| 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} |
