| 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 |
| 71 |
> |
bodies.\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 |
| 208 |
|
The thermodynamic behaviors and properties of Molecular Dynamics |
| 209 |
|
simulation are governed by the principle of Statistical Mechanics. |
| 210 |
|
The following section will give a brief introduction to some of the |
| 211 |
< |
Statistical Mechanics concepts and theorem presented in this |
| 211 |
> |
Statistical Mechanics concepts and theorems presented in this |
| 212 |
|
dissertation. |
| 213 |
|
|
| 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 |
|
|
| 308 |
– |
There are several different types of ensembles with different |
| 309 |
– |
statistical characteristics. As a function of macroscopic |
| 310 |
– |
parameters, such as temperature \textit{etc}, partition function can |
| 311 |
– |
be used to describe the statistical properties of a system in |
| 312 |
– |
thermodynamic equilibrium. |
| 313 |
– |
|
| 314 |
– |
As an ensemble of systems, each of which is known to be thermally |
| 315 |
– |
isolated and conserve energy, Microcanonical ensemble(NVE) has a |
| 316 |
– |
partition function like, |
| 317 |
– |
\begin{equation} |
| 318 |
– |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 319 |
– |
\end{equation} |
| 320 |
– |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
| 321 |
– |
can share its energy with a large heat reservoir. The distribution |
| 322 |
– |
of the total energy amongst the possible dynamical states is given |
| 323 |
– |
by the partition function, |
| 324 |
– |
\begin{equation} |
| 325 |
– |
\Omega (N,V,T) = e^{ - \beta A} |
| 326 |
– |
\label{introEquation:NVTPartition} |
| 327 |
– |
\end{equation} |
| 328 |
– |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 329 |
– |
TS$. Since most experiment are carried out under constant pressure |
| 330 |
– |
condition, isothermal-isobaric ensemble(NPT) play a very important |
| 331 |
– |
role in molecular simulation. The isothermal-isobaric ensemble allow |
| 332 |
– |
the system to exchange energy with a heat bath of temperature $T$ |
| 333 |
– |
and to change the volume as well. Its partition function is given as |
| 334 |
– |
\begin{equation} |
| 335 |
– |
\Delta (N,P,T) = - e^{\beta G}. |
| 336 |
– |
\label{introEquation:NPTPartition} |
| 337 |
– |
\end{equation} |
| 338 |
– |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
| 339 |
– |
|
| 286 |
|
\subsection{\label{introSection:liouville}Liouville's theorem} |
| 287 |
|
|
| 288 |
< |
The Liouville's theorem is the foundation on which statistical |
| 289 |
< |
mechanics rests. It describes the time evolution of phase space |
| 288 |
> |
Liouville's theorem is the foundation on which statistical mechanics |
| 289 |
> |
rests. It describes the time evolution of the phase space |
| 290 |
|
distribution function. In order to calculate the rate of change of |
| 291 |
< |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
| 292 |
< |
consider the two faces perpendicular to the $q_1$ axis, which are |
| 293 |
< |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
| 294 |
< |
leaving the opposite face is given by the expression, |
| 291 |
> |
$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider |
| 292 |
> |
the two faces perpendicular to the $q_1$ axis, which are located at |
| 293 |
> |
$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the |
| 294 |
> |
opposite face is given by the expression, |
| 295 |
|
\begin{equation} |
| 296 |
|
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
| 297 |
|
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
| 315 |
|
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
| 316 |
|
\end{equation} |
| 317 |
|
which cancels the first terms of the right hand side. Furthermore, |
| 318 |
< |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
| 318 |
> |
dividing $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
| 319 |
|
p_f $ in both sides, we can write out Liouville's theorem in a |
| 320 |
|
simple form, |
| 321 |
|
\begin{equation} |
| 324 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
| 325 |
|
\label{introEquation:liouvilleTheorem} |
| 326 |
|
\end{equation} |
| 381 |
– |
|
| 327 |
|
Liouville's theorem states that the distribution function is |
| 328 |
|
constant along any trajectory in phase space. In classical |
| 329 |
< |
statistical mechanics, since the number of particles in the system |
| 330 |
< |
is huge, we may be able to believe the system is stationary, |
| 329 |
> |
statistical mechanics, since the number of system copies in an |
| 330 |
> |
ensemble is huge and constant, we can assume the local density has |
| 331 |
> |
no reason (other than classical mechanics) to change, |
| 332 |
|
\begin{equation} |
| 333 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
| 334 |
|
\label{introEquation:stationary} |
| 337 |
|
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
| 338 |
|
distribution, |
| 339 |
|
\begin{equation} |
| 340 |
< |
\rho \propto e^{ - \beta H} |
| 340 |
> |
\rho \propto e^{ - \beta H}. |
| 341 |
|
\label{introEquation:densityAndHamiltonian} |
| 342 |
|
\end{equation} |
| 343 |
|
|
| 344 |
< |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
| 344 |
> |
\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}} |
| 345 |
|
Lets consider a region in the phase space, |
| 346 |
|
\begin{equation} |
| 347 |
|
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
| 348 |
|
\end{equation} |
| 349 |
|
If this region is small enough, the density $\rho$ can be regarded |
| 350 |
< |
as uniform over the whole phase space. Thus, the number of phase |
| 351 |
< |
points inside this region is given by, |
| 352 |
< |
\begin{equation} |
| 353 |
< |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
| 354 |
< |
dp_1 } ..dp_f. |
| 409 |
< |
\end{equation} |
| 410 |
< |
|
| 411 |
< |
\begin{equation} |
| 412 |
< |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
| 350 |
> |
as uniform over the whole integral. Thus, the number of phase points |
| 351 |
> |
inside this region is given by, |
| 352 |
> |
\begin{eqnarray} |
| 353 |
> |
\delta N &=& \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f,\\ |
| 354 |
> |
\frac{{d(\delta N)}}{{dt}} &=& \frac{{d\rho }}{{dt}}\delta v + \rho |
| 355 |
|
\frac{d}{{dt}}(\delta v) = 0. |
| 356 |
< |
\end{equation} |
| 357 |
< |
With the help of stationary assumption |
| 358 |
< |
(\ref{introEquation:stationary}), we obtain the principle of the |
| 359 |
< |
\emph{conservation of extension in phase space}, |
| 356 |
> |
\end{eqnarray} |
| 357 |
> |
With the help of the stationary assumption |
| 358 |
> |
(Eq.~\ref{introEquation:stationary}), we obtain the principle of |
| 359 |
> |
\emph{conservation of volume in phase space}, |
| 360 |
|
\begin{equation} |
| 361 |
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
| 362 |
|
...dq_f dp_1 } ..dp_f = 0. |
| 363 |
|
\label{introEquation:volumePreserving} |
| 364 |
|
\end{equation} |
| 365 |
|
|
| 366 |
< |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
| 366 |
> |
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
| 367 |
|
|
| 368 |
< |
Liouville's theorem can be expresses in a variety of different forms |
| 368 |
> |
Liouville's theorem can be expressed in a variety of different forms |
| 369 |
|
which are convenient within different contexts. For any two function |
| 370 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
| 371 |
< |
bracket ${F, G}$ is defined as |
| 371 |
> |
bracket $\{F,G\}$ is defined as |
| 372 |
|
\begin{equation} |
| 373 |
|
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
| 374 |
|
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
| 376 |
|
q_i }}} \right)}. |
| 377 |
|
\label{introEquation:poissonBracket} |
| 378 |
|
\end{equation} |
| 379 |
< |
Substituting equations of motion in Hamiltonian formalism( |
| 380 |
< |
\ref{introEquation:motionHamiltonianCoordinate} , |
| 381 |
< |
\ref{introEquation:motionHamiltonianMomentum} ) into |
| 382 |
< |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
| 383 |
< |
theorem using Poisson bracket notion, |
| 379 |
> |
Substituting equations of motion in Hamiltonian formalism |
| 380 |
> |
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
| 381 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into |
| 382 |
> |
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
| 383 |
> |
Liouville's theorem using Poisson bracket notion, |
| 384 |
|
\begin{equation} |
| 385 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
| 386 |
|
{\rho ,H} \right\}. |
| 399 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
| 400 |
|
\label{introEquation:liouvilleTheoremInOperator} |
| 401 |
|
\end{equation} |
| 402 |
< |
|
| 402 |
> |
which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$. |
| 403 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
| 404 |
|
|
| 405 |
|
Various thermodynamic properties can be calculated from Molecular |
| 406 |
|
Dynamics simulation. By comparing experimental values with the |
| 407 |
|
calculated properties, one can determine the accuracy of the |
| 408 |
< |
simulation and the quality of the underlying model. However, both of |
| 409 |
< |
experiment and computer simulation are usually performed during a |
| 408 |
> |
simulation and the quality of the underlying model. However, both |
| 409 |
> |
experiments and computer simulations are usually performed during a |
| 410 |
|
certain time interval and the measurements are averaged over a |
| 411 |
< |
period of them which is different from the average behavior of |
| 412 |
< |
many-body system in Statistical Mechanics. Fortunately, Ergodic |
| 413 |
< |
Hypothesis is proposed to make a connection between time average and |
| 414 |
< |
ensemble average. It states that time average and average over the |
| 415 |
< |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
| 411 |
> |
period of time which is different from the average behavior of |
| 412 |
> |
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 413 |
> |
Hypothesis makes a connection between time average and the ensemble |
| 414 |
> |
average. It states that the time average and average over the |
| 415 |
> |
statistical ensemble are identical:\cite{Frenkel1996, Leach2001} |
| 416 |
|
\begin{equation} |
| 417 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 418 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 421 |
|
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
| 422 |
|
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
| 423 |
|
distribution function. If an observation is averaged over a |
| 424 |
< |
sufficiently long time (longer than relaxation time), all accessible |
| 425 |
< |
microstates in phase space are assumed to be equally probed, giving |
| 426 |
< |
a properly weighted statistical average. This allows the researcher |
| 427 |
< |
freedom of choice when deciding how best to measure a given |
| 428 |
< |
observable. In case an ensemble averaged approach sounds most |
| 429 |
< |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
| 424 |
> |
sufficiently long time (longer than the relaxation time), all |
| 425 |
> |
accessible microstates in phase space are assumed to be equally |
| 426 |
> |
probed, giving a properly weighted statistical average. This allows |
| 427 |
> |
the researcher freedom of choice when deciding how best to measure a |
| 428 |
> |
given observable. In case an ensemble averaged approach sounds most |
| 429 |
> |
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
| 430 |
|
utilized. Or if the system lends itself to a time averaging |
| 431 |
|
approach, the Molecular Dynamics techniques in |
| 432 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 433 |
< |
choice\cite{Frenkel1996}. |
| 433 |
> |
choice.\cite{Frenkel1996} |
| 434 |
|
|
| 435 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 436 |
< |
A variety of numerical integrators were proposed to simulate the |
| 437 |
< |
motions. They usually begin with an initial conditionals and move |
| 438 |
< |
the objects in the direction governed by the differential equations. |
| 439 |
< |
However, most of them ignore the hidden physical law contained |
| 440 |
< |
within the equations. Since 1990, geometric integrators, which |
| 441 |
< |
preserve various phase-flow invariants such as symplectic structure, |
| 442 |
< |
volume and time reversal symmetry, are developed to address this |
| 443 |
< |
issue. The velocity verlet method, which happens to be a simple |
| 444 |
< |
example of symplectic integrator, continues to gain its popularity |
| 445 |
< |
in molecular dynamics community. This fact can be partly explained |
| 446 |
< |
by its geometric nature. |
| 436 |
> |
A variety of numerical integrators have been proposed to simulate |
| 437 |
> |
the motions of atoms in MD simulation. They usually begin with |
| 438 |
> |
initial conditions and move the objects in the direction governed by |
| 439 |
> |
the differential equations. However, most of them ignore the hidden |
| 440 |
> |
physical laws contained within the equations. Since 1990, geometric |
| 441 |
> |
integrators, which preserve various phase-flow invariants such as |
| 442 |
> |
symplectic structure, volume and time reversal symmetry, were |
| 443 |
> |
developed to address this issue.\cite{Dullweber1997, McLachlan1998, |
| 444 |
> |
Leimkuhler1999} The velocity Verlet method, which happens to be a |
| 445 |
> |
simple example of symplectic integrator, continues to gain |
| 446 |
> |
popularity in the molecular dynamics community. This fact can be |
| 447 |
> |
partly explained by its geometric nature. |
| 448 |
|
|
| 449 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 450 |
< |
A \emph{manifold} is an abstract mathematical space. It locally |
| 451 |
< |
looks like Euclidean space, but when viewed globally, it may have |
| 452 |
< |
more complicate structure. A good example of manifold is the surface |
| 453 |
< |
of Earth. It seems to be flat locally, but it is round if viewed as |
| 454 |
< |
a whole. A \emph{differentiable manifold} (also known as |
| 455 |
< |
\emph{smooth manifold}) is a manifold with an open cover in which |
| 456 |
< |
the covering neighborhoods are all smoothly isomorphic to one |
| 457 |
< |
another. In other words,it is possible to apply calculus on |
| 458 |
< |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
| 516 |
< |
defined as a pair $(M, \omega)$ which consisting of a |
| 517 |
< |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 449 |
> |
\subsection{\label{introSection:symplecticManifold}Manifolds and Bundles} |
| 450 |
> |
A \emph{manifold} is an abstract mathematical space. It looks |
| 451 |
> |
locally like Euclidean space, but when viewed globally, it may have |
| 452 |
> |
more complicated structure. A good example of manifold is the |
| 453 |
> |
surface of Earth. It seems to be flat locally, but it is round if |
| 454 |
> |
viewed as a whole. A \emph{differentiable manifold} (also known as |
| 455 |
> |
\emph{smooth manifold}) is a manifold on which it is possible to |
| 456 |
> |
apply calculus.\cite{Hirsch1997} A \emph{symplectic manifold} is |
| 457 |
> |
defined as a pair $(M, \omega)$ which consists of a |
| 458 |
> |
\emph{differentiable manifold} $M$ and a close, non-degenerate, |
| 459 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 460 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
| 461 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 462 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 463 |
< |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
| 464 |
< |
example of symplectic form. |
| 463 |
> |
$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in |
| 464 |
> |
vector field is an example of symplectic form. |
| 465 |
> |
Given vector spaces $V$ and $W$ over same field $F$, $f: V \to W$ is a linear transformation if |
| 466 |
> |
\begin{eqnarray*} |
| 467 |
> |
f(x+y) & = & f(x) + f(y) \\ |
| 468 |
> |
f(ax) & = & af(x) |
| 469 |
> |
\end{eqnarray*} |
| 470 |
> |
are always satisfied for any two vectors $x$ and $y$ in $V$ and any scalar $a$ in $F$. One can define the dual vector space $V^*$ of $V$ if any two built-in linear transformations $\phi$ and $\psi$ in $V^*$ satisfy the following definition of addition and scalar multiplication: |
| 471 |
> |
\begin{eqnarray*} |
| 472 |
> |
(\phi+\psi)(x) & = & \phi(x)+\psi(x) \\ |
| 473 |
> |
(a\phi)(x) & = & a \phi(x) |
| 474 |
> |
\end{eqnarray*} |
| 475 |
> |
for all $a$ in $F$ and $x$ in $V$. For a manifold $M$, one can define a tangent vector of a tangent space $TM_q$ at every point $q$ |
| 476 |
> |
\begin{equation} |
| 477 |
> |
\dot q = \mathop {\lim }\limits_{t \to 0} \frac{{\phi (t) - \phi (0)}}{t} |
| 478 |
> |
\end{equation} |
| 479 |
> |
where $\phi(0)=q$ and $\phi(t) \in M$. One may also define a cotangent space $T^*M_q$ as the dual space of the tangent space $TM_q$. The tangent space and the cotangent space are isomorphic to each other, since they are both real vector spaces with same dimension. |
| 480 |
> |
The union of tangent spaces at every point of $M$ is called the tangent bundle of $M$ and is denoted by $TM$, while cotangent bundle $T^*M$ is defined as the union of the cotangent spaces to $M$.\cite{Jost2002} For a Hamiltonian system with configuration manifold $V$, the $(q,\dot q)$ phase space is the tangent bundle of the configuration manifold $V$, while the cotangent bundle is represented by $(q,p)$. |
| 481 |
|
|
| 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 |
– |
|
| 482 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 483 |
|
|
| 484 |
< |
For a ordinary differential system defined as |
| 484 |
> |
For an ordinary differential system defined as |
| 485 |
|
\begin{equation} |
| 486 |
|
\dot x = f(x) |
| 487 |
|
\end{equation} |
| 488 |
< |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
| 488 |
> |
where $x = x(q,p)$, this system is a canonical Hamiltonian, if |
| 489 |
> |
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian |
| 490 |
> |
function and $J$ is the skew-symmetric matrix |
| 491 |
|
\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} |
| 492 |
|
J = \left( {\begin{array}{*{20}c} |
| 493 |
|
0 & I \\ |
| 494 |
|
{ - I} & 0 \\ |
| 498 |
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
| 499 |
|
system can be rewritten as, |
| 500 |
|
\begin{equation} |
| 501 |
< |
\frac{d}{{dt}}x = J\nabla _x H(x) |
| 501 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x). |
| 502 |
|
\label{introEquation:compactHamiltonian} |
| 503 |
|
\end{equation}In this case, $f$ is |
| 504 |
< |
called a \emph{Hamiltonian vector field}. |
| 505 |
< |
|
| 568 |
< |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
| 504 |
> |
called a \emph{Hamiltonian vector field}. Another generalization of |
| 505 |
> |
Hamiltonian dynamics is Poisson Dynamics,\cite{Olver1986} |
| 506 |
|
\begin{equation} |
| 507 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 508 |
|
\end{equation} |
| 509 |
< |
The most obvious change being that matrix $J$ now depends on $x$. |
| 510 |
< |
The free rigid body is an example of Poisson system (actually a |
| 574 |
< |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
| 575 |
< |
energy. |
| 576 |
< |
\begin{equation} |
| 577 |
< |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 578 |
< |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 579 |
< |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 580 |
< |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 581 |
< |
\end{array}} \right) |
| 582 |
< |
\end{equation} |
| 509 |
> |
where the most obvious change being that matrix $J$ now depends on |
| 510 |
> |
$x$. |
| 511 |
|
|
| 512 |
< |
\begin{equation} |
| 585 |
< |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
| 586 |
< |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
| 587 |
< |
\end{equation} |
| 512 |
> |
\subsection{\label{introSection:exactFlow}Exact Propagator} |
| 513 |
|
|
| 514 |
< |
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 515 |
< |
|
| 591 |
< |
Let $x(t)$ be the exact solution of the ODE system, |
| 514 |
> |
Let $x(t)$ be the exact solution of the ODE |
| 515 |
> |
system, |
| 516 |
|
\begin{equation} |
| 517 |
< |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
| 518 |
< |
\end{equation} |
| 519 |
< |
The exact flow(solution) $\varphi_\tau$ is defined by |
| 520 |
< |
\[ |
| 521 |
< |
x(t+\tau) =\varphi_\tau(x(t)) |
| 517 |
> |
\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE} |
| 518 |
> |
\end{equation} we can |
| 519 |
> |
define its exact propagator $\varphi_\tau$: |
| 520 |
> |
\[ x(t+\tau) |
| 521 |
> |
=\varphi_\tau(x(t)) |
| 522 |
|
\] |
| 523 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 524 |
< |
space to itself. The flow has the continuous group property, |
| 524 |
> |
space to itself. The propagator has the continuous group property, |
| 525 |
|
\begin{equation} |
| 526 |
|
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
| 527 |
|
+ \tau _2 } . |
| 530 |
|
\begin{equation} |
| 531 |
|
\varphi _\tau \circ \varphi _{ - \tau } = I |
| 532 |
|
\end{equation} |
| 533 |
< |
Therefore, the exact flow is self-adjoint, |
| 533 |
> |
Therefore, the exact propagator is self-adjoint, |
| 534 |
|
\begin{equation} |
| 535 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 536 |
|
\end{equation} |
| 537 |
< |
The exact flow can also be written in terms of the of an operator, |
| 537 |
> |
In most cases, it is not easy to find the exact propagator |
| 538 |
> |
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
| 539 |
> |
which is usually called an integrator. The order of an integrator |
| 540 |
> |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
| 541 |
> |
order $p$, |
| 542 |
|
\begin{equation} |
| 543 |
< |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 616 |
< |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 617 |
< |
\label{introEquation:exponentialOperator} |
| 543 |
> |
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 544 |
|
\end{equation} |
| 545 |
|
|
| 620 |
– |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
| 621 |
– |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
| 622 |
– |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
| 623 |
– |
the Taylor series of $\psi_\tau$ agree to order $p$, |
| 624 |
– |
\begin{equation} |
| 625 |
– |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 626 |
– |
\end{equation} |
| 627 |
– |
|
| 546 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 547 |
|
|
| 548 |
< |
The hidden geometric properties of ODE and its flow play important |
| 549 |
< |
roles in numerical studies. Many of them can be found in systems |
| 550 |
< |
which occur naturally in applications. |
| 551 |
< |
|
| 552 |
< |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 635 |
< |
a \emph{symplectic} flow if it satisfies, |
| 548 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
| 549 |
> |
ODE and its propagator play important roles in numerical studies. |
| 550 |
> |
Many of them can be found in systems which occur naturally in |
| 551 |
> |
applications. Let $\varphi$ be the propagator of Hamiltonian vector |
| 552 |
> |
field, $\varphi$ is a \emph{symplectic} propagator if it satisfies, |
| 553 |
|
\begin{equation} |
| 554 |
|
{\varphi '}^T J \varphi ' = J. |
| 555 |
|
\end{equation} |
| 556 |
|
According to Liouville's theorem, the symplectic volume is invariant |
| 557 |
< |
under a Hamiltonian flow, which is the basis for classical |
| 558 |
< |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
| 559 |
< |
field on a symplectic manifold can be shown to be a |
| 557 |
> |
under a Hamiltonian propagator, which is the basis for classical |
| 558 |
> |
statistical mechanics. Furthermore, the propagator of a Hamiltonian |
| 559 |
> |
vector field on a symplectic manifold can be shown to be a |
| 560 |
|
symplectomorphism. As to the Poisson system, |
| 561 |
|
\begin{equation} |
| 562 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 563 |
|
\end{equation} |
| 564 |
< |
is the property must be preserved by the integrator. |
| 565 |
< |
|
| 566 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
| 567 |
< |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
| 568 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
| 569 |
< |
be volume-preserving. |
| 570 |
< |
|
| 654 |
< |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
| 655 |
< |
will result in a new system, |
| 564 |
> |
is the property that must be preserved by the integrator. It is |
| 565 |
> |
possible to construct a \emph{volume-preserving} propagator for a |
| 566 |
> |
source free ODE ($ \nabla \cdot f = 0 $), if the propagator |
| 567 |
> |
satisfies $ \det d\varphi = 1$. One can show easily that a |
| 568 |
> |
symplectic propagator will be volume-preserving. Changing the |
| 569 |
> |
variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will |
| 570 |
> |
result in a new system, |
| 571 |
|
\[ |
| 572 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 573 |
|
\] |
| 574 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 575 |
< |
In other words, the flow of this vector field is reversible if and |
| 576 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 577 |
< |
|
| 578 |
< |
A \emph{first integral}, or conserved quantity of a general |
| 579 |
< |
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 665 |
< |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 575 |
> |
In other words, the propagator of this vector field is reversible if |
| 576 |
> |
and only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
| 577 |
> |
conserved quantity of a general differential function is a function |
| 578 |
> |
$ G:R^{2d} \to R^d $ which is constant for all solutions of the ODE |
| 579 |
> |
$\frac{{dx}}{{dt}} = f(x)$ , |
| 580 |
|
\[ |
| 581 |
|
\frac{{dG(x(t))}}{{dt}} = 0. |
| 582 |
|
\] |
| 583 |
< |
Using chain rule, one may obtain, |
| 583 |
> |
Using the chain rule, one may obtain, |
| 584 |
|
\[ |
| 585 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
| 585 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G, |
| 586 |
|
\] |
| 587 |
< |
which is the condition for conserving \emph{first integral}. For a |
| 588 |
< |
canonical Hamiltonian system, the time evolution of an arbitrary |
| 589 |
< |
smooth function $G$ is given by, |
| 590 |
< |
\begin{equation} |
| 591 |
< |
\begin{array}{c} |
| 592 |
< |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 679 |
< |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 680 |
< |
\end{array} |
| 587 |
> |
which is the condition for conserved quantities. For a canonical |
| 588 |
> |
Hamiltonian system, the time evolution of an arbitrary smooth |
| 589 |
> |
function $G$ is given by, |
| 590 |
> |
\begin{eqnarray} |
| 591 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
| 592 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
| 593 |
|
\label{introEquation:firstIntegral1} |
| 594 |
< |
\end{equation} |
| 595 |
< |
Using poisson bracket notion, Equation |
| 596 |
< |
\ref{introEquation:firstIntegral1} can be rewritten as |
| 594 |
> |
\end{eqnarray} |
| 595 |
> |
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1} |
| 596 |
> |
can be rewritten as |
| 597 |
|
\[ |
| 598 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 599 |
|
\] |
| 600 |
< |
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 601 |
< |
integral} of a Hamiltonian system is |
| 602 |
< |
\[ |
| 603 |
< |
\left\{ {G,H} \right\} = 0. |
| 604 |
< |
\] |
| 605 |
< |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 606 |
< |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 695 |
< |
0$. |
| 600 |
> |
Therefore, the sufficient condition for $G$ to be a conserved |
| 601 |
> |
quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As |
| 602 |
> |
is well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 603 |
> |
is a conserved quantity, which is due to the fact $\{ H,H\} = 0$. |
| 604 |
> |
When designing any numerical methods, one should always try to |
| 605 |
> |
preserve the structural properties of the original ODE and its |
| 606 |
> |
propagator. |
| 607 |
|
|
| 697 |
– |
|
| 698 |
– |
When designing any numerical methods, one should always try to |
| 699 |
– |
preserve the structural properties of the original ODE and its flow. |
| 700 |
– |
|
| 608 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 609 |
|
A lot of well established and very effective numerical methods have |
| 610 |
< |
been successful precisely because of their symplecticities even |
| 610 |
> |
been successful precisely because of their symplectic nature even |
| 611 |
|
though this fact was not recognized when they were first |
| 612 |
< |
constructed. The most famous example is leapfrog methods in |
| 613 |
< |
molecular dynamics. In general, symplectic integrators can be |
| 612 |
> |
constructed. The most famous example is the Verlet-leapfrog method |
| 613 |
> |
in molecular dynamics. In general, symplectic integrators can be |
| 614 |
|
constructed using one of four different methods. |
| 615 |
|
\begin{enumerate} |
| 616 |
|
\item Generating functions |
| 618 |
|
\item Runge-Kutta methods |
| 619 |
|
\item Splitting methods |
| 620 |
|
\end{enumerate} |
| 621 |
+ |
Generating functions\cite{Channell1990} tend to lead to methods |
| 622 |
+ |
which are cumbersome and difficult to use. In dissipative systems, |
| 623 |
+ |
variational methods can capture the decay of energy |
| 624 |
+ |
accurately.\cite{Kane2000} Since they are geometrically unstable |
| 625 |
+ |
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 626 |
+ |
methods are not suitable for Hamiltonian |
| 627 |
+ |
system.\cite{Cartwright1992} Recently, various high-order explicit |
| 628 |
+ |
Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to |
| 629 |
+ |
overcome this instability. However, due to computational penalty |
| 630 |
+ |
involved in implementing the Runge-Kutta methods, they have not |
| 631 |
+ |
attracted much attention from the Molecular Dynamics community. |
| 632 |
+ |
Instead, splitting methods have been widely accepted since they |
| 633 |
+ |
exploit natural decompositions of the system.\cite{McLachlan1998, |
| 634 |
+ |
Tuckerman1992} |
| 635 |
|
|
| 636 |
< |
Generating function tends to lead to methods which are cumbersome |
| 716 |
< |
and difficult to use. In dissipative systems, variational methods |
| 717 |
< |
can capture the decay of energy accurately. Since their |
| 718 |
< |
geometrically unstable nature against non-Hamiltonian perturbations, |
| 719 |
< |
ordinary implicit Runge-Kutta methods are not suitable for |
| 720 |
< |
Hamiltonian system. Recently, various high-order explicit |
| 721 |
< |
Runge--Kutta methods have been developed to overcome this |
| 722 |
< |
instability. However, due to computational penalty involved in |
| 723 |
< |
implementing the Runge-Kutta methods, they do not attract too much |
| 724 |
< |
attention from Molecular Dynamics community. Instead, splitting have |
| 725 |
< |
been widely accepted since they exploit natural decompositions of |
| 726 |
< |
the system\cite{Tuckerman92}. |
| 636 |
> |
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 637 |
|
|
| 728 |
– |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
| 729 |
– |
|
| 638 |
|
The main idea behind splitting methods is to decompose the discrete |
| 639 |
< |
$\varphi_h$ as a composition of simpler flows, |
| 639 |
> |
$\varphi_h$ as a composition of simpler propagators, |
| 640 |
|
\begin{equation} |
| 641 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 642 |
|
\varphi _{h_n } |
| 643 |
|
\label{introEquation:FlowDecomposition} |
| 644 |
|
\end{equation} |
| 645 |
< |
where each of the sub-flow is chosen such that each represent a |
| 646 |
< |
simpler integration of the system. |
| 647 |
< |
|
| 740 |
< |
Suppose that a Hamiltonian system takes the form, |
| 645 |
> |
where each of the sub-propagator is chosen such that each represent |
| 646 |
> |
a simpler integration of the system. Suppose that a Hamiltonian |
| 647 |
> |
system takes the form, |
| 648 |
|
\[ |
| 649 |
|
H = H_1 + H_2. |
| 650 |
|
\] |
| 651 |
|
Here, $H_1$ and $H_2$ may represent different physical processes of |
| 652 |
|
the system. For instance, they may relate to kinetic and potential |
| 653 |
|
energy respectively, which is a natural decomposition of the |
| 654 |
< |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
| 655 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
| 656 |
< |
order is then given by the Lie-Trotter formula |
| 654 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact |
| 655 |
> |
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
| 656 |
> |
simple first order expression is then given by the Lie-Trotter |
| 657 |
> |
formula\cite{Trotter1959} |
| 658 |
|
\begin{equation} |
| 659 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 660 |
|
\label{introEquation:firstOrderSplitting} |
| 663 |
|
continuous $\varphi _i$ over a time $h$. By definition, as |
| 664 |
|
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
| 665 |
|
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
| 666 |
< |
It is easy to show that any composition of symplectic flows yields a |
| 667 |
< |
symplectic map, |
| 666 |
> |
It is easy to show that any composition of symplectic propagators |
| 667 |
> |
yields a symplectic map, |
| 668 |
|
\begin{equation} |
| 669 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 670 |
|
'\phi ' = \phi '^T J\phi ' = J, |
| 672 |
|
\end{equation} |
| 673 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
| 674 |
|
splitting in this context automatically generates a symplectic map. |
| 675 |
< |
|
| 676 |
< |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
| 677 |
< |
introduces local errors proportional to $h^2$, while Strang |
| 678 |
< |
splitting gives a second-order decomposition, |
| 675 |
> |
The Lie-Trotter |
| 676 |
> |
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
| 677 |
> |
local errors proportional to $h^2$, while the Strang splitting gives |
| 678 |
> |
a second-order decomposition,\cite{Strang1968} |
| 679 |
|
\begin{equation} |
| 680 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 681 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 682 |
|
\end{equation} |
| 683 |
< |
which has a local error proportional to $h^3$. Sprang splitting's |
| 684 |
< |
popularity in molecular simulation community attribute to its |
| 685 |
< |
symmetric property, |
| 683 |
> |
which has a local error proportional to $h^3$. The Strang |
| 684 |
> |
splitting's popularity in molecular simulation community attribute |
| 685 |
> |
to its symmetric property, |
| 686 |
|
\begin{equation} |
| 687 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 688 |
|
\label{introEquation:timeReversible} |
| 689 |
|
\end{equation} |
| 690 |
|
|
| 691 |
< |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
| 691 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
| 692 |
|
The classical equation for a system consisting of interacting |
| 693 |
|
particles can be written in Hamiltonian form, |
| 694 |
|
\[ |
| 695 |
|
H = T + V |
| 696 |
|
\] |
| 697 |
|
where $T$ is the kinetic energy and $V$ is the potential energy. |
| 698 |
< |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
| 698 |
> |
Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one |
| 699 |
|
obtains the following: |
| 700 |
|
\begin{align} |
| 701 |
|
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 708 |
|
\end{align} |
| 709 |
|
where $F(t)$ is the force at time $t$. This integration scheme is |
| 710 |
|
known as \emph{velocity verlet} which is |
| 711 |
< |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
| 712 |
< |
time-reversible(\ref{introEquation:timeReversible}) and |
| 713 |
< |
volume-preserving (\ref{introEquation:volumePreserving}). These |
| 711 |
> |
symplectic(Eq.~\ref{introEquation:SymplecticFlowComposition}), |
| 712 |
> |
time-reversible(Eq.~\ref{introEquation:timeReversible}) and |
| 713 |
> |
volume-preserving (Eq.~\ref{introEquation:volumePreserving}). These |
| 714 |
|
geometric properties attribute to its long-time stability and its |
| 715 |
|
popularity in the community. However, the most commonly used |
| 716 |
|
velocity verlet integration scheme is written as below, |
| 722 |
|
\label{introEquation:Lp9b}\\% |
| 723 |
|
% |
| 724 |
|
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 725 |
< |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
| 725 |
> |
\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c} |
| 726 |
|
\end{align} |
| 727 |
|
From the preceding splitting, one can see that the integration of |
| 728 |
|
the equations of motion would follow: |
| 731 |
|
|
| 732 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 733 |
|
|
| 734 |
< |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 734 |
> |
\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move. |
| 735 |
|
|
| 736 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 737 |
|
\end{enumerate} |
| 738 |
< |
|
| 739 |
< |
Simply switching the order of splitting and composing, a new |
| 740 |
< |
integrator, the \emph{position verlet} integrator, can be generated, |
| 738 |
> |
By simply switching the order of the propagators in the splitting |
| 739 |
> |
and composing a new integrator, the \emph{position verlet} |
| 740 |
> |
integrator, can be generated, |
| 741 |
|
\begin{align} |
| 742 |
|
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
| 743 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
| 745 |
|
% |
| 746 |
|
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 747 |
|
q(\Delta t)} \right]. % |
| 748 |
< |
\label{introEquation:positionVerlet1} |
| 748 |
> |
\label{introEquation:positionVerlet2} |
| 749 |
|
\end{align} |
| 750 |
|
|
| 751 |
< |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
| 751 |
> |
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 752 |
|
|
| 753 |
< |
Baker-Campbell-Hausdorff formula can be used to determine the local |
| 754 |
< |
error of splitting method in terms of commutator of the |
| 755 |
< |
operators(\ref{introEquation:exponentialOperator}) associated with |
| 756 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
| 757 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
| 753 |
> |
The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used |
| 754 |
> |
to determine the local error of a splitting method in terms of the |
| 755 |
> |
commutator of the |
| 756 |
> |
operators associated |
| 757 |
> |
with the sub-propagator. For operators $hX$ and $hY$ which are |
| 758 |
> |
associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we |
| 759 |
> |
have |
| 760 |
|
\begin{equation} |
| 761 |
|
\exp (hX + hY) = \exp (hZ) |
| 762 |
|
\end{equation} |
| 765 |
|
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
| 766 |
|
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
| 767 |
|
\end{equation} |
| 768 |
< |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
| 768 |
> |
Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by |
| 769 |
|
\[ |
| 770 |
|
[X,Y] = XY - YX . |
| 771 |
|
\] |
| 772 |
< |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 773 |
< |
can obtain |
| 772 |
> |
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
| 773 |
> |
to the Strang splitting, we can obtain |
| 774 |
|
\begin{eqnarray*} |
| 775 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 776 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 777 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 778 |
< |
\ldots ) |
| 775 |
> |
\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 \\ |
| 776 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 777 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots |
| 778 |
> |
). |
| 779 |
|
\end{eqnarray*} |
| 780 |
< |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 781 |
< |
error of Spring splitting is proportional to $h^3$. The same |
| 782 |
< |
procedure can be applied to general splitting, of the form |
| 780 |
> |
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
| 781 |
> |
error of Strang splitting is proportional to $h^3$. The same |
| 782 |
> |
procedure can be applied to a general splitting of the form |
| 783 |
|
\begin{equation} |
| 784 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 785 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 786 |
|
\end{equation} |
| 787 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
| 788 |
< |
order method. Yoshida proposed an elegant way to compose higher |
| 789 |
< |
order methods based on symmetric splitting. Given a symmetric second |
| 790 |
< |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
| 791 |
< |
method can be constructed by composing, |
| 787 |
> |
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 788 |
> |
order methods. Yoshida proposed an elegant way to compose higher |
| 789 |
> |
order methods based on symmetric splitting.\cite{Yoshida1990} Given |
| 790 |
> |
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 791 |
> |
fourth-order symmetric method can be constructed by composing, |
| 792 |
|
\[ |
| 793 |
|
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
| 794 |
|
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
| 798 |
|
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
| 799 |
|
\begin{equation} |
| 800 |
|
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
| 801 |
< |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
| 801 |
> |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)}, |
| 802 |
|
\end{equation} |
| 803 |
< |
, if the weights are chosen as |
| 803 |
> |
if the weights are chosen as |
| 804 |
|
\[ |
| 805 |
|
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
| 806 |
|
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
| 808 |
|
|
| 809 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 810 |
|
|
| 811 |
< |
As a special discipline of molecular modeling, Molecular dynamics |
| 812 |
< |
has proven to be a powerful tool for studying the functions of |
| 813 |
< |
biological systems, providing structural, thermodynamic and |
| 814 |
< |
dynamical information. |
| 811 |
> |
As one of the principal tools of molecular modeling, Molecular |
| 812 |
> |
dynamics has proven to be a powerful tool for studying the functions |
| 813 |
> |
of biological systems, providing structural, thermodynamic and |
| 814 |
> |
dynamical information. The basic idea of molecular dynamics is that |
| 815 |
> |
macroscopic properties are related to microscopic behavior and |
| 816 |
> |
microscopic behavior can be calculated from the trajectories in |
| 817 |
> |
simulations. For instance, instantaneous temperature of a |
| 818 |
> |
Hamiltonian system of $N$ particles can be measured by |
| 819 |
> |
\[ |
| 820 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
| 821 |
> |
\] |
| 822 |
> |
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
| 823 |
> |
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
| 824 |
> |
the Boltzman constant. |
| 825 |
|
|
| 826 |
< |
\subsection{\label{introSec:mdInit}Initialization} |
| 826 |
> |
A typical molecular dynamics run consists of three essential steps: |
| 827 |
> |
\begin{enumerate} |
| 828 |
> |
\item Initialization |
| 829 |
> |
\begin{enumerate} |
| 830 |
> |
\item Preliminary preparation |
| 831 |
> |
\item Minimization |
| 832 |
> |
\item Heating |
| 833 |
> |
\item Equilibration |
| 834 |
> |
\end{enumerate} |
| 835 |
> |
\item Production |
| 836 |
> |
\item Analysis |
| 837 |
> |
\end{enumerate} |
| 838 |
> |
These three individual steps will be covered in the following |
| 839 |
> |
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 840 |
> |
initialization of a simulation. Sec.~\ref{introSection:production} |
| 841 |
> |
discusses issues of production runs. |
| 842 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 843 |
> |
analysis of trajectories. |
| 844 |
|
|
| 845 |
< |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 845 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 846 |
|
|
| 847 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 847 |
> |
\subsubsection{\textbf{Preliminary preparation}} |
| 848 |
|
|
| 849 |
< |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 849 |
> |
When selecting the starting structure of a molecule for molecular |
| 850 |
> |
simulation, one may retrieve its Cartesian coordinates from public |
| 851 |
> |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
| 852 |
> |
thousands of crystal structures of molecules are discovered every |
| 853 |
> |
year, many more remain unknown due to the difficulties of |
| 854 |
> |
purification and crystallization. Even for molecules with known |
| 855 |
> |
structures, some important information is missing. For example, a |
| 856 |
> |
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 857 |
> |
be added. Moreover, in order to include electrostatic interactions, |
| 858 |
> |
one may need to specify the partial charges for individual atoms. |
| 859 |
> |
Under some circumstances, we may even need to prepare the system in |
| 860 |
> |
a special configuration. For instance, when studying transport |
| 861 |
> |
phenomenon in membrane systems, we may prepare the lipids in a |
| 862 |
> |
bilayer structure instead of placing lipids randomly in solvent, |
| 863 |
> |
since we are not interested in the slow self-aggregation process. |
| 864 |
|
|
| 865 |
< |
Rigid bodies are frequently involved in the modeling of different |
| 915 |
< |
areas, from engineering, physics, to chemistry. For example, |
| 916 |
< |
missiles and vehicle are usually modeled by rigid bodies. The |
| 917 |
< |
movement of the objects in 3D gaming engine or other physics |
| 918 |
< |
simulator is governed by the rigid body dynamics. In molecular |
| 919 |
< |
simulation, rigid body is used to simplify the model in |
| 920 |
< |
protein-protein docking study{\cite{Gray03}}. |
| 865 |
> |
\subsubsection{\textbf{Minimization}} |
| 866 |
|
|
| 867 |
< |
It is very important to develop stable and efficient methods to |
| 868 |
< |
integrate the equations of motion of orientational degrees of |
| 869 |
< |
freedom. Euler angles are the nature choice to describe the |
| 870 |
< |
rotational degrees of freedom. However, due to its singularity, the |
| 871 |
< |
numerical integration of corresponding equations of motion is very |
| 872 |
< |
inefficient and inaccurate. Although an alternative integrator using |
| 873 |
< |
different sets of Euler angles can overcome this difficulty\cite{}, |
| 874 |
< |
the computational penalty and the lost of angular momentum |
| 875 |
< |
conservation still remain. A singularity free representation |
| 876 |
< |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
| 877 |
< |
this approach suffer from the nonseparable Hamiltonian resulted from |
| 878 |
< |
quaternion representation, which prevents the symplectic algorithm |
| 879 |
< |
to be utilized. Another different approach is to apply holonomic |
| 880 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
| 881 |
< |
moves independently under the normal forces deriving from potential |
| 882 |
< |
energy and constraint forces which are used to guarantee the |
| 883 |
< |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
| 884 |
< |
algorithm converge very slowly when the number of constraint |
| 885 |
< |
increases. |
| 867 |
> |
It is quite possible that some of molecules in the system from |
| 868 |
> |
preliminary preparation may be overlapping with each other. This |
| 869 |
> |
close proximity leads to high initial potential energy which |
| 870 |
> |
consequently jeopardizes any molecular dynamics simulations. To |
| 871 |
> |
remove these steric overlaps, one typically performs energy |
| 872 |
> |
minimization to find a more reasonable conformation. Several energy |
| 873 |
> |
minimization methods have been developed to exploit the energy |
| 874 |
> |
surface and to locate the local minimum. While converging slowly |
| 875 |
> |
near the minimum, the steepest descent method is extremely robust when |
| 876 |
> |
systems are strongly anharmonic. Thus, it is often used to refine |
| 877 |
> |
structures from crystallographic data. Relying on the Hessian, |
| 878 |
> |
advanced methods like Newton-Raphson converge rapidly to a local |
| 879 |
> |
minimum, but become unstable if the energy surface is far from |
| 880 |
> |
quadratic. Another factor that must be taken into account, when |
| 881 |
> |
choosing energy minimization method, is the size of the system. |
| 882 |
> |
Steepest descent and conjugate gradient can deal with models of any |
| 883 |
> |
size. Because of the limits on computer memory to store the hessian |
| 884 |
> |
matrix and the computing power needed to diagonalize these matrices, |
| 885 |
> |
most Newton-Raphson methods can not be used with very large systems. |
| 886 |
|
|
| 887 |
< |
The break through in geometric literature suggests that, in order to |
| 943 |
< |
develop a long-term integration scheme, one should preserve the |
| 944 |
< |
symplectic structure of the flow. Introducing conjugate momentum to |
| 945 |
< |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
| 946 |
< |
symplectic integrator, RSHAKE, was proposed to evolve the |
| 947 |
< |
Hamiltonian system in a constraint manifold by iteratively |
| 948 |
< |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
| 949 |
< |
method using quaternion representation was developed by Omelyan. |
| 950 |
< |
However, both of these methods are iterative and inefficient. In |
| 951 |
< |
this section, we will present a symplectic Lie-Poisson integrator |
| 952 |
< |
for rigid body developed by Dullweber and his coworkers\cite{}. |
| 887 |
> |
\subsubsection{\textbf{Heating}} |
| 888 |
|
|
| 889 |
< |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 889 |
> |
Typically, heating is performed by assigning random velocities |
| 890 |
> |
according to a Maxwell-Boltzman distribution for a desired |
| 891 |
> |
temperature. Beginning at a lower temperature and gradually |
| 892 |
> |
increasing the temperature by assigning larger random velocities, we |
| 893 |
> |
end up setting the temperature of the system to a final temperature |
| 894 |
> |
at which the simulation will be conducted. In the heating phase, we |
| 895 |
> |
should also keep the system from drifting or rotating as a whole. To |
| 896 |
> |
do this, the net linear momentum and angular momentum of the system |
| 897 |
> |
is shifted to zero after each resampling from the Maxwell -Boltzman |
| 898 |
> |
distribution. |
| 899 |
|
|
| 900 |
< |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 900 |
> |
\subsubsection{\textbf{Equilibration}} |
| 901 |
> |
|
| 902 |
> |
The purpose of equilibration is to allow the system to evolve |
| 903 |
> |
spontaneously for a period of time and reach equilibrium. The |
| 904 |
> |
procedure is continued until various statistical properties, such as |
| 905 |
> |
temperature, pressure, energy, volume and other structural |
| 906 |
> |
properties \textit{etc}, become independent of time. Strictly |
| 907 |
> |
speaking, minimization and heating are not necessary, provided the |
| 908 |
> |
equilibration process is long enough. However, these steps can serve |
| 909 |
> |
as a mean to arrive at an equilibrated structure in an effective |
| 910 |
> |
way. |
| 911 |
> |
|
| 912 |
> |
\subsection{\label{introSection:production}Production} |
| 913 |
> |
|
| 914 |
> |
The production run is the most important step of the simulation, in |
| 915 |
> |
which the equilibrated structure is used as a starting point and the |
| 916 |
> |
motions of the molecules are collected for later analysis. In order |
| 917 |
> |
to capture the macroscopic properties of the system, the molecular |
| 918 |
> |
dynamics simulation must be performed by sampling correctly and |
| 919 |
> |
efficiently from the relevant thermodynamic ensemble. |
| 920 |
> |
|
| 921 |
> |
The most expensive part of a molecular dynamics simulation is the |
| 922 |
> |
calculation of non-bonded forces, such as van der Waals force and |
| 923 |
> |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
| 924 |
> |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
| 925 |
> |
which makes large simulations prohibitive in the absence of any |
| 926 |
> |
algorithmic tricks. A natural approach to avoid system size issues |
| 927 |
> |
is to represent the bulk behavior by a finite number of the |
| 928 |
> |
particles. However, this approach will suffer from surface effects |
| 929 |
> |
at the edges of the simulation. To offset this, \textit{Periodic |
| 930 |
> |
boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to |
| 931 |
> |
simulate bulk properties with a relatively small number of |
| 932 |
> |
particles. In this method, the simulation box is replicated |
| 933 |
> |
throughout space to form an infinite lattice. During the simulation, |
| 934 |
> |
when a particle moves in the primary cell, its image in other cells |
| 935 |
> |
move in exactly the same direction with exactly the same |
| 936 |
> |
orientation. Thus, as a particle leaves the primary cell, one of its |
| 937 |
> |
images will enter through the opposite face. |
| 938 |
> |
\begin{figure} |
| 939 |
> |
\centering |
| 940 |
> |
\includegraphics[width=\linewidth]{pbc.eps} |
| 941 |
> |
\caption[An illustration of periodic boundary conditions]{A 2-D |
| 942 |
> |
illustration of periodic boundary conditions. As one particle leaves |
| 943 |
> |
the left of the simulation box, an image of it enters the right.} |
| 944 |
> |
\label{introFig:pbc} |
| 945 |
> |
\end{figure} |
| 946 |
|
|
| 947 |
+ |
%cutoff and minimum image convention |
| 948 |
+ |
Another important technique to improve the efficiency of force |
| 949 |
+ |
evaluation is to apply spherical cutoffs where particles farther |
| 950 |
+ |
than a predetermined distance are not included in the |
| 951 |
+ |
calculation.\cite{Frenkel1996} The use of a cutoff radius will cause |
| 952 |
+ |
a discontinuity in the potential energy curve. Fortunately, one can |
| 953 |
+ |
shift a simple radial potential to ensure the potential curve go |
| 954 |
+ |
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 955 |
+ |
well for Lennard-Jones interaction because of its short range |
| 956 |
+ |
nature. However, simply truncating the electrostatic interaction |
| 957 |
+ |
with the use of cutoffs has been shown to lead to severe artifacts |
| 958 |
+ |
in simulations. The Ewald summation, in which the slowly decaying |
| 959 |
+ |
Coulomb potential is transformed into direct and reciprocal sums |
| 960 |
+ |
with rapid and absolute convergence, has proved to minimize the |
| 961 |
+ |
periodicity artifacts in liquid simulations. Taking advantage of |
| 962 |
+ |
fast Fourier transform (FFT) techniques for calculating discrete |
| 963 |
+ |
Fourier transforms, the particle mesh-based |
| 964 |
+ |
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 965 |
+ |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
| 966 |
+ |
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
| 967 |
+ |
which treats Coulombic interactions exactly at short range, and |
| 968 |
+ |
approximate the potential at long range through multipolar |
| 969 |
+ |
expansion. In spite of their wide acceptance at the molecular |
| 970 |
+ |
simulation community, these two methods are difficult to implement |
| 971 |
+ |
correctly and efficiently. Instead, we use a damped and |
| 972 |
+ |
charge-neutralized Coulomb potential method developed by Wolf and |
| 973 |
+ |
his coworkers.\cite{Wolf1999} The shifted Coulomb potential for |
| 974 |
+ |
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 975 |
|
\begin{equation} |
| 976 |
< |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 977 |
< |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 978 |
< |
\label{introEquation:RBHamiltonian} |
| 976 |
> |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 977 |
> |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 978 |
> |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
| 979 |
> |
r_{ij})}{r_{ij}}\right\}, \label{introEquation:shiftedCoulomb} |
| 980 |
|
\end{equation} |
| 981 |
< |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 982 |
< |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 983 |
< |
$J$, a diagonal matrix, is defined by |
| 981 |
> |
where $\alpha$ is the convergence parameter. Due to the lack of |
| 982 |
> |
inherent periodicity and rapid convergence,this method is extremely |
| 983 |
> |
efficient and easy to implement. |
| 984 |
> |
\begin{figure} |
| 985 |
> |
\centering |
| 986 |
> |
\includegraphics[width=\linewidth]{shifted_coulomb.eps} |
| 987 |
> |
\caption[An illustration of shifted Coulomb potential]{An |
| 988 |
> |
illustration of shifted Coulomb potential.} |
| 989 |
> |
\label{introFigure:shiftedCoulomb} |
| 990 |
> |
\end{figure} |
| 991 |
> |
|
| 992 |
> |
\subsection{\label{introSection:Analysis} Analysis} |
| 993 |
> |
|
| 994 |
> |
According to the principles of |
| 995 |
> |
Statistical Mechanics in |
| 996 |
> |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 997 |
> |
thermodynamic properties, analyze fluctuations of structural |
| 998 |
> |
parameters, and investigate time-dependent processes of the molecule |
| 999 |
> |
from the trajectories. |
| 1000 |
> |
|
| 1001 |
> |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
| 1002 |
> |
|
| 1003 |
> |
Thermodynamic properties, which can be expressed in terms of some |
| 1004 |
> |
function of the coordinates and momenta of all particles in the |
| 1005 |
> |
system, can be directly computed from molecular dynamics. The usual |
| 1006 |
> |
way to measure the pressure is based on virial theorem of Clausius |
| 1007 |
> |
which states that the virial is equal to $-3Nk_BT$. For a system |
| 1008 |
> |
with forces between particles, the total virial, $W$, contains the |
| 1009 |
> |
contribution from external pressure and interaction between the |
| 1010 |
> |
particles: |
| 1011 |
|
\[ |
| 1012 |
< |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1012 |
> |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
| 1013 |
> |
f_{ij} } } \right\rangle |
| 1014 |
|
\] |
| 1015 |
< |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1016 |
< |
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 1015 |
> |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
| 1016 |
> |
distance $r_{ij}$. Thus, the expression for the pressure is given |
| 1017 |
> |
by: |
| 1018 |
|
\begin{equation} |
| 1019 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 1019 |
> |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
| 1020 |
> |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
| 1021 |
|
\end{equation} |
| 1022 |
< |
which is used to ensure rotation matrix's orthogonality. |
| 1023 |
< |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 1024 |
< |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1022 |
> |
|
| 1023 |
> |
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
| 1024 |
> |
|
| 1025 |
> |
Structural Properties of a simple fluid can be described by a set of |
| 1026 |
> |
distribution functions. Among these functions,the \emph{pair |
| 1027 |
> |
distribution function}, also known as \emph{radial distribution |
| 1028 |
> |
function}, is of most fundamental importance to liquid theory. |
| 1029 |
> |
Experimentally, pair distribution functions can be gathered by |
| 1030 |
> |
Fourier transforming raw data from a series of neutron diffraction |
| 1031 |
> |
experiments and integrating over the surface |
| 1032 |
> |
factor.\cite{Powles1973} The experimental results can serve as a |
| 1033 |
> |
criterion to justify the correctness of a liquid model. Moreover, |
| 1034 |
> |
various equilibrium thermodynamic and structural properties can also |
| 1035 |
> |
be expressed in terms of the radial distribution |
| 1036 |
> |
function.\cite{Allen1987} The pair distribution functions $g(r)$ |
| 1037 |
> |
gives the probability that a particle $i$ will be located at a |
| 1038 |
> |
distance $r$ from a another particle $j$ in the system |
| 1039 |
|
\begin{equation} |
| 1040 |
< |
Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0 . \\ |
| 1041 |
< |
\label{introEquation:RBFirstOrderConstraint} |
| 1040 |
> |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1041 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
| 1042 |
> |
(r)}{\rho}. |
| 1043 |
|
\end{equation} |
| 1044 |
+ |
Note that the delta function can be replaced by a histogram in |
| 1045 |
+ |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
| 1046 |
+ |
the height of these peaks gradually decreases to 1 as the liquid of |
| 1047 |
+ |
large distance approaches the bulk density. |
| 1048 |
|
|
| 982 |
– |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 983 |
– |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 984 |
– |
the equations of motion, |
| 985 |
– |
\[ |
| 986 |
– |
\begin{array}{c} |
| 987 |
– |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 988 |
– |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 989 |
– |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 990 |
– |
\frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 991 |
– |
\end{array} |
| 992 |
– |
\] |
| 1049 |
|
|
| 1050 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
| 1051 |
+ |
Properties}} |
| 1052 |
|
|
| 1053 |
< |
\[ |
| 1054 |
< |
M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} |
| 1055 |
< |
\right\} . |
| 1056 |
< |
\] |
| 1053 |
> |
Time-dependent properties are usually calculated using \emph{time |
| 1054 |
> |
correlation functions}, which correlate random variables $A$ and $B$ |
| 1055 |
> |
at two different times, |
| 1056 |
> |
\begin{equation} |
| 1057 |
> |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
| 1058 |
> |
\label{introEquation:timeCorrelationFunction} |
| 1059 |
> |
\end{equation} |
| 1060 |
> |
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1061 |
> |
functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation |
| 1062 |
> |
function which is directly related to transport properties of |
| 1063 |
> |
molecular liquids: |
| 1064 |
> |
\begin{equation} |
| 1065 |
> |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
| 1066 |
> |
\right\rangle } dt |
| 1067 |
> |
\end{equation} |
| 1068 |
> |
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1069 |
> |
function, which is averaged over time origins and over all the |
| 1070 |
> |
atoms, the dipole autocorrelation functions is calculated for the |
| 1071 |
> |
entire system. The dipole autocorrelation function is given by: |
| 1072 |
> |
\begin{equation} |
| 1073 |
> |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1074 |
> |
\right\rangle |
| 1075 |
> |
\end{equation} |
| 1076 |
> |
Here $u_{tot}$ is the net dipole of the entire system and is given |
| 1077 |
> |
by |
| 1078 |
> |
\begin{equation} |
| 1079 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
| 1080 |
> |
\end{equation} |
| 1081 |
> |
In principle, many time correlation functions can be related to |
| 1082 |
> |
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1083 |
> |
scattering spectra of molecular liquids. In practice, one can |
| 1084 |
> |
extract the IR spectrum from the intensity of the molecular dipole |
| 1085 |
> |
fluctuation at each frequency using the following relationship: |
| 1086 |
> |
\begin{equation} |
| 1087 |
> |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1088 |
> |
i2\pi vt} dt}. |
| 1089 |
> |
\end{equation} |
| 1090 |
|
|
| 1091 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 1091 |
> |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1092 |
|
|
| 1093 |
< |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations} |
| 1093 |
> |
Rigid bodies are frequently involved in the modeling of different |
| 1094 |
> |
areas, including engineering, physics and chemistry. For example, |
| 1095 |
> |
missiles and vehicles are usually modeled by rigid bodies. The |
| 1096 |
> |
movement of the objects in 3D gaming engines or other physics |
| 1097 |
> |
simulators is governed by rigid body dynamics. In molecular |
| 1098 |
> |
simulations, rigid bodies are used to simplify protein-protein |
| 1099 |
> |
docking studies.\cite{Gray2003} |
| 1100 |
|
|
| 1101 |
+ |
It is very important to develop stable and efficient methods to |
| 1102 |
+ |
integrate the equations of motion for orientational degrees of |
| 1103 |
+ |
freedom. Euler angles are the natural choice to describe the |
| 1104 |
+ |
rotational degrees of freedom. However, due to $\frac {1}{sin |
| 1105 |
+ |
\theta}$ singularities, the numerical integration of corresponding |
| 1106 |
+ |
equations of these motion is very inefficient and inaccurate. |
| 1107 |
+ |
Although an alternative integrator using multiple sets of Euler |
| 1108 |
+ |
angles can overcome this difficulty\cite{Barojas1973}, the |
| 1109 |
+ |
computational penalty and the loss of angular momentum conservation |
| 1110 |
+ |
still remain. A singularity-free representation utilizing |
| 1111 |
+ |
quaternions was developed by Evans in 1977.\cite{Evans1977} |
| 1112 |
+ |
Unfortunately, this approach used a nonseparable Hamiltonian |
| 1113 |
+ |
resulting from the quaternion representation, which prevented the |
| 1114 |
+ |
symplectic algorithm from being utilized. Another different approach |
| 1115 |
+ |
is to apply holonomic constraints to the atoms belonging to the |
| 1116 |
+ |
rigid body. Each atom moves independently under the normal forces |
| 1117 |
+ |
deriving from potential energy and constraint forces which are used |
| 1118 |
+ |
to guarantee the rigidness. However, due to their iterative nature, |
| 1119 |
+ |
the SHAKE and Rattle algorithms also converge very slowly when the |
| 1120 |
+ |
number of constraints increases.\cite{Ryckaert1977, Andersen1983} |
| 1121 |
|
|
| 1122 |
< |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1122 |
> |
A break-through in geometric literature suggests that, in order to |
| 1123 |
> |
develop a long-term integration scheme, one should preserve the |
| 1124 |
> |
symplectic structure of the propagator. By introducing a conjugate |
| 1125 |
> |
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
| 1126 |
> |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
| 1127 |
> |
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1128 |
> |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 1129 |
> |
An alternative method using the quaternion representation was |
| 1130 |
> |
developed by Omelyan.\cite{Omelyan1998} However, both of these |
| 1131 |
> |
methods are iterative and inefficient. In this section, we descibe a |
| 1132 |
> |
symplectic Lie-Poisson integrator for rigid bodies developed by |
| 1133 |
> |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1134 |
|
|
| 1135 |
< |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 1136 |
< |
|
| 1009 |
< |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 1010 |
< |
|
| 1135 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1136 |
> |
The Hamiltonian of a rigid body is given by |
| 1137 |
|
\begin{equation} |
| 1138 |
< |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1139 |
< |
\label{introEquation:bathGLE} |
| 1138 |
> |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1139 |
> |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1140 |
> |
\label{introEquation:RBHamiltonian} |
| 1141 |
|
\end{equation} |
| 1142 |
< |
where $H_B$ is harmonic bath Hamiltonian, |
| 1142 |
> |
Here, $q$ and $Q$ are the position vector and rotation matrix for |
| 1143 |
> |
the rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , |
| 1144 |
> |
and $J$, a diagonal matrix, is defined by |
| 1145 |
|
\[ |
| 1146 |
< |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1018 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
| 1146 |
> |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1147 |
|
\] |
| 1148 |
< |
and $\Delta U$ is bilinear system-bath coupling, |
| 1149 |
< |
\[ |
| 1150 |
< |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1151 |
< |
\] |
| 1152 |
< |
Completing the square, |
| 1148 |
> |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1149 |
> |
constrained Hamiltonian equation is subjected to a holonomic |
| 1150 |
> |
constraint, |
| 1151 |
> |
\begin{equation} |
| 1152 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1153 |
> |
\end{equation} |
| 1154 |
> |
which is used to ensure the rotation matrix's unitarity. Using |
| 1155 |
> |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~ |
| 1156 |
> |
\ref{introEquation:motionHamiltonianMomentum}, one can write down |
| 1157 |
> |
the equations of motion, |
| 1158 |
> |
\begin{eqnarray} |
| 1159 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
| 1160 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\ |
| 1161 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
| 1162 |
> |
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1163 |
> |
\end{eqnarray} |
| 1164 |
> |
Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and |
| 1165 |
> |
using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1166 |
> |
\begin{equation} |
| 1167 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1168 |
> |
\label{introEquation:RBFirstOrderConstraint} |
| 1169 |
> |
\end{equation} |
| 1170 |
> |
In general, there are two ways to satisfy the holonomic constraints. |
| 1171 |
> |
We can use a constraint force provided by a Lagrange multiplier on |
| 1172 |
> |
the normal manifold to keep the motion on the constraint space. Or |
| 1173 |
> |
we can simply evolve the system on the constraint manifold. These |
| 1174 |
> |
two methods have been proved to be equivalent. The holonomic |
| 1175 |
> |
constraint and equations of motions define a constraint manifold for |
| 1176 |
> |
rigid bodies |
| 1177 |
|
\[ |
| 1178 |
< |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
| 1179 |
< |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1028 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1029 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
| 1030 |
< |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1178 |
> |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1179 |
> |
\right\}. |
| 1180 |
|
\] |
| 1181 |
< |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
| 1181 |
> |
Unfortunately, this constraint manifold is not $T^* SO(3)$ which is |
| 1182 |
> |
a symplectic manifold on Lie rotation group $SO(3)$. However, it |
| 1183 |
> |
turns out that under symplectic transformation, the cotangent space |
| 1184 |
> |
and the phase space are diffeomorphic. By introducing |
| 1185 |
|
\[ |
| 1186 |
< |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1035 |
< |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1036 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1037 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1186 |
> |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1187 |
|
\] |
| 1188 |
< |
where |
| 1188 |
> |
the mechanical system subjected to a holonomic constraint manifold $M$ |
| 1189 |
> |
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1190 |
|
\[ |
| 1191 |
< |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1192 |
< |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1191 |
> |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1192 |
> |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1193 |
|
\] |
| 1194 |
< |
Since the first two terms of the new Hamiltonian depend only on the |
| 1195 |
< |
system coordinates, we can get the equations of motion for |
| 1196 |
< |
Generalized Langevin Dynamics by Hamilton's equations |
| 1197 |
< |
\ref{introEquation:motionHamiltonianCoordinate, |
| 1198 |
< |
introEquation:motionHamiltonianMomentum}, |
| 1199 |
< |
\begin{align} |
| 1200 |
< |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 1051 |
< |
&= m\ddot x |
| 1052 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
| 1053 |
< |
\label{introEquation:Lp5} |
| 1054 |
< |
\end{align} |
| 1055 |
< |
, and |
| 1056 |
< |
\begin{align} |
| 1057 |
< |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
| 1058 |
< |
&= m\ddot x_\alpha |
| 1059 |
< |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
| 1060 |
< |
\end{align} |
| 1061 |
< |
|
| 1062 |
< |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
| 1063 |
< |
|
| 1194 |
> |
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1195 |
> |
the rigid body, its corresponding lab fixed vector $X_i^{lab}$ is |
| 1196 |
> |
given as |
| 1197 |
> |
\begin{equation} |
| 1198 |
> |
X_i^{lab} = Q X_i + q. |
| 1199 |
> |
\end{equation} |
| 1200 |
> |
Therefore, potential energy $V(q,Q)$ is defined by |
| 1201 |
|
\[ |
| 1202 |
< |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
| 1202 |
> |
V(q,Q) = V(Q X_0 + q). |
| 1203 |
|
\] |
| 1204 |
< |
|
| 1204 |
> |
Hence, the force and torque are given by |
| 1205 |
|
\[ |
| 1206 |
< |
L(x + y) = L(x) + L(y) |
| 1206 |
> |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, |
| 1207 |
|
\] |
| 1208 |
< |
|
| 1208 |
> |
and |
| 1209 |
|
\[ |
| 1210 |
< |
L(ax) = aL(x) |
| 1210 |
> |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1211 |
|
\] |
| 1212 |
< |
|
| 1212 |
> |
respectively. As a common choice to describe the rotation dynamics |
| 1213 |
> |
of the rigid body, the angular momentum on the body fixed frame $\Pi |
| 1214 |
> |
= Q^t P$ is introduced to rewrite the equations of motion, |
| 1215 |
> |
\begin{equation} |
| 1216 |
> |
\begin{array}{l} |
| 1217 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda, \\ |
| 1218 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1}, \\ |
| 1219 |
> |
\end{array} |
| 1220 |
> |
\label{introEqaution:RBMotionPI} |
| 1221 |
> |
\end{equation} |
| 1222 |
> |
as well as holonomic constraints $\Pi J^{ - 1} + J^{ - 1} \Pi ^t = |
| 1223 |
> |
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a |
| 1224 |
> |
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism, |
| 1225 |
> |
\begin{equation} |
| 1226 |
> |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1227 |
> |
{\begin{array}{*{20}c} |
| 1228 |
> |
0 & { - v_3 } & {v_2 } \\ |
| 1229 |
> |
{v_3 } & 0 & { - v_1 } \\ |
| 1230 |
> |
{ - v_2 } & {v_1 } & 0 \\ |
| 1231 |
> |
\end{array}} \right), |
| 1232 |
> |
\label{introEquation:hatmapIsomorphism} |
| 1233 |
> |
\end{equation} |
| 1234 |
> |
will let us associate the matrix products with traditional vector |
| 1235 |
> |
operations |
| 1236 |
|
\[ |
| 1237 |
< |
L(\dot x) = pL(x) - px(0) |
| 1237 |
> |
\hat vu = v \times u. |
| 1238 |
|
\] |
| 1239 |
< |
|
| 1239 |
> |
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1240 |
> |
matrix, |
| 1241 |
> |
\begin{eqnarray} |
| 1242 |
> |
(\dot \Pi - \dot \Pi ^T )&= &(\Pi - \Pi ^T )(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
| 1243 |
> |
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1244 |
> |
(\Lambda - \Lambda ^T ). \label{introEquation:skewMatrixPI} |
| 1245 |
> |
\end{eqnarray} |
| 1246 |
> |
Since $\Lambda$ is symmetric, the last term of |
| 1247 |
> |
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
| 1248 |
> |
Lagrange multiplier $\Lambda$ is absent from the equations of |
| 1249 |
> |
motion. This unique property eliminates the requirement of |
| 1250 |
> |
iterations which can not be avoided in other methods.\cite{Kol1997, |
| 1251 |
> |
Omelyan1998} Applying the hat-map isomorphism, we obtain the |
| 1252 |
> |
equation of motion for angular momentum |
| 1253 |
> |
\begin{equation} |
| 1254 |
> |
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1255 |
> |
F_i (r,Q)} \right) \times X_i }. |
| 1256 |
> |
\label{introEquation:bodyAngularMotion} |
| 1257 |
> |
\end{equation} |
| 1258 |
> |
In the same manner, the equation of motion for rotation matrix is |
| 1259 |
> |
given by |
| 1260 |
|
\[ |
| 1261 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
| 1261 |
> |
\dot Q = Qskew(I^{ - 1} \pi ). |
| 1262 |
|
\] |
| 1263 |
|
|
| 1264 |
< |
\[ |
| 1265 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
| 1086 |
< |
\] |
| 1264 |
> |
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1265 |
> |
Lie-Poisson Integrator for Free Rigid Bodies} |
| 1266 |
|
|
| 1267 |
< |
Some relatively important transformation, |
| 1268 |
< |
\[ |
| 1269 |
< |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
| 1267 |
> |
If there are no external forces exerted on the rigid body, the only |
| 1268 |
> |
contribution to the rotational motion is from the kinetic energy |
| 1269 |
> |
(the first term of \ref{introEquation:bodyAngularMotion}). The free |
| 1270 |
> |
rigid body is an example of a Lie-Poisson system with Hamiltonian |
| 1271 |
> |
function |
| 1272 |
> |
\begin{equation} |
| 1273 |
> |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1274 |
> |
\label{introEquation:rotationalKineticRB} |
| 1275 |
> |
\end{equation} |
| 1276 |
> |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
| 1277 |
> |
Lie-Poisson structure matrix, |
| 1278 |
> |
\begin{equation} |
| 1279 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 1280 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1281 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1282 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1283 |
> |
\end{array}} \right). |
| 1284 |
> |
\end{equation} |
| 1285 |
> |
Thus, the dynamics of free rigid body is governed by |
| 1286 |
> |
\begin{equation} |
| 1287 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
| 1288 |
> |
\end{equation} |
| 1289 |
> |
One may notice that each $T_i^r$ in |
| 1290 |
> |
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly. |
| 1291 |
> |
For instance, the equations of motion due to $T_1^r$ are given by |
| 1292 |
> |
\begin{equation} |
| 1293 |
> |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
| 1294 |
> |
\label{introEqaution:RBMotionSingleTerm} |
| 1295 |
> |
\end{equation} |
| 1296 |
> |
with |
| 1297 |
> |
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1298 |
> |
0 & 0 & 0 \\ |
| 1299 |
> |
0 & 0 & {\pi _1 } \\ |
| 1300 |
> |
0 & { - \pi _1 } & 0 \\ |
| 1301 |
> |
\end{array}} \right). |
| 1302 |
|
\] |
| 1303 |
< |
|
| 1303 |
> |
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is |
| 1304 |
|
\[ |
| 1305 |
< |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
| 1305 |
> |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
| 1306 |
> |
Q(0)e^{\Delta tR_1 } |
| 1307 |
|
\] |
| 1308 |
< |
|
| 1308 |
> |
with |
| 1309 |
|
\[ |
| 1310 |
< |
L(1) = \frac{1}{p} |
| 1310 |
> |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
| 1311 |
> |
0 & 0 & 0 \\ |
| 1312 |
> |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
| 1313 |
> |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1314 |
> |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1315 |
> |
\] |
| 1316 |
> |
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1317 |
> |
tR_1 }$, we can use the Cayley transformation to obtain a |
| 1318 |
> |
single-aixs propagator, |
| 1319 |
> |
\begin{eqnarray*} |
| 1320 |
> |
e^{\Delta tR_1 } & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta |
| 1321 |
> |
tR_1 ) \\ |
| 1322 |
> |
% |
| 1323 |
> |
& \approx & \left( \begin{array}{ccc} |
| 1324 |
> |
1 & 0 & 0 \\ |
| 1325 |
> |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
| 1326 |
> |
\theta^2 / 4} \\ |
| 1327 |
> |
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
| 1328 |
> |
\theta^2 / 4} |
| 1329 |
> |
\end{array} |
| 1330 |
> |
\right). |
| 1331 |
> |
\end{eqnarray*} |
| 1332 |
> |
The propagators for $T_2^r$ and $T_3^r$ can be found in the same |
| 1333 |
> |
manner. In order to construct a second-order symplectic method, we |
| 1334 |
> |
split the angular kinetic Hamiltonian function into five terms |
| 1335 |
> |
\[ |
| 1336 |
> |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1337 |
> |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1338 |
> |
(\pi _1 ). |
| 1339 |
|
\] |
| 1340 |
+ |
By concatenating the propagators corresponding to these five terms, |
| 1341 |
+ |
we can obtain an symplectic integrator, |
| 1342 |
+ |
\[ |
| 1343 |
+ |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1344 |
+ |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1345 |
+ |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1346 |
+ |
_1 }. |
| 1347 |
+ |
\] |
| 1348 |
+ |
The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by |
| 1349 |
+ |
\[ |
| 1350 |
+ |
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1351 |
+ |
). |
| 1352 |
+ |
\] |
| 1353 |
+ |
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1354 |
+ |
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1355 |
+ |
conserved quantity in Poisson system. We can easily verify that the |
| 1356 |
+ |
norm of the angular momentum, $\parallel \pi |
| 1357 |
+ |
\parallel$, is a \emph{Casimir}.\cite{McLachlan1993} Let $F(\pi ) = S(\frac{{\parallel |
| 1358 |
+ |
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1359 |
+ |
then by the chain rule |
| 1360 |
+ |
\[ |
| 1361 |
+ |
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1362 |
+ |
}}{2})\pi. |
| 1363 |
+ |
\] |
| 1364 |
+ |
Thus, $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel |
| 1365 |
+ |
\pi |
| 1366 |
+ |
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1367 |
+ |
Lie-Poisson integrator is found to be both extremely efficient and |
| 1368 |
+ |
stable. These properties can be explained by the fact the small |
| 1369 |
+ |
angle approximation is used and the norm of the angular momentum is |
| 1370 |
+ |
conserved. |
| 1371 |
|
|
| 1372 |
< |
First, the bath coordinates, |
| 1372 |
> |
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
| 1373 |
> |
Splitting for Rigid Body} |
| 1374 |
> |
|
| 1375 |
> |
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1376 |
> |
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations |
| 1377 |
> |
of motion corresponding to potential energy and kinetic energy are |
| 1378 |
> |
listed in Table~\ref{introTable:rbEquations}. |
| 1379 |
> |
\begin{table} |
| 1380 |
> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1381 |
> |
\label{introTable:rbEquations} |
| 1382 |
> |
\begin{center} |
| 1383 |
> |
\begin{tabular}{|l|l|} |
| 1384 |
> |
\hline |
| 1385 |
> |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 1386 |
> |
Potential & Kinetic \\ |
| 1387 |
> |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
| 1388 |
> |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
| 1389 |
> |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
| 1390 |
> |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
| 1391 |
> |
\hline |
| 1392 |
> |
\end{tabular} |
| 1393 |
> |
\end{center} |
| 1394 |
> |
\end{table} |
| 1395 |
> |
A second-order symplectic method is now obtained by the composition |
| 1396 |
> |
of the position and velocity propagators, |
| 1397 |
|
\[ |
| 1398 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
| 1399 |
< |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
| 1105 |
< |
}}L(x) |
| 1398 |
> |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1399 |
> |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1400 |
|
\] |
| 1401 |
+ |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 1402 |
+ |
sub-propagators which corresponding to force and torque |
| 1403 |
+ |
respectively, |
| 1404 |
|
\[ |
| 1405 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
| 1406 |
< |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
| 1405 |
> |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1406 |
> |
_{\Delta t/2,\tau }. |
| 1407 |
|
\] |
| 1408 |
< |
Then, the system coordinates, |
| 1409 |
< |
\begin{align} |
| 1410 |
< |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1411 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
| 1412 |
< |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
| 1413 |
< |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
| 1414 |
< |
}}\omega _\alpha ^2 L(x)} \right\}} |
| 1415 |
< |
% |
| 1416 |
< |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1417 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
| 1418 |
< |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
| 1419 |
< |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
| 1420 |
< |
\end{align} |
| 1421 |
< |
Then, the inverse transform, |
| 1408 |
> |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1409 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order |
| 1410 |
> |
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the |
| 1411 |
> |
kinetic energy can be separated to translational kinetic term, $T^t |
| 1412 |
> |
(p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1413 |
> |
\begin{equation} |
| 1414 |
> |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1415 |
> |
\end{equation} |
| 1416 |
> |
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1417 |
> |
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore, |
| 1418 |
> |
the corresponding propagators are given by |
| 1419 |
> |
\[ |
| 1420 |
> |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1421 |
> |
_{\Delta t,T^r }. |
| 1422 |
> |
\] |
| 1423 |
> |
Finally, we obtain the overall symplectic propagators for freely |
| 1424 |
> |
moving rigid bodies |
| 1425 |
> |
\begin{eqnarray} |
| 1426 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
| 1427 |
> |
& & \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\\ |
| 1428 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} . |
| 1429 |
> |
\label{introEquation:overallRBFlowMaps} |
| 1430 |
> |
\end{eqnarray} |
| 1431 |
|
|
| 1432 |
< |
\begin{align} |
| 1433 |
< |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
| 1432 |
> |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1433 |
> |
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1434 |
> |
mimics a simple heat bath with stochastic and dissipative forces, |
| 1435 |
> |
has been applied in a variety of studies. This section will review |
| 1436 |
> |
the theory of Langevin dynamics. A brief derivation of the generalized |
| 1437 |
> |
Langevin equation will be given first. Following that, we will |
| 1438 |
> |
discuss the physical meaning of the terms appearing in the equation. |
| 1439 |
> |
|
| 1440 |
> |
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1441 |
> |
|
| 1442 |
> |
A harmonic bath model, in which an effective set of harmonic |
| 1443 |
> |
oscillators are used to mimic the effect of a linearly responding |
| 1444 |
> |
environment, has been widely used in quantum chemistry and |
| 1445 |
> |
statistical mechanics. One of the successful applications of |
| 1446 |
> |
Harmonic bath model is the derivation of the Generalized Langevin |
| 1447 |
> |
Dynamics (GLE). Consider a system, in which the degree of |
| 1448 |
> |
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1449 |
> |
Hamiltonian of the form |
| 1450 |
> |
\begin{equation} |
| 1451 |
> |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1452 |
> |
\label{introEquation:bathGLE}. |
| 1453 |
> |
\end{equation} |
| 1454 |
> |
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated |
| 1455 |
> |
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
| 1456 |
> |
\[ |
| 1457 |
> |
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1458 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha x_\alpha ^2 } |
| 1459 |
> |
\right\}} |
| 1460 |
> |
\] |
| 1461 |
> |
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1462 |
> |
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
| 1463 |
> |
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath |
| 1464 |
> |
coupling, |
| 1465 |
> |
\[ |
| 1466 |
> |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1467 |
> |
\] |
| 1468 |
> |
where $g_\alpha$ are the coupling constants between the bath |
| 1469 |
> |
coordinates ($x_ \alpha$) and the system coordinate ($x$). |
| 1470 |
> |
Introducing |
| 1471 |
> |
\[ |
| 1472 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1473 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1474 |
> |
\] |
| 1475 |
> |
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as |
| 1476 |
> |
\[ |
| 1477 |
> |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1478 |
> |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1479 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1480 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}}. |
| 1481 |
> |
\] |
| 1482 |
> |
Since the first two terms of the new Hamiltonian depend only on the |
| 1483 |
> |
system coordinates, we can get the equations of motion for |
| 1484 |
> |
Generalized Langevin Dynamics by Hamilton's equations, |
| 1485 |
> |
\begin{equation} |
| 1486 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1487 |
> |
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
| 1488 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)}, |
| 1489 |
> |
\label{introEquation:coorMotionGLE} |
| 1490 |
> |
\end{equation} |
| 1491 |
> |
and |
| 1492 |
> |
\begin{equation} |
| 1493 |
> |
m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - |
| 1494 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1495 |
> |
\label{introEquation:bathMotionGLE} |
| 1496 |
> |
\end{equation} |
| 1497 |
> |
In order to derive an equation for $x$, the dynamics of the bath |
| 1498 |
> |
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1499 |
> |
transform which is particularly useful in solving linear ordinary |
| 1500 |
> |
differential equations,the Laplace transform is the appropriate tool |
| 1501 |
> |
to solve this problem. The basic idea is to transform the difficult |
| 1502 |
> |
differential equations into simple algebra problems which can be |
| 1503 |
> |
solved easily. Then, by applying the inverse Laplace transform, we |
| 1504 |
> |
can retrieve the solutions of the original problems. Let $f(t)$ be a |
| 1505 |
> |
function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$ |
| 1506 |
> |
is a new function defined as |
| 1507 |
> |
\[ |
| 1508 |
> |
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1509 |
> |
\] |
| 1510 |
> |
where $p$ is real and $L$ is called the Laplace Transform |
| 1511 |
> |
Operator. Below are some important properties of the Laplace transform |
| 1512 |
> |
\begin{eqnarray*} |
| 1513 |
> |
L(x + y) & = & L(x) + L(y) \\ |
| 1514 |
> |
L(ax) & = & aL(x) \\ |
| 1515 |
> |
L(\dot x) & = & pL(x) - px(0) \\ |
| 1516 |
> |
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
| 1517 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
| 1518 |
> |
\end{eqnarray*} |
| 1519 |
> |
Applying the Laplace transform to the bath coordinates, we obtain |
| 1520 |
> |
\begin{eqnarray*} |
| 1521 |
> |
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), \\ |
| 1522 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
| 1523 |
> |
\end{eqnarray*} |
| 1524 |
> |
In the same way, the system coordinates become |
| 1525 |
> |
\begin{eqnarray*} |
| 1526 |
> |
mL(\ddot x) & = & |
| 1527 |
> |
- \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\}} \\ |
| 1528 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}. |
| 1529 |
> |
\end{eqnarray*} |
| 1530 |
> |
With the help of some relatively important inverse Laplace |
| 1531 |
> |
transformations: |
| 1532 |
> |
\[ |
| 1533 |
> |
\begin{array}{c} |
| 1534 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ |
| 1535 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ |
| 1536 |
> |
L(1) = \frac{1}{p} \\ |
| 1537 |
> |
\end{array} |
| 1538 |
> |
\] |
| 1539 |
> |
we obtain |
| 1540 |
> |
\begin{eqnarray*} |
| 1541 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
| 1542 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1543 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
| 1544 |
< |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
| 1545 |
< |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
| 1546 |
< |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 1547 |
< |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
| 1544 |
> |
_\alpha t)\dot x(t - \tau )d\tau } } \right\}} \\ |
| 1545 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1546 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1547 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1548 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}}\\ |
| 1549 |
|
% |
| 1550 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1551 |
< |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1552 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 1553 |
< |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
| 1554 |
< |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
| 1555 |
< |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
| 1556 |
< |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
| 1557 |
< |
(\omega _\alpha t)} \right\}} |
| 1558 |
< |
\end{align} |
| 1559 |
< |
|
| 1550 |
> |
& = & - |
| 1551 |
> |
\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha |
| 1552 |
> |
= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha |
| 1553 |
> |
^2 }}} \right)\cos (\omega _\alpha |
| 1554 |
> |
t)\dot x(t - \tau )d} \tau } \\ |
| 1555 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1556 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1557 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1558 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1559 |
> |
\end{eqnarray*} |
| 1560 |
> |
Introducing a \emph{dynamic friction kernel} |
| 1561 |
|
\begin{equation} |
| 1562 |
+ |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1563 |
+ |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 1564 |
+ |
\label{introEquation:dynamicFrictionKernelDefinition} |
| 1565 |
+ |
\end{equation} |
| 1566 |
+ |
and \emph{a random force} |
| 1567 |
+ |
\begin{equation} |
| 1568 |
+ |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 1569 |
+ |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 1570 |
+ |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 1571 |
+ |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), |
| 1572 |
+ |
\label{introEquation:randomForceDefinition} |
| 1573 |
+ |
\end{equation} |
| 1574 |
+ |
the equation of motion can be rewritten as |
| 1575 |
+ |
\begin{equation} |
| 1576 |
|
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
| 1577 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1578 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1579 |
|
\end{equation} |
| 1580 |
< |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
| 1581 |
< |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
| 1580 |
> |
which is known as the \emph{generalized Langevin equation} (GLE). |
| 1581 |
> |
|
| 1582 |
> |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
| 1583 |
> |
|
| 1584 |
> |
One may notice that $R(t)$ depends only on initial conditions, which |
| 1585 |
> |
implies it is completely deterministic within the context of a |
| 1586 |
> |
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1587 |
> |
uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)} |
| 1588 |
> |
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
| 1589 |
> |
0.$ This property is what we expect from a truly random process. As |
| 1590 |
> |
long as the model chosen for $R(t)$ was a gaussian distribution in |
| 1591 |
> |
general, the stochastic nature of the GLE still remains. |
| 1592 |
> |
%dynamic friction kernel |
| 1593 |
> |
The convolution integral |
| 1594 |
|
\[ |
| 1595 |
< |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1154 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 1595 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } |
| 1596 |
|
\] |
| 1597 |
< |
For an infinite harmonic bath, we can use the spectral density and |
| 1598 |
< |
an integral over frequencies. |
| 1599 |
< |
|
| 1597 |
> |
depends on the entire history of the evolution of $x$, which implies |
| 1598 |
> |
that the bath retains memory of previous motions. In other words, |
| 1599 |
> |
the bath requires a finite time to respond to change in the motion |
| 1600 |
> |
of the system. For a sluggish bath which responds slowly to changes |
| 1601 |
> |
in the system coordinate, we may regard $\xi(t)$ as a constant |
| 1602 |
> |
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes |
| 1603 |
|
\[ |
| 1604 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 1161 |
< |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 1162 |
< |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 1163 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
| 1604 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
| 1605 |
|
\] |
| 1606 |
< |
The random forces depend only on initial conditions. |
| 1166 |
< |
|
| 1167 |
< |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 1168 |
< |
So we can define a new set of coordinates, |
| 1606 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1607 |
|
\[ |
| 1608 |
< |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1609 |
< |
^2 }}x(0) |
| 1608 |
> |
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1609 |
> |
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1610 |
|
\] |
| 1611 |
< |
This makes |
| 1611 |
> |
which can be used to describe the effect of dynamic caging in |
| 1612 |
> |
viscous solvents. The other extreme is the bath that responds |
| 1613 |
> |
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
| 1614 |
> |
taken as a $delta$ function in time: |
| 1615 |
|
\[ |
| 1616 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
| 1616 |
> |
\xi (t) = 2\xi _0 \delta (t). |
| 1617 |
|
\] |
| 1618 |
< |
And since the $q$ coordinates are harmonic oscillators, |
| 1618 |
> |
Hence, the convolution integral becomes |
| 1619 |
|
\[ |
| 1620 |
< |
\begin{array}{l} |
| 1621 |
< |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1181 |
< |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1182 |
< |
\end{array} |
| 1620 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1621 |
> |
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1622 |
|
\] |
| 1623 |
+ |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes the |
| 1624 |
+ |
Langevin equation |
| 1625 |
+ |
\begin{equation} |
| 1626 |
+ |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1627 |
+ |
x(t) + R(t) \label{introEquation:LangevinEquation}. |
| 1628 |
+ |
\end{equation} |
| 1629 |
+ |
The static friction coefficient $\xi _0$ can either be calculated |
| 1630 |
+ |
from spectral density or be determined by Stokes' law for regular |
| 1631 |
+ |
shaped particles. A brief review on calculating friction tensors for |
| 1632 |
+ |
arbitrary shaped particles is given in |
| 1633 |
+ |
Sec.~\ref{introSection:frictionTensor}. |
| 1634 |
|
|
| 1635 |
< |
\begin{align} |
| 1186 |
< |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
| 1187 |
< |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
| 1188 |
< |
(t)q_\beta (0)} \right\rangle } } |
| 1189 |
< |
% |
| 1190 |
< |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
| 1191 |
< |
\right\rangle \cos (\omega _\alpha t)} |
| 1192 |
< |
% |
| 1193 |
< |
&= kT\xi (t) |
| 1194 |
< |
\end{align} |
| 1635 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1636 |
|
|
| 1637 |
+ |
Defining a new set of coordinates |
| 1638 |
+ |
\[ |
| 1639 |
+ |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1640 |
+ |
^2 }}x(0), |
| 1641 |
+ |
\] |
| 1642 |
+ |
we can rewrite $R(t)$ as |
| 1643 |
+ |
\[ |
| 1644 |
+ |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1645 |
+ |
\] |
| 1646 |
+ |
And since the $q$ coordinates are harmonic oscillators, |
| 1647 |
+ |
\begin{eqnarray*} |
| 1648 |
+ |
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1649 |
+ |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1650 |
+ |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1651 |
+ |
\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 } } \\ |
| 1652 |
+ |
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1653 |
+ |
& = &kT\xi (t) |
| 1654 |
+ |
\end{eqnarray*} |
| 1655 |
+ |
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1656 |
|
\begin{equation} |
| 1657 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 1658 |
< |
\label{introEquation:secondFluctuationDissipation} |
| 1658 |
> |
\label{introEquation:secondFluctuationDissipation}, |
| 1659 |
|
\end{equation} |
| 1660 |
< |
|
| 1661 |
< |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
| 1202 |
< |
|
| 1203 |
< |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1204 |
< |
\subsection{\label{introSection:analyticalApproach}Analytical |
| 1205 |
< |
Approach} |
| 1206 |
< |
|
| 1207 |
< |
\subsection{\label{introSection:approximationApproach}Approximation |
| 1208 |
< |
Approach} |
| 1209 |
< |
|
| 1210 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1211 |
< |
Body} |
| 1212 |
< |
|
| 1213 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 1660 |
> |
which acts as a constraint on the possible ways in which one can |
| 1661 |
> |
model the random force and friction kernel. |
