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# Line 570 | Line 570 | The free rigid body is an example of Poisson system (a
570   \dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
571   \end{equation}
572   The most obvious change being that matrix $J$ now depends on $x$.
573 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}
583
584 \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}
573  
574   \subsection{\label{introSection:exactFlow}Exact Flow}
575  
# Line 950 | Line 935 | coworkers\cite{Dullweber1997}.
935   However, both of these methods are iterative and inefficient. In
936   this section, we will present a symplectic Lie-Poisson integrator
937   for rigid body developed by Dullweber and his
938 < coworkers\cite{Dullweber1997}.
954 <
955 < \subsection{\label{introSection:lieAlgebra}Lie Algebra}
938 > coworkers\cite{Dullweber1997} in depth.
939  
940   \subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body}
941 <
941 > The motion of the rigid body is Hamiltonian with the Hamiltonian
942 > function
943   \begin{equation}
944   H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
945   V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
# Line 1027 | Line 1011 | Hence,
1011   \[
1012   V(q,Q) = V(Q X_0 + q).
1013   \]
1014 < Hence,
1014 > Hence, the force and torque are given by
1015   \[
1016 < \nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}
1016 > \nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
1017   \]
1018 <
1018 > and
1019   \[
1020   \nabla _Q V(q,Q) = F(q,Q)X_i^t
1021   \]
1022 + respectively.
1023  
1024   As a common choice to describe the rotation dynamics of the rigid
1025   body, angular momentum on body frame $\Pi  = Q^t P$ is introduced to
# Line 1080 | Line 1065 | Since $\Lambda$ is symmetric, the last term of Equatio
1065   (\Lambda  - \Lambda ^T ) . \label{introEquation:skewMatrixPI}
1066   \end{equation}
1067   Since $\Lambda$ is symmetric, the last term of Equation
1068 < \ref{introEquation:skewMatrixPI}, which implies the Lagrange
1069 < multiplier $\Lambda$ is ignored in the integration.
1068 > \ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
1069 > multiplier $\Lambda$ is absent from the equations of motion. This
1070 > unique property eliminate the requirement of iterations which can
1071 > not be avoided in other methods\cite{}.
1072  
1073 < Hence, applying hat-map isomorphism, we obtain the equation of
1074 < motion for angular momentum on body frame
1075 < \[
1076 < \dot \pi  = \pi  \times I^{ - 1} \pi  + Q^T \sum\limits_i {F_i (r,Q)
1077 < \times X_i }
1078 < \]
1073 > Applying hat-map isomorphism, we obtain the equation of motion for
1074 > angular momentum on body frame
1075 > \begin{equation}
1076 > \dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
1077 > F_i (r,Q)} \right) \times X_i }.
1078 > \label{introEquation:bodyAngularMotion}
1079 > \end{equation}
1080   In the same manner, the equation of motion for rotation matrix is
1081   given by
1082   \[
1083 < \dot Q = Qskew(M^{ - 1} \pi )
1083 > \dot Q = Qskew(I^{ - 1} \pi )
1084   \]
1085  
1086 < The free rigid body equation is an example of a non-canonical
1087 < Hamiltonian system.
1086 > \subsection{\label{introSection:SymplecticFreeRB}Symplectic
1087 > Lie-Poisson Integrator for Free Rigid Body}
1088  
1089 < \subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Integration of Euler Equations}
1090 <
1091 < \[
1092 < \varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
1093 < _{\Delta t,T}  \circ \varphi _{\Delta t/2,V}
1094 < \]
1089 > If there is not external forces exerted on the rigid body, the only
1090 > contribution to the rotational is from the kinetic potential (the
1091 > first term of \ref{ introEquation:bodyAngularMotion}). The free
1092 > rigid body is an example of Lie-Poisson system with Hamiltonian
1093 > function
1094 > \begin{equation}
1095 > T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
1096 > \label{introEquation:rotationalKineticRB}
1097 > \end{equation}
1098 > where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
1099 > Lie-Poisson structure matrix,
1100 > \begin{equation}
1101 > J(\pi ) = \left( {\begin{array}{*{20}c}
1102 >   0 & {\pi _3 } & { - \pi _2 }  \\
1103 >   { - \pi _3 } & 0 & {\pi _1 }  \\
1104 >   {\pi _2 } & { - \pi _1 } & 0  \\
1105 > \end{array}} \right)
1106 > \end{equation}
1107 > Thus, the dynamics of free rigid body is governed by
1108 > \begin{equation}
1109 > \frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
1110 > \end{equation}
1111  
1112 + One may notice that each $T_i^r$ in Equation
1113 + \ref{introEquation:rotationalKineticRB} can be solved exactly. For
1114 + instance, the equations of motion due to $T_1^r$ are given by
1115 + \begin{equation}
1116 + \frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
1117 + \label{introEqaution:RBMotionSingleTerm}
1118 + \end{equation}
1119 + where
1120 + \[ R_1  = \left( {\begin{array}{*{20}c}
1121 +   0 & 0 & 0  \\
1122 +   0 & 0 & {\pi _1 }  \\
1123 +   0 & { - \pi _1 } & 0  \\
1124 + \end{array}} \right).
1125 + \]
1126 + The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is
1127   \[
1128 < \varphi _{\Delta t,T}  = \varphi _{\Delta t,R}  \circ \varphi
1129 < _{\Delta t,\pi }
1128 > \pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
1129 > Q(0)e^{\Delta tR_1 }
1130   \]
1131 <
1131 > with
1132   \[
1133 < \varphi _{\Delta t,\pi }  = \varphi _{\Delta t/2,\pi _1 }  \circ
1134 < \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
1135 < \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1136 < _1 }
1133 > e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
1134 >   0 & 0 & 0  \\
1135 >   0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
1136 >   0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
1137 > \end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
1138   \]
1139 <
1139 > To reduce the cost of computing expensive functions in e^{\Delta
1140 > tR_1 }, we can use Cayley transformation,
1141   \[
1142 < \varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
1143 < _{\Delta t/2,\tau }
1142 > e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
1143 > )
1144 > \]
1145 >
1146 > The flow maps for $T_2^r$ and $T_2^r$ can be found in the same
1147 > manner.
1148 >
1149 > In order to construct a second-order symplectic method, we split the
1150 > angular kinetic Hamiltonian function can into five terms
1151 > \[
1152 > T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
1153 > ) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
1154 > (\pi _1 )
1155 > \].
1156 > Concatenating flows corresponding to these five terms, we can obtain
1157 > an symplectic integrator,
1158 > \[
1159 > \varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
1160 > \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
1161 > \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1162 > _1 }.
1163   \]
1164  
1165 + The non-canonical Lie-Poisson bracket ${F, G}$ of two function
1166 + $F(\pi )$ and $G(\pi )$ is defined by
1167 + \[
1168 + \{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
1169 + )
1170 + \]
1171 + If the Poisson bracket of a function $F$ with an arbitrary smooth
1172 + function $G$ is zero, $F$ is a \emph{Casimir}, which is the
1173 + conserved quantity in Poisson system. We can easily verify that the
1174 + norm of the angular momentum, $\parallel \pi
1175 + \parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel
1176 + \pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
1177 + then by the chain rule
1178 + \[
1179 + \nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
1180 + }}{2})\pi
1181 + \]
1182 + Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
1183 + \parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
1184 + Lie-Poisson integrator is found to be extremely efficient and stable
1185 + which can be explained by the fact the small angle approximation is
1186 + used and the norm of the angular momentum is conserved.
1187  
1188 < \section{\label{introSection:langevinDynamics}Langevin Dynamics}
1188 > \subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
1189 > Splitting for Rigid Body}
1190  
1191 < \subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
1191 > The Hamiltonian of rigid body can be separated in terms of kinetic
1192 > energy and potential energy,
1193 > \[
1194 > H = T(p,\pi ) + V(q,Q)
1195 > \]
1196 > The equations of motion corresponding to potential energy and
1197 > kinetic energy are listed in the below table,
1198 > \begin{center}
1199 > \begin{tabular}{|l|l|}
1200 >  \hline
1201 >  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
1202 >  Potential & Kinetic \\
1203 >  $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
1204 >  $\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
1205 >  $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
1206 >  $ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
1207 >  \hline
1208 > \end{tabular}
1209 > \end{center}
1210 > A second-order symplectic method is now obtained by the composition
1211 > of the flow maps,
1212 > \[
1213 > \varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
1214 > _{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
1215 > \]
1216 > Moreover, \varphi _{\Delta t/2,V} can be divided into two sub-flows
1217 > which corresponding to force and torque respectively,
1218 > \[
1219 > \varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
1220 > _{\Delta t/2,\tau }.
1221 > \]
1222 > Since the associated operators of $\varphi _{\Delta t/2,F} $ and
1223 > $\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition
1224 > order inside \varphi _{\Delta t/2,V} does not matter.
1225  
1226 + Furthermore, kinetic potential can be separated to translational
1227 + kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$,
1228 + \begin{equation}
1229 + T(p,\pi ) =T^t (p) + T^r (\pi ).
1230 + \end{equation}
1231 + where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
1232 + defined by \ref{introEquation:rotationalKineticRB}. Therefore, the
1233 + corresponding flow maps are given by
1234 + \[
1235 + \varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
1236 + _{\Delta t,T^r }.
1237 + \]
1238 + Finally, we obtain the overall symplectic flow maps for free moving
1239 + rigid body
1240 + \begin{equation}
1241 + \begin{array}{c}
1242 + \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
1243 +  \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 }  \\
1244 +  \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
1245 + \end{array}
1246 + \label{introEquation:overallRBFlowMaps}
1247 + \end{equation}
1248 +
1249 + \section{\label{introSection:langevinDynamics}Langevin Dynamics}
1250 + As an alternative to newtonian dynamics, Langevin dynamics, which
1251 + mimics a simple heat bath with stochastic and dissipative forces,
1252 + has been applied in a variety of studies. This section will review
1253 + the theory of Langevin dynamics simulation. A brief derivation of
1254 + generalized Langevin Dynamics will be given first. Follow that, we
1255 + will discuss the physical meaning of the terms appearing in the
1256 + equation as well as the calculation of friction tensor from
1257 + hydrodynamics theory.
1258 +
1259   \subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
1260  
1261   \begin{equation}
# Line 1319 | Line 1448 | And since the $q$ coordinates are harmonic oscillators
1448   \label{introEquation:secondFluctuationDissipation}
1449   \end{equation}
1450  
1322 \section{\label{introSection:hydroynamics}Hydrodynamics}
1323
1451   \subsection{\label{introSection:frictionTensor} Friction Tensor}
1452 < \subsection{\label{introSection:analyticalApproach}Analytical
1453 < Approach}
1454 <
1455 < \subsection{\label{introSection:approximationApproach}Approximation
1456 < Approach}
1452 > Theoretically, the friction kernel can be determined using velocity
1453 > autocorrelation function. However, this approach become impractical
1454 > when the system become more and more complicate. Instead, various
1455 > approaches based on hydrodynamics have been developed to calculate
1456 > the friction coefficients. The friction effect is isotropic in
1457 > Equation, \zeta can be taken as a scalar. In general, friction
1458 > tensor \Xi is a $6\times 6$ matrix given by
1459 > \[
1460 > \Xi  = \left( {\begin{array}{*{20}c}
1461 >   {\Xi _{}^{tt} } & {\Xi _{}^{rt} }  \\
1462 >   {\Xi _{}^{tr} } & {\Xi _{}^{rr} }  \\
1463 > \end{array}} \right).
1464 > \]
1465 > Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction
1466 > tensor and rotational friction tensor respectively, while ${\Xi^{tr}
1467 > }$ is translation-rotation coupling tensor and $ {\Xi^{rt} }$ is
1468 > rotation-translation coupling tensor.
1469  
1470 < \subsection{\label{introSection:centersRigidBody}Centers of Rigid
1471 < Body}
1470 > \[
1471 > \left( \begin{array}{l}
1472 > F_t  \\
1473 > \tau  \\
1474 > \end{array} \right) =  - \left( {\begin{array}{*{20}c}
1475 >   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
1476 >   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
1477 > \end{array}} \right)\left( \begin{array}{l}
1478 > v \\
1479 > w \\
1480 > \end{array} \right)
1481 > \]
1482  
1483 < \section{\label{introSection:correlationFunctions}Correlation Functions}
1483 > \subsubsection{\label{introSection:analyticalApproach}The Friction Tensor for Regular Shape}
1484 > For a spherical particle, the translational and rotational friction
1485 > constant can be calculated from Stoke's law,
1486 > \[
1487 > \Xi ^{tt}  = \left( {\begin{array}{*{20}c}
1488 >   {6\pi \eta R} & 0 & 0  \\
1489 >   0 & {6\pi \eta R} & 0  \\
1490 >   0 & 0 & {6\pi \eta R}  \\
1491 > \end{array}} \right)
1492 > \]
1493 > and
1494 > \[
1495 > \Xi ^{rr}  = \left( {\begin{array}{*{20}c}
1496 >   {8\pi \eta R^3 } & 0 & 0  \\
1497 >   0 & {8\pi \eta R^3 } & 0  \\
1498 >   0 & 0 & {8\pi \eta R^3 }  \\
1499 > \end{array}} \right)
1500 > \]
1501 > where $\eta$ is the viscosity of the solvent and $R$ is the
1502 > hydrodynamics radius.
1503 >
1504 > Other non-spherical particles have more complex properties.
1505 >
1506 > \[
1507 > S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
1508 > } }}{b}
1509 > \]
1510 >
1511 >
1512 > \[
1513 > S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
1514 > }}{a}
1515 > \]
1516 >
1517 > \[
1518 > \begin{array}{l}
1519 > \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
1520 > \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
1521 > \end{array}
1522 > \]
1523 >
1524 > \[
1525 > \begin{array}{l}
1526 > \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
1527 > \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
1528 > \end{array}
1529 > \]
1530 >
1531 >
1532 > \subsubsection{\label{introSection:approximationApproach}The Friction Tensor for Arbitrary Shape}
1533 > Unlike spherical and other regular shaped molecules, there is not
1534 > analytical solution for friction tensor of any arbitrary shaped
1535 > rigid molecules. The ellipsoid of revolution model and general
1536 > triaxial ellipsoid model have been used to approximate the
1537 > hydrodynamic properties of rigid bodies. However, since the mapping
1538 > from all possible ellipsoidal space, $r$-space, to all possible
1539 > combination of rotational diffusion coefficients, $D$-space is not
1540 > unique\cite{Wegener79} as well as the intrinsic coupling between
1541 > translational and rotational motion of rigid body\cite{}, general
1542 > ellipsoid is not always suitable for modeling arbitrarily shaped
1543 > rigid molecule. A number of studies have been devoted to determine
1544 > the friction tensor for irregularly shaped rigid bodies using more
1545 > advanced method\cite{} where the molecule of interest was modeled by
1546 > combinations of spheres(beads)\cite{} and the hydrodynamics
1547 > properties of the molecule can be calculated using the hydrodynamic
1548 > interaction tensor. Let us consider a rigid assembly of $N$ beads
1549 > immersed in a continuous medium. Due to hydrodynamics interaction,
1550 > the ``net'' velocity of $i$th bead, $v'_i$ is different than its
1551 > unperturbed velocity $v_i$,
1552 > \[
1553 > v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
1554 > \]
1555 > where $F_i$ is the frictional force, and $T_{ij}$ is the
1556 > hydrodynamic interaction tensor. The friction force of $i$th bead is
1557 > proportional to its ``net'' velocity
1558 > \begin{equation}
1559 > F_i  = \zeta _i v_i  - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }.
1560 > \label{introEquation:tensorExpression}
1561 > \end{equation}
1562 > This equation is the basis for deriving the hydrodynamic tensor. In
1563 > 1930, Oseen and Burgers gave a simple solution to Equation
1564 > \ref{introEquation:tensorExpression}
1565 > \begin{equation}
1566 > T_{ij}  = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij}
1567 > R_{ij}^T }}{{R_{ij}^2 }}} \right).
1568 > \label{introEquation:oseenTensor}
1569 > \end{equation}
1570 > Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
1571 > A second order expression for element of different size was
1572 > introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de
1573 > la Torre and Bloomfield,
1574 > \begin{equation}
1575 > T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
1576 > \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
1577 > _i^2  + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} -
1578 > \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
1579 > \label{introEquation:RPTensorNonOverlapped}
1580 > \end{equation}
1581 > Both of the Equation \ref{introEquation:oseenTensor} and Equation
1582 > \ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij}
1583 > \ge \sigma _i  + \sigma _j$. An alternative expression for
1584 > overlapping beads with the same radius, $\sigma$, is given by
1585 > \begin{equation}
1586 > T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
1587 > \frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I +
1588 > \frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
1589 > \label{introEquation:RPTensorOverlapped}
1590 > \end{equation}
1591 >
1592 > %Bead Modeling
1593 >
1594 > \[
1595 > B = \left( {\begin{array}{*{20}c}
1596 >   {T_{11} } &  \ldots  & {T_{1N} }  \\
1597 >    \vdots  &  \ddots  &  \vdots   \\
1598 >   {T_{N1} } &  \cdots  & {T_{NN} }  \\
1599 > \end{array}} \right)
1600 > \]
1601 >
1602 > \[
1603 > C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
1604 >   {C_{11} } &  \ldots  & {C_{1N} }  \\
1605 >    \vdots  &  \ddots  &  \vdots   \\
1606 >   {C_{N1} } &  \cdots  & {C_{NN} }  \\
1607 > \end{array}} \right)
1608 > \]
1609 >
1610 > \begin{equation}
1611 > \begin{array}{l}
1612 > \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\
1613 > \Xi _{}^{tr}  = \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
1614 > \Xi _{}^{rr}  =  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j  \\
1615 > \end{array}
1616 > \end{equation}
1617 > where
1618 > \[
1619 > U_i  = \left( {\begin{array}{*{20}c}
1620 >   0 & { - z_i } & {y_i }  \\
1621 >   {z_i } & 0 & { - x_i }  \\
1622 >   { - y_i } & {x_i } & 0  \\
1623 > \end{array}} \right)
1624 > \]
1625 >
1626 > \[
1627 > r_{OR}  = \left( \begin{array}{l}
1628 > x_{OR}  \\
1629 > y_{OR}  \\
1630 > z_{OR}  \\
1631 > \end{array} \right) = \left( {\begin{array}{*{20}c}
1632 >   {\Xi _{yy}^{rr}  + \Xi _{zz}^{rr} } & { - \Xi _{xy}^{rr} } & { - \Xi _{xz}^{rr} }  \\
1633 >   { - \Xi _{yx}^{rr} } & {\Xi _{zz}^{rr}  + \Xi _{xx}^{rr} } & { - \Xi _{yz}^{rr} }  \\
1634 >   { - \Xi _{zx}^{rr} } & { - \Xi _{yz}^{rr} } & {\Xi _{xx}^{rr}  + \Xi _{yy}^{rr} }  \\
1635 > \end{array}} \right)^{ - 1} \left( \begin{array}{l}
1636 > \Xi _{yz}^{tr}  - \Xi _{zy}^{tr}  \\
1637 > \Xi _{zx}^{tr}  - \Xi _{xz}^{tr}  \\
1638 > \Xi _{xy}^{tr}  - \Xi _{yx}^{tr}  \\
1639 > \end{array} \right)
1640 > \]
1641 >
1642 > \[
1643 > U_{OR}  = \left( {\begin{array}{*{20}c}
1644 >   0 & { - z_{OR} } & {y_{OR} }  \\
1645 >   {z_i } & 0 & { - x_{OR} }  \\
1646 >   { - y_{OR} } & {x_{OR} } & 0  \\
1647 > \end{array}} \right)
1648 > \]
1649 >
1650 > \[
1651 > \begin{array}{l}
1652 > \Xi _R^{tt}  = \Xi _{}^{tt}  \\
1653 > \Xi _R^{tr}  = \Xi _R^{rt}  = \Xi _{}^{tr}  - U_{OR} \Xi _{}^{tt}  \\
1654 > \Xi _R^{rr}  = \Xi _{}^{rr}  - U_{OR} \Xi _{}^{tt} U_{OR}  + \Xi _{}^{tr} U_{OR}  - U_{OR} \Xi _{}^{tr} ^{^T }  \\
1655 > \end{array}
1656 > \]
1657 >
1658 > \[
1659 > D_R  = \left( {\begin{array}{*{20}c}
1660 >   {D_R^{tt} } & {D_R^{rt} }  \\
1661 >   {D_R^{tr} } & {D_R^{rr} }  \\
1662 > \end{array}} \right) = k_b T\left( {\begin{array}{*{20}c}
1663 >   {\Xi _R^{tt} } & {\Xi _R^{rt} }  \\
1664 >   {\Xi _R^{tr} } & {\Xi _R^{rr} }  \\
1665 > \end{array}} \right)^{ - 1}
1666 > \]
1667 >
1668 >
1669 > %Approximation Methods
1670 >
1671 > %\section{\label{introSection:correlationFunctions}Correlation Functions}

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