |
\end{array} |
1554 |
< |
\] |
1643 |
< |
|
1553 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}. |
1554 |
> |
\end{eqnarray*} |
1555 |
|
With the help of some relatively important inverse Laplace |
1556 |
|
transformations: |
1557 |
|
\[ |
1561 |
|
L(1) = \frac{1}{p} \\ |
1562 |
|
\end{array} |
1563 |
|
\] |
1564 |
< |
, we obtain |
1565 |
< |
\begin{align} |
1566 |
< |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
1564 |
> |
we obtain |
1565 |
> |
\begin{eqnarray*} |
1566 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
1567 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1568 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
1569 |
< |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
1570 |
< |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
1571 |
< |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
1572 |
< |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
1573 |
< |
% |
1574 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1569 |
> |
_\alpha t)\dot x(t - \tau )d\tau } } \right\}} \\ |
1570 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1571 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1572 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1573 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1574 |
> |
\end{eqnarray*} |
1575 |
> |
\begin{eqnarray*} |
1576 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1577 |
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1578 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1579 |
< |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
1580 |
< |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
1581 |
< |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
1582 |
< |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
1583 |
< |
(\omega _\alpha t)} \right\}} |
1584 |
< |
\end{align} |
1672 |
< |
|
1579 |
> |
t)\dot x(t - \tau )d} \tau } \\ |
1580 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1581 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1582 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1583 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1584 |
> |
\end{eqnarray*} |
1585 |
|
Introducing a \emph{dynamic friction kernel} |
1586 |
|
\begin{equation} |
1587 |
|
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1604 |
|
\end{equation} |
1605 |
|
which is known as the \emph{generalized Langevin equation}. |
1606 |
|
|
1607 |
< |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel} |
1607 |
> |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
1608 |
|
|
1609 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
1610 |
|
implies it is completely deterministic within the context of a |
1611 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
1612 |
< |
uncorrelated to $x$ and $\dot x$, |
1613 |
< |
\[ |
1614 |
< |
\begin{array}{l} |
1615 |
< |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
1616 |
< |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
1705 |
< |
\end{array} |
1706 |
< |
\] |
1707 |
< |
This property is what we expect from a truly random process. As long |
1708 |
< |
as the model, which is gaussian distribution in general, chosen for |
1709 |
< |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
1710 |
< |
still remains. |
1711 |
< |
|
1612 |
> |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
1613 |
> |
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
1614 |
> |
0.$ This property is what we expect from a truly random process. As |
1615 |
> |
long as the model chosen for $R(t)$ was a gaussian distribution in |
1616 |
> |
general, the stochastic nature of the GLE still remains. |
1617 |
|
%dynamic friction kernel |
1618 |
|
The convolution integral |
1619 |
|
\[ |
1628 |
|
\[ |
1629 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
1630 |
|
\] |
1631 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1631 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1632 |
|
\[ |
1633 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
1634 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
1635 |
|
\] |
1636 |
< |
which can be used to describe dynamic caging effect. The other |
1637 |
< |
extreme is the bath that responds infinitely quickly to motions in |
1638 |
< |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
1639 |
< |
time: |
1636 |
> |
which can be used to describe the effect of dynamic caging in |
1637 |
> |
viscous solvents. The other extreme is the bath that responds |
1638 |
> |
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
1639 |
> |
taken as a $delta$ function in time: |
1640 |
|
\[ |
1641 |
|
\xi (t) = 2\xi _0 \delta (t) |
1642 |
|
\] |
1645 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
1646 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
1647 |
|
\] |
1648 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1648 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1649 |
|
\begin{equation} |
1650 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
1651 |
|
x(t) + R(t) \label{introEquation:LangevinEquation} |
1652 |
|
\end{equation} |
1653 |
|
which is known as the Langevin equation. The static friction |
1654 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
1655 |
< |
or be determined by Stokes' law for regular shaped particles.A |
1655 |
> |
or be determined by Stokes' law for regular shaped particles. A |
1656 |
|
briefly review on calculating friction tensor for arbitrary shaped |
1657 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
1658 |
|
|
1659 |
< |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
1659 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
1660 |
|
|
1661 |
< |
Defining a new set of coordinates, |
1661 |
> |
Defining a new set of coordinates |
1662 |
|
\[ |
1663 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1664 |
< |
^2 }}x(0) |
1665 |
< |
\], |
1664 |
> |
^2 }}x(0), |
1665 |
> |
\] |
1666 |
|
we can rewrite $R(T)$ as |
1667 |
|
\[ |
1668 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1669 |
|
\] |
1670 |
|
And since the $q$ coordinates are harmonic oscillators, |
1671 |
< |
\[ |
1672 |
< |
\begin{array}{c} |
1673 |
< |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1674 |
< |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1675 |
< |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1676 |
< |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1677 |
< |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1678 |
< |
= kT\xi (t) \\ |
1774 |
< |
\end{array} |
1775 |
< |
\] |
1671 |
> |
\begin{eqnarray*} |
1672 |
> |
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1673 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1674 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1675 |
> |
\left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1676 |
> |
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1677 |
> |
& = &kT\xi (t) \\ |
1678 |
> |
\end{eqnarray*} |
1679 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1680 |
|
\begin{equation} |
1681 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1682 |
< |
\label{introEquation:secondFluctuationDissipation}. |
1682 |
> |
\label{introEquation:secondFluctuationDissipation}, |
1683 |
|
\end{equation} |
1684 |
< |
In effect, it acts as a constraint on the possible ways in which one |
1685 |
< |
can model the random force and friction kernel. |
1783 |
< |
|
1784 |
< |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1785 |
< |
Theoretically, the friction kernel can be determined using velocity |
1786 |
< |
autocorrelation function. However, this approach become impractical |
1787 |
< |
when the system become more and more complicate. Instead, various |
1788 |
< |
approaches based on hydrodynamics have been developed to calculate |
1789 |
< |
the friction coefficients. The friction effect is isotropic in |
1790 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
1791 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
1792 |
< |
\[ |
1793 |
< |
\Xi = \left( {\begin{array}{*{20}c} |
1794 |
< |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
1795 |
< |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
1796 |
< |
\end{array}} \right). |
1797 |
< |
\] |
1798 |
< |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
1799 |
< |
tensor and rotational resistance (friction) tensor respectively, |
1800 |
< |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
1801 |
< |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
1802 |
< |
particle moves in a fluid, it may experience friction force or |
1803 |
< |
torque along the opposite direction of the velocity or angular |
1804 |
< |
velocity, |
1805 |
< |
\[ |
1806 |
< |
\left( \begin{array}{l} |
1807 |
< |
F_R \\ |
1808 |
< |
\tau _R \\ |
1809 |
< |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
1810 |
< |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
1811 |
< |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
1812 |
< |
\end{array}} \right)\left( \begin{array}{l} |
1813 |
< |
v \\ |
1814 |
< |
w \\ |
1815 |
< |
\end{array} \right) |
1816 |
< |
\] |
1817 |
< |
where $F_r$ is the friction force and $\tau _R$ is the friction |
1818 |
< |
toque. |
1819 |
< |
|
1820 |
< |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
1821 |
< |
|
1822 |
< |
For a spherical particle, the translational and rotational friction |
1823 |
< |
constant can be calculated from Stoke's law, |
1824 |
< |
\[ |
1825 |
< |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
1826 |
< |
{6\pi \eta R} & 0 & 0 \\ |
1827 |
< |
0 & {6\pi \eta R} & 0 \\ |
1828 |
< |
0 & 0 & {6\pi \eta R} \\ |
1829 |
< |
\end{array}} \right) |
1830 |
< |
\] |
1831 |
< |
and |
1832 |
< |
\[ |
1833 |
< |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
1834 |
< |
{8\pi \eta R^3 } & 0 & 0 \\ |
1835 |
< |
0 & {8\pi \eta R^3 } & 0 \\ |
1836 |
< |
0 & 0 & {8\pi \eta R^3 } \\ |
1837 |
< |
\end{array}} \right) |
1838 |
< |
\] |
1839 |
< |
where $\eta$ is the viscosity of the solvent and $R$ is the |
1840 |
< |
hydrodynamics radius. |
1841 |
< |
|
1842 |
< |
Other non-spherical shape, such as cylinder and ellipsoid |
1843 |
< |
\textit{etc}, are widely used as reference for developing new |
1844 |
< |
hydrodynamics theory, because their properties can be calculated |
1845 |
< |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
1846 |
< |
also called a triaxial ellipsoid, which is given in Cartesian |
1847 |
< |
coordinates by |
1848 |
< |
\[ |
1849 |
< |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
1850 |
< |
}} = 1 |
1851 |
< |
\] |
1852 |
< |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
1853 |
< |
due to the complexity of the elliptic integral, only the ellipsoid |
1854 |
< |
with the restriction of two axes having to be equal, \textit{i.e.} |
1855 |
< |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
1856 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
1857 |
< |
\[ |
1858 |
< |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
1859 |
< |
} }}{b}, |
1860 |
< |
\] |
1861 |
< |
and oblate, |
1862 |
< |
\[ |
1863 |
< |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
1864 |
< |
}}{a} |
1865 |
< |
\], |
1866 |
< |
one can write down the translational and rotational resistance |
1867 |
< |
tensors |
1868 |
< |
\[ |
1869 |
< |
\begin{array}{l} |
1870 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
1871 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
1872 |
< |
\end{array}, |
1873 |
< |
\] |
1874 |
< |
and |
1875 |
< |
\[ |
1876 |
< |
\begin{array}{l} |
1877 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
1878 |
< |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
1879 |
< |
\end{array}. |
1880 |
< |
\] |
1881 |
< |
|
1882 |
< |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
1883 |
< |
|
1884 |
< |
Unlike spherical and other regular shaped molecules, there is not |
1885 |
< |
analytical solution for friction tensor of any arbitrary shaped |
1886 |
< |
rigid molecules. The ellipsoid of revolution model and general |
1887 |
< |
triaxial ellipsoid model have been used to approximate the |
1888 |
< |
hydrodynamic properties of rigid bodies. However, since the mapping |
1889 |
< |
from all possible ellipsoidal space, $r$-space, to all possible |
1890 |
< |
combination of rotational diffusion coefficients, $D$-space is not |
1891 |
< |
unique\cite{Wegener79} as well as the intrinsic coupling between |
1892 |
< |
translational and rotational motion of rigid body\cite{}, general |
1893 |
< |
ellipsoid is not always suitable for modeling arbitrarily shaped |
1894 |
< |
rigid molecule. A number of studies have been devoted to determine |
1895 |
< |
the friction tensor for irregularly shaped rigid bodies using more |
1896 |
< |
advanced method\cite{} where the molecule of interest was modeled by |
1897 |
< |
combinations of spheres(beads)\cite{} and the hydrodynamics |
1898 |
< |
properties of the molecule can be calculated using the hydrodynamic |
1899 |
< |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
1900 |
< |
immersed in a continuous medium. Due to hydrodynamics interaction, |
1901 |
< |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
1902 |
< |
unperturbed velocity $v_i$, |
1903 |
< |
\[ |
1904 |
< |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
1905 |
< |
\] |
1906 |
< |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
1907 |
< |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
1908 |
< |
proportional to its ``net'' velocity |
1909 |
< |
\begin{equation} |
1910 |
< |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
1911 |
< |
\label{introEquation:tensorExpression} |
1912 |
< |
\end{equation} |
1913 |
< |
This equation is the basis for deriving the hydrodynamic tensor. In |
1914 |
< |
1930, Oseen and Burgers gave a simple solution to Equation |
1915 |
< |
\ref{introEquation:tensorExpression} |
1916 |
< |
\begin{equation} |
1917 |
< |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
1918 |
< |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
1919 |
< |
\label{introEquation:oseenTensor} |
1920 |
< |
\end{equation} |
1921 |
< |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
1922 |
< |
A second order expression for element of different size was |
1923 |
< |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
1924 |
< |
la Torre and Bloomfield, |
1925 |
< |
\begin{equation} |
1926 |
< |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
1927 |
< |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
1928 |
< |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
1929 |
< |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
1930 |
< |
\label{introEquation:RPTensorNonOverlapped} |
1931 |
< |
\end{equation} |
1932 |
< |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
1933 |
< |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
1934 |
< |
\ge \sigma _i + \sigma _j$. An alternative expression for |
1935 |
< |
overlapping beads with the same radius, $\sigma$, is given by |
1936 |
< |
\begin{equation} |
1937 |
< |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
1938 |
< |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
1939 |
< |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
1940 |
< |
\label{introEquation:RPTensorOverlapped} |
1941 |
< |
\end{equation} |
1942 |
< |
|
1943 |
< |
To calculate the resistance tensor at an arbitrary origin $O$, we |
1944 |
< |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
1945 |
< |
$B_{ij}$ blocks |
1946 |
< |
\begin{equation} |
1947 |
< |
B = \left( {\begin{array}{*{20}c} |
1948 |
< |
{B_{11} } & \ldots & {B_{1N} } \\ |
1949 |
< |
\vdots & \ddots & \vdots \\ |
1950 |
< |
{B_{N1} } & \cdots & {B_{NN} } \\ |
1951 |
< |
\end{array}} \right), |
1952 |
< |
\end{equation} |
1953 |
< |
where $B_{ij}$ is given by |
1954 |
< |
\[ |
1955 |
< |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
1956 |
< |
)T_{ij} |
1957 |
< |
\] |
1958 |
< |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
1959 |
< |
$B$, we obtain |
1960 |
< |
|
1961 |
< |
\[ |
1962 |
< |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
1963 |
< |
{C_{11} } & \ldots & {C_{1N} } \\ |
1964 |
< |
\vdots & \ddots & \vdots \\ |
1965 |
< |
{C_{N1} } & \cdots & {C_{NN} } \\ |
1966 |
< |
\end{array}} \right) |
1967 |
< |
\] |
1968 |
< |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
1969 |
< |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
1970 |
< |
\[ |
1971 |
< |
U_i = \left( {\begin{array}{*{20}c} |
1972 |
< |
0 & { - z_i } & {y_i } \\ |
1973 |
< |
{z_i } & 0 & { - x_i } \\ |
1974 |
< |
{ - y_i } & {x_i } & 0 \\ |
1975 |
< |
\end{array}} \right) |
1976 |
< |
\] |
1977 |
< |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
1978 |
< |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
1979 |
< |
arbitrary origin $O$ can be written as |
1980 |
< |
\begin{equation} |
1981 |
< |
\begin{array}{l} |
1982 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
1983 |
< |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
1984 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
1985 |
< |
\end{array} |
1986 |
< |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
1987 |
< |
\end{equation} |
1988 |
< |
|
1989 |
< |
The resistance tensor depends on the origin to which they refer. The |
1990 |
< |
proper location for applying friction force is the center of |
1991 |
< |
resistance (reaction), at which the trace of rotational resistance |
1992 |
< |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
1993 |
< |
resistance is defined as an unique point of the rigid body at which |
1994 |
< |
the translation-rotation coupling tensor are symmetric, |
1995 |
< |
\begin{equation} |
1996 |
< |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
1997 |
< |
\label{introEquation:definitionCR} |
1998 |
< |
\end{equation} |
1999 |
< |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
2000 |
< |
we can easily find out that the translational resistance tensor is |
2001 |
< |
origin independent, while the rotational resistance tensor and |
2002 |
< |
translation-rotation coupling resistance tensor depend on the |
2003 |
< |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
2004 |
< |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
2005 |
< |
obtain the resistance tensor at $P$ by |
2006 |
< |
\begin{equation} |
2007 |
< |
\begin{array}{l} |
2008 |
< |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
2009 |
< |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
2010 |
< |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
2011 |
< |
\end{array} |
2012 |
< |
\label{introEquation:resistanceTensorTransformation} |
2013 |
< |
\end{equation} |
2014 |
< |
where |
2015 |
< |
\[ |
2016 |
< |
U_{OP} = \left( {\begin{array}{*{20}c} |
2017 |
< |
0 & { - z_{OP} } & {y_{OP} } \\ |
2018 |
< |
{z_i } & 0 & { - x_{OP} } \\ |
2019 |
< |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
2020 |
< |
\end{array}} \right) |
2021 |
< |
\] |
2022 |
< |
Using Equations \ref{introEquation:definitionCR} and |
2023 |
< |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
2024 |
< |
the position of center of resistance, |
2025 |
< |
\[ |
2026 |
< |
\left( \begin{array}{l} |
2027 |
< |
x_{OR} \\ |
2028 |
< |
y_{OR} \\ |
2029 |
< |
z_{OR} \\ |
2030 |
< |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
2031 |
< |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
2032 |
< |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
2033 |
< |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
2034 |
< |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
2035 |
< |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
2036 |
< |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
2037 |
< |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
2038 |
< |
\end{array} \right). |
2039 |
< |
\] |
2040 |
< |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
2041 |
< |
joining center of resistance $R$ and origin $O$. |
1684 |
> |
which acts as a constraint on the possible ways in which one can |
1685 |
> |
model the random force and friction kernel. |