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} |
319 |
< |
\label{introEqaution:NVEPartition}. |
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 |
770 |
|
splitting gives a second-order decomposition, |
771 |
|
\begin{equation} |
772 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
773 |
< |
_{1,h/2} , |
775 |
< |
\label{introEqaution:secondOrderSplitting} |
773 |
> |
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
774 |
|
\end{equation} |
775 |
|
which has a local error proportional to $h^3$. Sprang splitting's |
776 |
|
popularity in molecular simulation community attribute to its |
953 |
|
|
954 |
|
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
955 |
|
|
956 |
+ |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
957 |
+ |
|
958 |
+ |
\begin{equation} |
959 |
+ |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
960 |
+ |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
961 |
+ |
\label{introEquation:RBHamiltonian} |
962 |
+ |
\end{equation} |
963 |
+ |
Here, $q$ and $Q$ are the position and rotation matrix for the |
964 |
+ |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
965 |
+ |
$J$, a diagonal matrix, is defined by |
966 |
+ |
\[ |
967 |
+ |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
968 |
+ |
\] |
969 |
+ |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
970 |
+ |
constrained Hamiltonian equation subjects to a holonomic constraint, |
971 |
+ |
\begin{equation} |
972 |
+ |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
973 |
+ |
\end{equation} |
974 |
+ |
which is used to ensure rotation matrix's orthogonality. |
975 |
+ |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
976 |
+ |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
977 |
+ |
\begin{equation} |
978 |
+ |
Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0 . \\ |
979 |
+ |
\label{introEquation:RBFirstOrderConstraint} |
980 |
+ |
\end{equation} |
981 |
+ |
|
982 |
+ |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
983 |
+ |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
984 |
+ |
the equations of motion, |
985 |
+ |
\[ |
986 |
+ |
\begin{array}{c} |
987 |
+ |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
988 |
+ |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
989 |
+ |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
990 |
+ |
\frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
991 |
+ |
\end{array} |
992 |
+ |
\] |
993 |
+ |
|
994 |
+ |
|
995 |
+ |
\[ |
996 |
+ |
M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} |
997 |
+ |
\right\} . |
998 |
+ |
\] |
999 |
+ |
|
1000 |
|
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
1001 |
|
|
1002 |
< |
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
1002 |
> |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations} |
1003 |
|
|
962 |
– |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
1004 |
|
|
1005 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1006 |
|
|
1209 |
|
|
1210 |
|
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
1211 |
|
Body} |
1212 |
+ |
|
1213 |
+ |
\section{\label{introSection:correlationFunctions}Correlation Functions} |