| 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 |
| 634 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 635 |
|
a \emph{symplectic} flow if it satisfies, |
| 636 |
|
\begin{equation} |
| 637 |
< |
'\varphi^T J '\varphi = J. |
| 637 |
> |
{\varphi '}^T J \varphi ' = J. |
| 638 |
|
\end{equation} |
| 639 |
|
According to Liouville's theorem, the symplectic volume is invariant |
| 640 |
|
under a Hamiltonian flow, which is the basis for classical |
| 642 |
|
field on a symplectic manifold can be shown to be a |
| 643 |
|
symplectomorphism. As to the Poisson system, |
| 644 |
|
\begin{equation} |
| 645 |
< |
'\varphi ^T J '\varphi = J \circ \varphi |
| 645 |
> |
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 646 |
|
\end{equation} |
| 647 |
|
is the property must be preserved by the integrator. |
| 648 |
|
|
| 660 |
|
In other words, the flow of this vector field is reversible if and |
| 661 |
|
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 662 |
|
|
| 663 |
< |
When designing any numerical methods, one should always try to |
| 663 |
> |
A \emph{first integral}, or conserved quantity of a general |
| 664 |
> |
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 665 |
> |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 666 |
> |
\[ |
| 667 |
> |
\frac{{dG(x(t))}}{{dt}} = 0. |
| 668 |
> |
\] |
| 669 |
> |
Using chain rule, one may obtain, |
| 670 |
> |
\[ |
| 671 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
| 672 |
> |
\] |
| 673 |
> |
which is the condition for conserving \emph{first integral}. For a |
| 674 |
> |
canonical Hamiltonian system, the time evolution of an arbitrary |
| 675 |
> |
smooth function $G$ is given by, |
| 676 |
> |
\begin{equation} |
| 677 |
> |
\begin{array}{c} |
| 678 |
> |
\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} |
| 681 |
> |
\label{introEquation:firstIntegral1} |
| 682 |
> |
\end{equation} |
| 683 |
> |
Using poisson bracket notion, Equation |
| 684 |
> |
\ref{introEquation:firstIntegral1} can be rewritten as |
| 685 |
> |
\[ |
| 686 |
> |
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 687 |
> |
\] |
| 688 |
> |
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 689 |
> |
integral} of a Hamiltonian system is |
| 690 |
> |
\[ |
| 691 |
> |
\left\{ {G,H} \right\} = 0. |
| 692 |
> |
\] |
| 693 |
> |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 694 |
> |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 695 |
> |
0$. |
| 696 |
> |
|
| 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 |
|
|
| 701 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 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 \cite{}. However, due to computational penalty involved |
| 723 |
< |
in implementing the Runge-Kutta methods, they do not attract too |
| 724 |
< |
much attention from Molecular Dynamics community. Instead, splitting |
| 725 |
< |
have been widely accepted since they exploit natural decompositions |
| 726 |
< |
of the system\cite{Tuckerman92}. |
| 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}. |
| 727 |
|
|
| 728 |
|
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
| 729 |
|
|
| 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} , |
| 740 |
< |
\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 |
| 777 |
|
symmetric property, |
| 778 |
|
\begin{equation} |
| 779 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 780 |
< |
\lable{introEquation:timeReversible} |
| 780 |
> |
\label{introEquation:timeReversible} |
| 781 |
|
\end{equation} |
| 782 |
|
|
| 783 |
|
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
| 835 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
| 836 |
|
\label{introEquation:positionVerlet1} \\% |
| 837 |
|
% |
| 838 |
< |
q(\Delta t) = q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 838 |
> |
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 839 |
|
q(\Delta t)} \right]. % |
| 840 |
|
\label{introEquation:positionVerlet1} |
| 841 |
|
\end{align} |
| 861 |
|
\] |
| 862 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 863 |
|
can obtain |
| 864 |
< |
\begin{eqnarray} |
| 864 |
> |
\begin{eqnarray*} |
| 865 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 866 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 + |
| 867 |
< |
h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 + \ldots ) |
| 868 |
< |
\end{eqnarray} |
| 866 |
> |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 867 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 868 |
> |
\ldots ) |
| 869 |
> |
\end{eqnarray*} |
| 870 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 871 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 872 |
|
procedure can be applied to general splitting, of the form |
| 905 |
|
|
| 906 |
|
\subsection{\label{introSec:mdInit}Initialization} |
| 907 |
|
|
| 908 |
+ |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 909 |
+ |
|
| 910 |
|
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 911 |
|
|
| 912 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 913 |
|
|
| 914 |
< |
A rigid body is a body in which the distance between any two given |
| 915 |
< |
points of a rigid body remains constant regardless of external |
| 916 |
< |
forces exerted on it. A rigid body therefore conserves its shape |
| 917 |
< |
during its motion. |
| 914 |
> |
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}}. |
| 921 |
|
|
| 922 |
< |
Applications of dynamics of rigid bodies. |
| 922 |
> |
It is very important to develop stable and efficient methods to |
| 923 |
> |
integrate the equations of motion of orientational degrees of |
| 924 |
> |
freedom. Euler angles are the nature choice to describe the |
| 925 |
> |
rotational degrees of freedom. However, due to its singularity, the |
| 926 |
> |
numerical integration of corresponding equations of motion is very |
| 927 |
> |
inefficient and inaccurate. Although an alternative integrator using |
| 928 |
> |
different sets of Euler angles can overcome this difficulty\cite{}, |
| 929 |
> |
the computational penalty and the lost of angular momentum |
| 930 |
> |
conservation still remain. A singularity free representation |
| 931 |
> |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
| 932 |
> |
this approach suffer from the nonseparable Hamiltonian resulted from |
| 933 |
> |
quaternion representation, which prevents the symplectic algorithm |
| 934 |
> |
to be utilized. Another different approach is to apply holonomic |
| 935 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
| 936 |
> |
moves independently under the normal forces deriving from potential |
| 937 |
> |
energy and constraint forces which are used to guarantee the |
| 938 |
> |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
| 939 |
> |
algorithm converge very slowly when the number of constraint |
| 940 |
> |
increases. |
| 941 |
|
|
| 942 |
+ |
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 |
| 953 |
+ |
coworkers\cite{Dullweber1997}. |
| 954 |
+ |
|
| 955 |
|
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 956 |
|
|
| 957 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 957 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 958 |
|
|
| 959 |
< |
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
| 959 |
> |
\begin{equation} |
| 960 |
> |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 961 |
> |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 962 |
> |
\label{introEquation:RBHamiltonian} |
| 963 |
> |
\end{equation} |
| 964 |
> |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 965 |
> |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 966 |
> |
$J$, a diagonal matrix, is defined by |
| 967 |
> |
\[ |
| 968 |
> |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 969 |
> |
\] |
| 970 |
> |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 971 |
> |
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 972 |
> |
\begin{equation} |
| 973 |
> |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 974 |
> |
\end{equation} |
| 975 |
> |
which is used to ensure rotation matrix's orthogonality. |
| 976 |
> |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 977 |
> |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 978 |
> |
\begin{equation} |
| 979 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 980 |
> |
\label{introEquation:RBFirstOrderConstraint} |
| 981 |
> |
\end{equation} |
| 982 |
|
|
| 983 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 983 |
> |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 984 |
> |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 985 |
> |
the equations of motion, |
| 986 |
> |
\[ |
| 987 |
> |
\begin{array}{c} |
| 988 |
> |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 989 |
> |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 990 |
> |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 991 |
> |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 992 |
> |
\end{array} |
| 993 |
> |
\] |
| 994 |
|
|
| 995 |
+ |
In general, there are two ways to satisfy the holonomic constraints. |
| 996 |
+ |
We can use constraint force provided by lagrange multiplier on the |
| 997 |
+ |
normal manifold to keep the motion on constraint space. Or we can |
| 998 |
+ |
simply evolve the system in constraint manifold. The two method are |
| 999 |
+ |
proved to be equivalent. The holonomic constraint and equations of |
| 1000 |
+ |
motions define a constraint manifold for rigid body |
| 1001 |
+ |
\[ |
| 1002 |
+ |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1003 |
+ |
\right\}. |
| 1004 |
+ |
\] |
| 1005 |
+ |
|
| 1006 |
+ |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1007 |
+ |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 1008 |
+ |
transformation, the cotangent space and the phase space are |
| 1009 |
+ |
diffeomorphic. Introducing |
| 1010 |
+ |
\[ |
| 1011 |
+ |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1012 |
+ |
\] |
| 1013 |
+ |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 1014 |
+ |
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1015 |
+ |
\[ |
| 1016 |
+ |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1017 |
+ |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1018 |
+ |
\] |
| 1019 |
+ |
|
| 1020 |
+ |
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1021 |
+ |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1022 |
+ |
given as |
| 1023 |
+ |
\begin{equation} |
| 1024 |
+ |
X_i^{lab} = Q X_i + q. |
| 1025 |
+ |
\end{equation} |
| 1026 |
+ |
Therefore, potential energy $V(q,Q)$ is defined by |
| 1027 |
+ |
\[ |
| 1028 |
+ |
V(q,Q) = V(Q X_0 + q). |
| 1029 |
+ |
\] |
| 1030 |
+ |
Hence, |
| 1031 |
+ |
\[ |
| 1032 |
+ |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)} |
| 1033 |
+ |
\] |
| 1034 |
+ |
|
| 1035 |
+ |
\[ |
| 1036 |
+ |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1037 |
+ |
\] |
| 1038 |
+ |
|
| 1039 |
+ |
As a common choice to describe the rotation dynamics of the rigid |
| 1040 |
+ |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1041 |
+ |
rewrite the equations of motion, |
| 1042 |
+ |
\begin{equation} |
| 1043 |
+ |
\begin{array}{l} |
| 1044 |
+ |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1045 |
+ |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1046 |
+ |
\end{array} |
| 1047 |
+ |
\label{introEqaution:RBMotionPI} |
| 1048 |
+ |
\end{equation} |
| 1049 |
+ |
, as well as holonomic constraints, |
| 1050 |
+ |
\[ |
| 1051 |
+ |
\begin{array}{l} |
| 1052 |
+ |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1053 |
+ |
Q^T Q = 1 \\ |
| 1054 |
+ |
\end{array} |
| 1055 |
+ |
\] |
| 1056 |
+ |
|
| 1057 |
+ |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1058 |
+ |
so(3)^ \star$, the hat-map isomorphism, |
| 1059 |
+ |
\begin{equation} |
| 1060 |
+ |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1061 |
+ |
{\begin{array}{*{20}c} |
| 1062 |
+ |
0 & { - v_3 } & {v_2 } \\ |
| 1063 |
+ |
{v_3 } & 0 & { - v_1 } \\ |
| 1064 |
+ |
{ - v_2 } & {v_1 } & 0 \\ |
| 1065 |
+ |
\end{array}} \right), |
| 1066 |
+ |
\label{introEquation:hatmapIsomorphism} |
| 1067 |
+ |
\end{equation} |
| 1068 |
+ |
will let us associate the matrix products with traditional vector |
| 1069 |
+ |
operations |
| 1070 |
+ |
\[ |
| 1071 |
+ |
\hat vu = v \times u |
| 1072 |
+ |
\] |
| 1073 |
+ |
|
| 1074 |
+ |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1075 |
+ |
matrix, |
| 1076 |
+ |
\begin{equation} |
| 1077 |
+ |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1078 |
+ |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1079 |
+ |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1080 |
+ |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1081 |
+ |
\end{equation} |
| 1082 |
+ |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1083 |
+ |
\ref{introEquation:skewMatrixPI}, which implies the Lagrange |
| 1084 |
+ |
multiplier $\Lambda$ is ignored in the integration. |
| 1085 |
+ |
|
| 1086 |
+ |
Hence, applying hat-map isomorphism, we obtain the equation of |
| 1087 |
+ |
motion for angular momentum on body frame |
| 1088 |
+ |
\[ |
| 1089 |
+ |
\dot \pi = \pi \times I^{ - 1} \pi + Q^T \sum\limits_i {F_i (r,Q) |
| 1090 |
+ |
\times X_i } |
| 1091 |
+ |
\] |
| 1092 |
+ |
In the same manner, the equation of motion for rotation matrix is |
| 1093 |
+ |
given by |
| 1094 |
+ |
\[ |
| 1095 |
+ |
\dot Q = Qskew(M^{ - 1} \pi ) |
| 1096 |
+ |
\] |
| 1097 |
+ |
|
| 1098 |
+ |
The free rigid body equation is an example of a non-canonical |
| 1099 |
+ |
Hamiltonian system. |
| 1100 |
+ |
|
| 1101 |
+ |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Integration of Euler Equations} |
| 1102 |
+ |
|
| 1103 |
+ |
\[ |
| 1104 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1105 |
+ |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V} |
| 1106 |
+ |
\] |
| 1107 |
+ |
|
| 1108 |
+ |
\[ |
| 1109 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,R} \circ \varphi |
| 1110 |
+ |
_{\Delta t,\pi } |
| 1111 |
+ |
\] |
| 1112 |
+ |
|
| 1113 |
+ |
\[ |
| 1114 |
+ |
\varphi _{\Delta t,\pi } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1115 |
+ |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1116 |
+ |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1117 |
+ |
_1 } |
| 1118 |
+ |
\] |
| 1119 |
+ |
|
| 1120 |
+ |
\[ |
| 1121 |
+ |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1122 |
+ |
_{\Delta t/2,\tau } |
| 1123 |
+ |
\] |
| 1124 |
+ |
|
| 1125 |
+ |
|
| 1126 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1127 |
|
|
| 1128 |
|
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 1330 |
|
|
| 1331 |
|
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1332 |
|
Body} |
| 1333 |
+ |
|
| 1334 |
+ |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
