| 62 |
|
\end{equation} |
| 63 |
|
If there are no external torques acting on a body, the angular |
| 64 |
|
momentum of it is conserved. The last conservation theorem state |
| 65 |
< |
that if all forces are conservative, Energy |
| 66 |
< |
\begin{equation}E = T + V \label{introEquation:energyConservation} |
| 65 |
> |
that if all forces are conservative, energy is conserved, |
| 66 |
> |
\begin{equation}E = T + V. \label{introEquation:energyConservation} |
| 67 |
|
\end{equation} |
| 68 |
< |
is conserved. All of these conserved quantities are |
| 69 |
< |
important factors to determine the quality of numerical integration |
| 70 |
< |
schemes for rigid bodies \cite{Dullweber1997}. |
| 68 |
> |
All of these conserved quantities are important factors to determine |
| 69 |
> |
the quality of numerical integration schemes for rigid bodies |
| 70 |
> |
\cite{Dullweber1997}. |
| 71 |
|
|
| 72 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 73 |
|
|
| 74 |
< |
Newtonian Mechanics suffers from two important limitations: motions |
| 75 |
< |
can only be described in cartesian coordinate systems. Moreover, it |
| 76 |
< |
becomes impossible to predict analytically the properties of the |
| 77 |
< |
system even if we know all of the details of the interaction. In |
| 78 |
< |
order to overcome some of the practical difficulties which arise in |
| 79 |
< |
attempts to apply Newton's equation to complex system, approximate |
| 80 |
< |
numerical procedures may be developed. |
| 74 |
> |
Newtonian Mechanics suffers from a important limitation: motions can |
| 75 |
> |
only be described in cartesian coordinate systems which make it |
| 76 |
> |
impossible to predict analytically the properties of the system even |
| 77 |
> |
if we know all of the details of the interaction. In order to |
| 78 |
> |
overcome some of the practical difficulties which arise in attempts |
| 79 |
> |
to apply Newton's equation to complex system, approximate numerical |
| 80 |
> |
procedures may be developed. |
| 81 |
|
|
| 82 |
|
\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
| 83 |
|
Principle}} |
| 88 |
|
trajectory, along which a dynamical system may move from one point |
| 89 |
|
to another within a specified time, is derived by finding the path |
| 90 |
|
which minimizes the time integral of the difference between the |
| 91 |
< |
kinetic, $K$, and potential energies, $U$. |
| 91 |
> |
kinetic, $K$, and potential energies, $U$, |
| 92 |
|
\begin{equation} |
| 93 |
< |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
| 93 |
> |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}. |
| 94 |
|
\label{introEquation:halmitonianPrinciple1} |
| 95 |
|
\end{equation} |
| 96 |
|
For simple mechanical systems, where the forces acting on the |
| 98 |
|
function $L$ can be defined as the difference between the kinetic |
| 99 |
|
energy of the system and its potential energy, |
| 100 |
|
\begin{equation} |
| 101 |
< |
L \equiv K - U = L(q_i ,\dot q_i ) , |
| 101 |
> |
L \equiv K - U = L(q_i ,\dot q_i ). |
| 102 |
|
\label{introEquation:lagrangianDef} |
| 103 |
|
\end{equation} |
| 104 |
< |
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
| 104 |
> |
Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
| 105 |
|
\begin{equation} |
| 106 |
< |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
| 106 |
> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} . |
| 107 |
|
\label{introEquation:halmitonianPrinciple2} |
| 108 |
|
\end{equation} |
| 109 |
|
|
| 148 |
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
| 149 |
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
| 150 |
|
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
| 151 |
< |
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
| 151 |
> |
L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1} |
| 152 |
|
\end{equation} |
| 153 |
< |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
| 154 |
< |
second and fourth terms in the parentheses cancel. Therefore, |
| 153 |
> |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second |
| 154 |
> |
and fourth terms in the parentheses cancel. Therefore, |
| 155 |
|
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
| 156 |
|
\begin{equation} |
| 157 |
|
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
| 158 |
< |
\right)} - \frac{{\partial L}}{{\partial t}}dt |
| 158 |
> |
\right)} - \frac{{\partial L}}{{\partial t}}dt . |
| 159 |
|
\label{introEquation:diffHamiltonian2} |
| 160 |
|
\end{equation} |
| 161 |
|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
| 174 |
|
t}} |
| 175 |
|
\label{introEquation:motionHamiltonianTime} |
| 176 |
|
\end{equation} |
| 177 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 177 |
> |
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 178 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
| 179 |
|
equation of motion. Due to their symmetrical formula, they are also |
| 180 |
|
known as the canonical equations of motions \cite{Goldstein2001}. |
| 188 |
|
statistical mechanics and quantum mechanics, since it treats the |
| 189 |
|
coordinate and its time derivative as independent variables and it |
| 190 |
|
only works with 1st-order differential equations\cite{Marion1990}. |
| 191 |
– |
|
| 191 |
|
In Newtonian Mechanics, a system described by conservative forces |
| 192 |
< |
conserves the total energy \ref{introEquation:energyConservation}. |
| 193 |
< |
It follows that Hamilton's equations of motion conserve the total |
| 194 |
< |
Hamiltonian. |
| 192 |
> |
conserves the total energy |
| 193 |
> |
(Eq.~\ref{introEquation:energyConservation}). It follows that |
| 194 |
> |
Hamilton's equations of motion conserve the total Hamiltonian |
| 195 |
|
\begin{equation} |
| 196 |
|
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
| 197 |
|
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
| 198 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
| 199 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
| 200 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
| 201 |
< |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
| 201 |
> |
q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian} |
| 202 |
|
\end{equation} |
| 203 |
|
|
| 204 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
| 219 |
|
momentum variables. Consider a dynamic system of $f$ particles in a |
| 220 |
|
cartesian space, where each of the $6f$ coordinates and momenta is |
| 221 |
|
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
| 222 |
< |
this system is a $6f$ dimensional space. A point, $x = (\rightarrow |
| 223 |
< |
q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow |
| 224 |
< |
p_f )$, with a unique set of values of $6f$ coordinates and momenta |
| 225 |
< |
is a phase space vector. |
| 222 |
> |
this system is a $6f$ dimensional space. A point, $x = |
| 223 |
> |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 224 |
> |
\over q} _1 , \ldots |
| 225 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 226 |
> |
\over q} _f |
| 227 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 228 |
> |
\over p} _1 \ldots |
| 229 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 230 |
> |
\over p} _f )$ , with a unique set of values of $6f$ coordinates and |
| 231 |
> |
momenta is a phase space vector. |
| 232 |
|
%%%fix me |
| 233 |
|
|
| 234 |
|
In statistical mechanics, the condition of an ensemble at any time |
| 290 |
|
is known to be thermally isolated and conserve energy, the |
| 291 |
|
Microcanonical ensemble (NVE) has a partition function like, |
| 292 |
|
\begin{equation} |
| 293 |
< |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 293 |
> |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
| 294 |
|
\end{equation} |
| 295 |
|
A canonical ensemble (NVT)is an ensemble of systems, each of which |
| 296 |
|
can share its energy with a large heat reservoir. The distribution |
| 297 |
|
of the total energy amongst the possible dynamical states is given |
| 298 |
|
by the partition function, |
| 299 |
|
\begin{equation} |
| 300 |
< |
\Omega (N,V,T) = e^{ - \beta A} |
| 300 |
> |
\Omega (N,V,T) = e^{ - \beta A}. |
| 301 |
|
\label{introEquation:NVTPartition} |
| 302 |
|
\end{equation} |
| 303 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 354 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
| 355 |
|
\label{introEquation:liouvilleTheorem} |
| 356 |
|
\end{equation} |
| 352 |
– |
|
| 357 |
|
Liouville's theorem states that the distribution function is |
| 358 |
|
constant along any trajectory in phase space. In classical |
| 359 |
|
statistical mechanics, since the number of members in an ensemble is |
| 495 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 496 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 497 |
|
$\omega(x, x) = 0$. The cross product operation in vector field is |
| 498 |
< |
an example of symplectic form. |
| 499 |
< |
|
| 500 |
< |
One of the motivations to study \emph{symplectic manifolds} in |
| 501 |
< |
Hamiltonian Mechanics is that a symplectic manifold can represent |
| 502 |
< |
all possible configurations of the system and the phase space of the |
| 503 |
< |
system can be described by it's cotangent bundle. Every symplectic |
| 504 |
< |
manifold is even dimensional. For instance, in Hamilton equations, |
| 501 |
< |
coordinate and momentum always appear in pairs. |
| 498 |
> |
an example of symplectic form. One of the motivations to study |
| 499 |
> |
\emph{symplectic manifolds} in Hamiltonian Mechanics is that a |
| 500 |
> |
symplectic manifold can represent all possible configurations of the |
| 501 |
> |
system and the phase space of the system can be described by it's |
| 502 |
> |
cotangent bundle. Every symplectic manifold is even dimensional. For |
| 503 |
> |
instance, in Hamilton equations, coordinate and momentum always |
| 504 |
> |
appear in pairs. |
| 505 |
|
|
| 506 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 507 |
|
|
| 528 |
|
\frac{d}{{dt}}x = J\nabla _x H(x) |
| 529 |
|
\label{introEquation:compactHamiltonian} |
| 530 |
|
\end{equation}In this case, $f$ is |
| 531 |
< |
called a \emph{Hamiltonian vector field}. |
| 532 |
< |
|
| 530 |
< |
Another generalization of Hamiltonian dynamics is Poisson |
| 531 |
< |
Dynamics\cite{Olver1986}, |
| 531 |
> |
called a \emph{Hamiltonian vector field}. Another generalization of |
| 532 |
> |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
| 533 |
|
\begin{equation} |
| 534 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 535 |
|
\end{equation} |
| 766 |
|
|
| 767 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 768 |
|
\end{enumerate} |
| 768 |
– |
|
| 769 |
|
By simply switching the order of the propagators in the splitting |
| 770 |
|
and composing a new integrator, the \emph{position verlet} |
| 771 |
|
integrator, can be generated, |
| 1068 |
|
to justify the correctness of a liquid model. Moreover, various |
| 1069 |
|
equilibrium thermodynamic and structural properties can also be |
| 1070 |
|
expressed in terms of radial distribution function \cite{Allen1987}. |
| 1071 |
– |
|
| 1071 |
|
The pair distribution functions $g(r)$ gives the probability that a |
| 1072 |
|
particle $i$ will be located at a distance $r$ from a another |
| 1073 |
|
particle $j$ in the system |
| 1195 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1196 |
|
\label{introEquation:RBFirstOrderConstraint} |
| 1197 |
|
\end{equation} |
| 1199 |
– |
|
| 1198 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1199 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1200 |
|
the equations of motion, |
| 1203 |
– |
|
| 1201 |
|
\begin{eqnarray} |
| 1202 |
|
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1203 |
|
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1204 |
|
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1205 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1206 |
|
\end{eqnarray} |
| 1210 |
– |
|
| 1207 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1208 |
|
We can use a constraint force provided by a Lagrange multiplier on |
| 1209 |
|
the normal manifold to keep the motion on constraint space. Or we |
| 1215 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1216 |
|
\right\}. |
| 1217 |
|
\] |
| 1222 |
– |
|
| 1218 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1219 |
|
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
| 1220 |
|
rotation group $SO(3)$. However, it turns out that under symplectic |
| 1229 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1230 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1231 |
|
\] |
| 1237 |
– |
|
| 1232 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1233 |
|
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1234 |
|
given as |
| 1247 |
|
\[ |
| 1248 |
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1249 |
|
\] |
| 1250 |
< |
respectively. |
| 1251 |
< |
|
| 1252 |
< |
As a common choice to describe the rotation dynamics of the rigid |
| 1259 |
< |
body, the angular momentum on the body fixed frame $\Pi = Q^t P$ is |
| 1260 |
< |
introduced to rewrite the equations of motion, |
| 1250 |
> |
respectively. As a common choice to describe the rotation dynamics |
| 1251 |
> |
of the rigid body, the angular momentum on the body fixed frame $\Pi |
| 1252 |
> |
= Q^t P$ is introduced to rewrite the equations of motion, |
| 1253 |
|
\begin{equation} |
| 1254 |
|
\begin{array}{l} |
| 1255 |
< |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1256 |
< |
\dot Q = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1255 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda, \\ |
| 1256 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1}, \\ |
| 1257 |
|
\end{array} |
| 1258 |
|
\label{introEqaution:RBMotionPI} |
| 1259 |
|
\end{equation} |
| 1260 |
< |
, as well as holonomic constraints, |
| 1260 |
> |
as well as holonomic constraints, |
| 1261 |
|
\[ |
| 1262 |
|
\begin{array}{l} |
| 1263 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1264 |
< |
Q^T Q = 1 \\ |
| 1263 |
> |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0, \\ |
| 1264 |
> |
Q^T Q = 1 .\\ |
| 1265 |
|
\end{array} |
| 1266 |
|
\] |
| 1275 |
– |
|
| 1267 |
|
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1268 |
|
so(3)^ \star$, the hat-map isomorphism, |
| 1269 |
|
\begin{equation} |
| 1278 |
|
will let us associate the matrix products with traditional vector |
| 1279 |
|
operations |
| 1280 |
|
\[ |
| 1281 |
< |
\hat vu = v \times u |
| 1281 |
> |
\hat vu = v \times u. |
| 1282 |
|
\] |
| 1283 |
< |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1283 |
> |
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1284 |
|
matrix, |
| 1285 |
< |
|
| 1286 |
< |
\begin{eqnarray*} |
| 1287 |
< |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
| 1288 |
< |
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
| 1285 |
> |
\begin{eqnarray} |
| 1286 |
> |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} &= &{\rm{ }}(\Pi - \Pi ^T ){\rm{ |
| 1287 |
> |
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
| 1288 |
> |
+ \sum\limits_i {[Q^T F_i |
| 1289 |
|
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
| 1290 |
|
\label{introEquation:skewMatrixPI} |
| 1291 |
< |
\end{eqnarray*} |
| 1292 |
< |
|
| 1293 |
< |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1294 |
< |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1295 |
< |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1296 |
< |
unique property eliminates the requirement of iterations which can |
| 1297 |
< |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
| 1298 |
< |
|
| 1308 |
< |
Applying the hat-map isomorphism, we obtain the equation of motion |
| 1309 |
< |
for angular momentum on body frame |
| 1291 |
> |
\end{eqnarray} |
| 1292 |
> |
Since $\Lambda$ is symmetric, the last term of |
| 1293 |
> |
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
| 1294 |
> |
Lagrange multiplier $\Lambda$ is absent from the equations of |
| 1295 |
> |
motion. This unique property eliminates the requirement of |
| 1296 |
> |
iterations which can not be avoided in other methods\cite{Kol1997, |
| 1297 |
> |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
| 1298 |
> |
equation of motion for angular momentum on body frame |
| 1299 |
|
\begin{equation} |
| 1300 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1301 |
|
F_i (r,Q)} \right) \times X_i }. |
| 1304 |
|
In the same manner, the equation of motion for rotation matrix is |
| 1305 |
|
given by |
| 1306 |
|
\[ |
| 1307 |
< |
\dot Q = Qskew(I^{ - 1} \pi ) |
| 1307 |
> |
\dot Q = Qskew(I^{ - 1} \pi ). |
| 1308 |
|
\] |
| 1309 |
|
|
| 1310 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1326 |
|
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1327 |
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1328 |
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1329 |
< |
\end{array}} \right) |
| 1329 |
> |
\end{array}} \right). |
| 1330 |
|
\end{equation} |
| 1331 |
|
Thus, the dynamics of free rigid body is governed by |
| 1332 |
|
\begin{equation} |
| 1333 |
< |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1333 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
| 1334 |
|
\end{equation} |
| 1346 |
– |
|
| 1335 |
|
One may notice that each $T_i^r$ in Equation |
| 1336 |
|
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1337 |
|
instance, the equations of motion due to $T_1^r$ are given by |
| 1364 |
|
propagator, |
| 1365 |
|
\[ |
| 1366 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1367 |
< |
) |
| 1367 |
> |
). |
| 1368 |
|
\] |
| 1369 |
|
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1370 |
|
manner. In order to construct a second-order symplectic method, we |
| 1382 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1383 |
|
_1 }. |
| 1384 |
|
\] |
| 1397 |
– |
|
| 1385 |
|
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1386 |
|
$F(\pi )$ and $G(\pi )$ is defined by |
| 1387 |
|
\[ |
| 1388 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1389 |
< |
) |
| 1389 |
> |
). |
| 1390 |
|
\] |
| 1391 |
|
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1392 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1397 |
|
then by the chain rule |
| 1398 |
|
\[ |
| 1399 |
|
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1400 |
< |
}}{2})\pi |
| 1400 |
> |
}}{2})\pi. |
| 1401 |
|
\] |
| 1402 |
< |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1402 |
> |
Thus, $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel |
| 1403 |
> |
\pi |
| 1404 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1405 |
|
Lie-Poisson integrator is found to be both extremely efficient and |
| 1406 |
|
stable. These properties can be explained by the fact the small |
| 1413 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1414 |
|
energy and potential energy, |
| 1415 |
|
\[ |
| 1416 |
< |
H = T(p,\pi ) + V(q,Q) |
| 1416 |
> |
H = T(p,\pi ) + V(q,Q). |
| 1417 |
|
\] |
| 1418 |
|
The equations of motion corresponding to potential energy and |
| 1419 |
|
kinetic energy are listed in the below table, |
| 1462 |
|
\] |
| 1463 |
|
Finally, we obtain the overall symplectic propagators for freely |
| 1464 |
|
moving rigid bodies |
| 1465 |
< |
\begin{equation} |
| 1466 |
< |
\begin{array}{c} |
| 1467 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1468 |
< |
\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 } \\ |
| 1481 |
< |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1482 |
< |
\end{array} |
| 1465 |
> |
\begin{eqnarray*} |
| 1466 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1467 |
> |
& & \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 } \\ |
| 1468 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1469 |
|
\label{introEquation:overallRBFlowMaps} |
| 1470 |
< |
\end{equation} |
| 1470 |
> |
\end{eqnarray*} |
| 1471 |
|
|
| 1472 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1473 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1513 |
|
\[ |
| 1514 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1515 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1516 |
< |
\] and combining the last two terms in Equation |
| 1517 |
< |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
| 1532 |
< |
Hamiltonian as |
| 1516 |
> |
\] |
| 1517 |
> |
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as |
| 1518 |
|
\[ |
| 1519 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1520 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1521 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1522 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1522 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}}. |
| 1523 |
|
\] |
| 1524 |
|
Since the first two terms of the new Hamiltonian depend only on the |
| 1525 |
|
system coordinates, we can get the equations of motion for |
| 1536 |
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1537 |
|
\label{introEquation:bathMotionGLE} |
| 1538 |
|
\end{equation} |
| 1554 |
– |
|
| 1539 |
|
In order to derive an equation for $x$, the dynamics of the bath |
| 1540 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1541 |
|
transform which is particularly useful in solving linear ordinary |
| 1544 |
|
differential equations into simple algebra problems which can be |
| 1545 |
|
solved easily. Then, by applying the inverse Laplace transform, also |
| 1546 |
|
known as the Bromwich integral, we can retrieve the solutions of the |
| 1547 |
< |
original problems. |
| 1548 |
< |
|
| 1565 |
< |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1566 |
< |
transform of f(t) is a new function defined as |
| 1547 |
> |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
| 1548 |
> |
$. The Laplace transform of f(t) is a new function defined as |
| 1549 |
|
\[ |
| 1550 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1551 |
|
\] |
| 1552 |
|
where $p$ is real and $L$ is called the Laplace Transform |
| 1553 |
|
Operator. Below are some important properties of Laplace transform |
| 1572 |
– |
|
| 1554 |
|
\begin{eqnarray*} |
| 1555 |
|
L(x + y) & = & L(x) + L(y) \\ |
| 1556 |
|
L(ax) & = & aL(x) \\ |
| 1558 |
|
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
| 1559 |
|
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
| 1560 |
|
\end{eqnarray*} |
| 1580 |
– |
|
| 1581 |
– |
|
| 1561 |
|
Applying the Laplace transform to the bath coordinates, we obtain |
| 1562 |
|
\begin{eqnarray*} |
| 1563 |
|
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) \\ |
| 1564 |
|
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1565 |
|
\end{eqnarray*} |
| 1587 |
– |
|
| 1566 |
|
By the same way, the system coordinates become |
| 1567 |
|
\begin{eqnarray*} |
| 1568 |
< |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1569 |
< |
& & \mbox{} - \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\}} \\ |
| 1568 |
> |
mL(\ddot x) & = & |
| 1569 |
> |
- \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\}} \\ |
| 1570 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} |
| 1571 |
|
\end{eqnarray*} |
| 1593 |
– |
|
| 1572 |
|
With the help of some relatively important inverse Laplace |
| 1573 |
|
transformations: |
| 1574 |
|
\[ |
| 1578 |
|
L(1) = \frac{1}{p} \\ |
| 1579 |
|
\end{array} |
| 1580 |
|
\] |
| 1581 |
< |
, we obtain |
| 1581 |
> |
we obtain |
| 1582 |
|
\begin{eqnarray*} |
| 1583 |
|
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
| 1584 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1651 |
|
\[ |
| 1652 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
| 1653 |
|
\] |
| 1654 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1654 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1655 |
|
\[ |
| 1656 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1657 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1668 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1669 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1670 |
|
\] |
| 1671 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1671 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1672 |
|
\begin{equation} |
| 1673 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1674 |
|
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1691 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1692 |
|
\] |
| 1693 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1716 |
– |
|
| 1694 |
|
\begin{eqnarray*} |
| 1695 |
|
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1696 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1699 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1700 |
|
& = &kT\xi (t) \\ |
| 1701 |
|
\end{eqnarray*} |
| 1725 |
– |
|
| 1702 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1703 |
|
\begin{equation} |
| 1704 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
