| 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} |
| 573 |
|
|
| 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} |
| 588 |
– |
|
| 574 |
|
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 575 |
|
|
| 576 |
|
Let $x(t)$ be the exact solution of the ODE system, |
| 822 |
|
% |
| 823 |
|
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 824 |
|
q(\Delta t)} \right]. % |
| 825 |
< |
\label{introEquation:positionVerlet1} |
| 825 |
> |
\label{introEquation:positionVerlet2} |
| 826 |
|
\end{align} |
| 827 |
|
|
| 828 |
|
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
| 888 |
|
biological systems, providing structural, thermodynamic and |
| 889 |
|
dynamical information. |
| 890 |
|
|
| 891 |
+ |
One of the principal tools for modeling proteins, nucleic acids and |
| 892 |
+ |
their complexes. Stability of proteins Folding of proteins. |
| 893 |
+ |
Molecular recognition by:proteins, DNA, RNA, lipids, hormones STP, |
| 894 |
+ |
etc. Enzyme reactions Rational design of biologically active |
| 895 |
+ |
molecules (drug design) Small and large-scale conformational |
| 896 |
+ |
changes. determination and construction of 3D structures (homology, |
| 897 |
+ |
Xray diffraction, NMR) Dynamic processes such as ion transport in |
| 898 |
+ |
biological systems. |
| 899 |
+ |
|
| 900 |
+ |
Macroscopic properties are related to microscopic behavior. |
| 901 |
+ |
|
| 902 |
+ |
Time dependent (and independent) microscopic behavior of a molecule |
| 903 |
+ |
can be calculated by molecular dynamics simulations. |
| 904 |
+ |
|
| 905 |
|
\subsection{\label{introSec:mdInit}Initialization} |
| 906 |
|
|
| 907 |
|
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 941 |
|
The break through in geometric literature suggests that, in order to |
| 942 |
|
develop a long-term integration scheme, one should preserve the |
| 943 |
|
symplectic structure of the flow. Introducing conjugate momentum to |
| 944 |
< |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
| 944 |
> |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
| 945 |
|
symplectic integrator, RSHAKE, was proposed to evolve the |
| 946 |
|
Hamiltonian system in a constraint manifold by iteratively |
| 947 |
< |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
| 947 |
> |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
| 948 |
|
method using quaternion representation was developed by Omelyan. |
| 949 |
|
However, both of these methods are iterative and inefficient. In |
| 950 |
|
this section, we will present a symplectic Lie-Poisson integrator |
| 951 |
|
for rigid body developed by Dullweber and his |
| 952 |
< |
coworkers\cite{Dullweber1997}. |
| 952 |
> |
coworkers\cite{Dullweber1997} in depth. |
| 953 |
|
|
| 955 |
– |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 956 |
– |
|
| 954 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 955 |
< |
|
| 955 |
> |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
| 956 |
> |
function |
| 957 |
|
\begin{equation} |
| 958 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 959 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1025 |
|
\[ |
| 1026 |
|
V(q,Q) = V(Q X_0 + q). |
| 1027 |
|
\] |
| 1028 |
< |
Hence, |
| 1028 |
> |
Hence, the force and torque are given by |
| 1029 |
|
\[ |
| 1030 |
< |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)} |
| 1030 |
> |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, |
| 1031 |
|
\] |
| 1032 |
< |
|
| 1032 |
> |
and |
| 1033 |
|
\[ |
| 1034 |
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1035 |
|
\] |
| 1036 |
+ |
respectively. |
| 1037 |
|
|
| 1038 |
|
As a common choice to describe the rotation dynamics of the rigid |
| 1039 |
|
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1079 |
|
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1080 |
|
\end{equation} |
| 1081 |
|
Since $\Lambda$ is symmetric, the last term of Equation |
| 1082 |
< |
\ref{introEquation:skewMatrixPI}, which implies the Lagrange |
| 1083 |
< |
multiplier $\Lambda$ is ignored in the integration. |
| 1082 |
> |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1083 |
> |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1084 |
> |
unique property eliminate the requirement of iterations which can |
| 1085 |
> |
not be avoided in other methods\cite{}. |
| 1086 |
|
|
| 1087 |
< |
Hence, applying hat-map isomorphism, we obtain the equation of |
| 1088 |
< |
motion for angular momentum on body frame |
| 1089 |
< |
\[ |
| 1090 |
< |
\dot \pi = \pi \times I^{ - 1} \pi + Q^T \sum\limits_i {F_i (r,Q) |
| 1091 |
< |
\times X_i } |
| 1092 |
< |
\] |
| 1087 |
> |
Applying hat-map isomorphism, we obtain the equation of motion for |
| 1088 |
> |
angular momentum on body frame |
| 1089 |
> |
\begin{equation} |
| 1090 |
> |
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1091 |
> |
F_i (r,Q)} \right) \times X_i }. |
| 1092 |
> |
\label{introEquation:bodyAngularMotion} |
| 1093 |
> |
\end{equation} |
| 1094 |
|
In the same manner, the equation of motion for rotation matrix is |
| 1095 |
|
given by |
| 1096 |
|
\[ |
| 1097 |
< |
\dot Q = Qskew(M^{ - 1} \pi ) |
| 1097 |
> |
\dot Q = Qskew(I^{ - 1} \pi ) |
| 1098 |
|
\] |
| 1099 |
|
|
| 1100 |
< |
The free rigid body equation is an example of a non-canonical |
| 1101 |
< |
Hamiltonian system. |
| 1100 |
> |
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1101 |
> |
Lie-Poisson Integrator for Free Rigid Body} |
| 1102 |
|
|
| 1103 |
< |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Integration of Euler Equations} |
| 1103 |
> |
If there is not external forces exerted on the rigid body, the only |
| 1104 |
> |
contribution to the rotational is from the kinetic potential (the |
| 1105 |
> |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
| 1106 |
> |
rigid body is an example of Lie-Poisson system with Hamiltonian |
| 1107 |
> |
function |
| 1108 |
> |
\begin{equation} |
| 1109 |
> |
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1110 |
> |
\label{introEquation:rotationalKineticRB} |
| 1111 |
> |
\end{equation} |
| 1112 |
> |
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
| 1113 |
> |
Lie-Poisson structure matrix, |
| 1114 |
> |
\begin{equation} |
| 1115 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 1116 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1117 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1118 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1119 |
> |
\end{array}} \right) |
| 1120 |
> |
\end{equation} |
| 1121 |
> |
Thus, the dynamics of free rigid body is governed by |
| 1122 |
> |
\begin{equation} |
| 1123 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1124 |
> |
\end{equation} |
| 1125 |
|
|
| 1126 |
< |
\[ |
| 1127 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1128 |
< |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V} |
| 1129 |
< |
\] |
| 1130 |
< |
|
| 1131 |
< |
\[ |
| 1132 |
< |
\varphi _{\Delta t,T} = \varphi _{\Delta t,R} \circ \varphi |
| 1133 |
< |
_{\Delta t,\pi } |
| 1126 |
> |
One may notice that each $T_i^r$ in Equation |
| 1127 |
> |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1128 |
> |
instance, the equations of motion due to $T_1^r$ are given by |
| 1129 |
> |
\begin{equation} |
| 1130 |
> |
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
| 1131 |
> |
\label{introEqaution:RBMotionSingleTerm} |
| 1132 |
> |
\end{equation} |
| 1133 |
> |
where |
| 1134 |
> |
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1135 |
> |
0 & 0 & 0 \\ |
| 1136 |
> |
0 & 0 & {\pi _1 } \\ |
| 1137 |
> |
0 & { - \pi _1 } & 0 \\ |
| 1138 |
> |
\end{array}} \right). |
| 1139 |
|
\] |
| 1140 |
+ |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1141 |
+ |
\[ |
| 1142 |
+ |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
| 1143 |
+ |
Q(0)e^{\Delta tR_1 } |
| 1144 |
+ |
\] |
| 1145 |
+ |
with |
| 1146 |
+ |
\[ |
| 1147 |
+ |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
| 1148 |
+ |
0 & 0 & 0 \\ |
| 1149 |
+ |
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
| 1150 |
+ |
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
| 1151 |
+ |
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1152 |
+ |
\] |
| 1153 |
+ |
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1154 |
+ |
tR_1 }$, we can use Cayley transformation, |
| 1155 |
+ |
\[ |
| 1156 |
+ |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1157 |
+ |
) |
| 1158 |
+ |
\] |
| 1159 |
|
|
| 1160 |
+ |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
| 1161 |
+ |
manner. |
| 1162 |
+ |
|
| 1163 |
+ |
In order to construct a second-order symplectic method, we split the |
| 1164 |
+ |
angular kinetic Hamiltonian function can into five terms |
| 1165 |
|
\[ |
| 1166 |
< |
\varphi _{\Delta t,\pi } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1166 |
> |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1167 |
> |
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1168 |
> |
(\pi _1 ) |
| 1169 |
> |
\]. |
| 1170 |
> |
Concatenating flows corresponding to these five terms, we can obtain |
| 1171 |
> |
an symplectic integrator, |
| 1172 |
> |
\[ |
| 1173 |
> |
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1174 |
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1175 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1176 |
< |
_1 } |
| 1176 |
> |
_1 }. |
| 1177 |
|
\] |
| 1178 |
|
|
| 1179 |
+ |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1180 |
+ |
$F(\pi )$ and $G(\pi )$ is defined by |
| 1181 |
|
\[ |
| 1182 |
+ |
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1183 |
+ |
) |
| 1184 |
+ |
\] |
| 1185 |
+ |
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1186 |
+ |
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1187 |
+ |
conserved quantity in Poisson system. We can easily verify that the |
| 1188 |
+ |
norm of the angular momentum, $\parallel \pi |
| 1189 |
+ |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1190 |
+ |
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1191 |
+ |
then by the chain rule |
| 1192 |
+ |
\[ |
| 1193 |
+ |
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1194 |
+ |
}}{2})\pi |
| 1195 |
+ |
\] |
| 1196 |
+ |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1197 |
+ |
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1198 |
+ |
Lie-Poisson integrator is found to be extremely efficient and stable |
| 1199 |
+ |
which can be explained by the fact the small angle approximation is |
| 1200 |
+ |
used and the norm of the angular momentum is conserved. |
| 1201 |
+ |
|
| 1202 |
+ |
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
| 1203 |
+ |
Splitting for Rigid Body} |
| 1204 |
+ |
|
| 1205 |
+ |
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1206 |
+ |
energy and potential energy, |
| 1207 |
+ |
\[ |
| 1208 |
+ |
H = T(p,\pi ) + V(q,Q) |
| 1209 |
+ |
\] |
| 1210 |
+ |
The equations of motion corresponding to potential energy and |
| 1211 |
+ |
kinetic energy are listed in the below table, |
| 1212 |
+ |
\begin{center} |
| 1213 |
+ |
\begin{tabular}{|l|l|} |
| 1214 |
+ |
\hline |
| 1215 |
+ |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 1216 |
+ |
Potential & Kinetic \\ |
| 1217 |
+ |
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
| 1218 |
+ |
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
| 1219 |
+ |
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
| 1220 |
+ |
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
| 1221 |
+ |
\hline |
| 1222 |
+ |
\end{tabular} |
| 1223 |
+ |
\end{center} |
| 1224 |
+ |
A second-order symplectic method is now obtained by the composition |
| 1225 |
+ |
of the flow maps, |
| 1226 |
+ |
\[ |
| 1227 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1228 |
+ |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1229 |
+ |
\] |
| 1230 |
+ |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 1231 |
+ |
sub-flows which corresponding to force and torque respectively, |
| 1232 |
+ |
\[ |
| 1233 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1234 |
< |
_{\Delta t/2,\tau } |
| 1234 |
> |
_{\Delta t/2,\tau }. |
| 1235 |
|
\] |
| 1236 |
+ |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1237 |
+ |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 1238 |
+ |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
| 1239 |
|
|
| 1240 |
+ |
Furthermore, kinetic potential can be separated to translational |
| 1241 |
+ |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1242 |
+ |
\begin{equation} |
| 1243 |
+ |
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1244 |
+ |
\end{equation} |
| 1245 |
+ |
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1246 |
+ |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1247 |
+ |
corresponding flow maps are given by |
| 1248 |
+ |
\[ |
| 1249 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1250 |
+ |
_{\Delta t,T^r }. |
| 1251 |
+ |
\] |
| 1252 |
+ |
Finally, we obtain the overall symplectic flow maps for free moving |
| 1253 |
+ |
rigid body |
| 1254 |
+ |
\begin{equation} |
| 1255 |
+ |
\begin{array}{c} |
| 1256 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1257 |
+ |
\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 } \\ |
| 1258 |
+ |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1259 |
+ |
\end{array} |
| 1260 |
+ |
\label{introEquation:overallRBFlowMaps} |
| 1261 |
+ |
\end{equation} |
| 1262 |
|
|
| 1263 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1264 |
+ |
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1265 |
+ |
mimics a simple heat bath with stochastic and dissipative forces, |
| 1266 |
+ |
has been applied in a variety of studies. This section will review |
| 1267 |
+ |
the theory of Langevin dynamics simulation. A brief derivation of |
| 1268 |
+ |
generalized Langevin equation will be given first. Follow that, we |
| 1269 |
+ |
will discuss the physical meaning of the terms appearing in the |
| 1270 |
+ |
equation as well as the calculation of friction tensor from |
| 1271 |
+ |
hydrodynamics theory. |
| 1272 |
|
|
| 1273 |
< |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 1273 |
> |
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1274 |
|
|
| 1275 |
< |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 1276 |
< |
|
| 1275 |
> |
Harmonic bath model, in which an effective set of harmonic |
| 1276 |
> |
oscillators are used to mimic the effect of a linearly responding |
| 1277 |
> |
environment, has been widely used in quantum chemistry and |
| 1278 |
> |
statistical mechanics. One of the successful applications of |
| 1279 |
> |
Harmonic bath model is the derivation of Deriving Generalized |
| 1280 |
> |
Langevin Dynamics. Lets consider a system, in which the degree of |
| 1281 |
> |
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1282 |
> |
Hamiltonian of the form |
| 1283 |
|
\begin{equation} |
| 1284 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1285 |
< |
\label{introEquation:bathGLE} |
| 1285 |
> |
\label{introEquation:bathGLE}. |
| 1286 |
|
\end{equation} |
| 1287 |
< |
where $H_B$ is harmonic bath Hamiltonian, |
| 1287 |
> |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
| 1288 |
> |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
| 1289 |
|
\[ |
| 1290 |
< |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1291 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
| 1290 |
> |
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1291 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1292 |
> |
\right\}} |
| 1293 |
|
\] |
| 1294 |
< |
and $\Delta U$ is bilinear system-bath coupling, |
| 1294 |
> |
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1295 |
> |
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
| 1296 |
> |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
| 1297 |
> |
coupling, |
| 1298 |
|
\[ |
| 1299 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1300 |
|
\] |
| 1301 |
< |
Completing the square, |
| 1301 |
> |
where $g_\alpha$ are the coupling constants between the bath and the |
| 1302 |
> |
coordinate $x$. Introducing |
| 1303 |
|
\[ |
| 1304 |
< |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
| 1305 |
< |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1306 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1307 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
| 1308 |
< |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1152 |
< |
\] |
| 1153 |
< |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
| 1304 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1305 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1306 |
> |
\] and combining the last two terms in Equation |
| 1307 |
> |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
| 1308 |
> |
Hamiltonian as |
| 1309 |
|
\[ |
| 1310 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1311 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1312 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1313 |
|
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1314 |
|
\] |
| 1160 |
– |
where |
| 1161 |
– |
\[ |
| 1162 |
– |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1163 |
– |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1164 |
– |
\] |
| 1315 |
|
Since the first two terms of the new Hamiltonian depend only on the |
| 1316 |
|
system coordinates, we can get the equations of motion for |
| 1317 |
|
Generalized Langevin Dynamics by Hamilton's equations |
| 1318 |
|
\ref{introEquation:motionHamiltonianCoordinate, |
| 1319 |
|
introEquation:motionHamiltonianMomentum}, |
| 1320 |
< |
\begin{align} |
| 1321 |
< |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 1322 |
< |
&= m\ddot x |
| 1323 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
| 1324 |
< |
\label{introEquation:Lp5} |
| 1325 |
< |
\end{align} |
| 1326 |
< |
, and |
| 1327 |
< |
\begin{align} |
| 1328 |
< |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
| 1329 |
< |
&= m\ddot x_\alpha |
| 1330 |
< |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
| 1331 |
< |
\end{align} |
| 1320 |
> |
\begin{equation} |
| 1321 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1322 |
> |
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
| 1323 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)}, |
| 1324 |
> |
\label{introEquation:coorMotionGLE} |
| 1325 |
> |
\end{equation} |
| 1326 |
> |
and |
| 1327 |
> |
\begin{equation} |
| 1328 |
> |
m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - |
| 1329 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1330 |
> |
\label{introEquation:bathMotionGLE} |
| 1331 |
> |
\end{equation} |
| 1332 |
|
|
| 1333 |
< |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
| 1333 |
> |
In order to derive an equation for $x$, the dynamics of the bath |
| 1334 |
> |
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1335 |
> |
transform which is particularly useful in solving linear ordinary |
| 1336 |
> |
differential equations, Laplace transform is the appropriate tool to |
| 1337 |
> |
solve this problem. The basic idea is to transform the difficult |
| 1338 |
> |
differential equations into simple algebra problems which can be |
| 1339 |
> |
solved easily. Then applying inverse Laplace transform, also known |
| 1340 |
> |
as the Bromwich integral, we can retrieve the solutions of the |
| 1341 |
> |
original problems. |
| 1342 |
|
|
| 1343 |
+ |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1344 |
+ |
transform of f(t) is a new function defined as |
| 1345 |
|
\[ |
| 1346 |
< |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
| 1346 |
> |
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1347 |
|
\] |
| 1348 |
+ |
where $p$ is real and $L$ is called the Laplace Transform |
| 1349 |
+ |
Operator. Below are some important properties of Laplace transform |
| 1350 |
+ |
\begin{equation} |
| 1351 |
+ |
\begin{array}{c} |
| 1352 |
+ |
L(x + y) = L(x) + L(y) \\ |
| 1353 |
+ |
L(ax) = aL(x) \\ |
| 1354 |
+ |
L(\dot x) = pL(x) - px(0) \\ |
| 1355 |
+ |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
| 1356 |
+ |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
| 1357 |
+ |
\end{array} |
| 1358 |
+ |
\end{equation} |
| 1359 |
|
|
| 1360 |
+ |
Applying Laplace transform to the bath coordinates, we obtain |
| 1361 |
|
\[ |
| 1362 |
< |
L(x + y) = L(x) + L(y) |
| 1362 |
> |
\begin{array}{c} |
| 1363 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
| 1364 |
> |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1365 |
> |
\end{array} |
| 1366 |
|
\] |
| 1367 |
< |
|
| 1367 |
> |
By the same way, the system coordinates become |
| 1368 |
|
\[ |
| 1369 |
< |
L(ax) = aL(x) |
| 1369 |
> |
\begin{array}{c} |
| 1370 |
> |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1371 |
> |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1372 |
> |
\end{array} |
| 1373 |
|
\] |
| 1374 |
|
|
| 1375 |
+ |
With the help of some relatively important inverse Laplace |
| 1376 |
+ |
transformations: |
| 1377 |
|
\[ |
| 1378 |
< |
L(\dot x) = pL(x) - px(0) |
| 1378 |
> |
\begin{array}{c} |
| 1379 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ |
| 1380 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ |
| 1381 |
> |
L(1) = \frac{1}{p} \\ |
| 1382 |
> |
\end{array} |
| 1383 |
|
\] |
| 1384 |
< |
|
| 1201 |
< |
\[ |
| 1202 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
| 1203 |
< |
\] |
| 1204 |
< |
|
| 1205 |
< |
\[ |
| 1206 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
| 1207 |
< |
\] |
| 1208 |
< |
|
| 1209 |
< |
Some relatively important transformation, |
| 1210 |
< |
\[ |
| 1211 |
< |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
| 1212 |
< |
\] |
| 1213 |
< |
|
| 1214 |
< |
\[ |
| 1215 |
< |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
| 1216 |
< |
\] |
| 1217 |
< |
|
| 1218 |
< |
\[ |
| 1219 |
< |
L(1) = \frac{1}{p} |
| 1220 |
< |
\] |
| 1221 |
< |
|
| 1222 |
< |
First, the bath coordinates, |
| 1223 |
< |
\[ |
| 1224 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
| 1225 |
< |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
| 1226 |
< |
}}L(x) |
| 1227 |
< |
\] |
| 1228 |
< |
\[ |
| 1229 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
| 1230 |
< |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
| 1231 |
< |
\] |
| 1232 |
< |
Then, the system coordinates, |
| 1233 |
< |
\begin{align} |
| 1234 |
< |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1235 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
| 1236 |
< |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
| 1237 |
< |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
| 1238 |
< |
}}\omega _\alpha ^2 L(x)} \right\}} |
| 1239 |
< |
% |
| 1240 |
< |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 1241 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
| 1242 |
< |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
| 1243 |
< |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
| 1244 |
< |
\end{align} |
| 1245 |
< |
Then, the inverse transform, |
| 1246 |
< |
|
| 1384 |
> |
, we obtain |
| 1385 |
|
\begin{align} |
| 1386 |
|
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
| 1387 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1401 |
|
(\omega _\alpha t)} \right\}} |
| 1402 |
|
\end{align} |
| 1403 |
|
|
| 1404 |
+ |
Introducing a \emph{dynamic friction kernel} |
| 1405 |
|
\begin{equation} |
| 1406 |
+ |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1407 |
+ |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 1408 |
+ |
\label{introEquation:dynamicFrictionKernelDefinition} |
| 1409 |
+ |
\end{equation} |
| 1410 |
+ |
and \emph{a random force} |
| 1411 |
+ |
\begin{equation} |
| 1412 |
+ |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 1413 |
+ |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 1414 |
+ |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 1415 |
+ |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), |
| 1416 |
+ |
\label{introEquation:randomForceDefinition} |
| 1417 |
+ |
\end{equation} |
| 1418 |
+ |
the equation of motion can be rewritten as |
| 1419 |
+ |
\begin{equation} |
| 1420 |
|
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
| 1421 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1422 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1423 |
|
\end{equation} |
| 1424 |
< |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
| 1425 |
< |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
| 1424 |
> |
which is known as the \emph{generalized Langevin equation}. |
| 1425 |
> |
|
| 1426 |
> |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel} |
| 1427 |
> |
|
| 1428 |
> |
One may notice that $R(t)$ depends only on initial conditions, which |
| 1429 |
> |
implies it is completely deterministic within the context of a |
| 1430 |
> |
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1431 |
> |
uncorrelated to $x$ and $\dot x$, |
| 1432 |
|
\[ |
| 1433 |
< |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1434 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 1433 |
> |
\begin{array}{l} |
| 1434 |
> |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
| 1435 |
> |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
| 1436 |
> |
\end{array} |
| 1437 |
|
\] |
| 1438 |
< |
For an infinite harmonic bath, we can use the spectral density and |
| 1439 |
< |
an integral over frequencies. |
| 1438 |
> |
This property is what we expect from a truly random process. As long |
| 1439 |
> |
as the model, which is gaussian distribution in general, chosen for |
| 1440 |
> |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
| 1441 |
> |
still remains. |
| 1442 |
|
|
| 1443 |
+ |
%dynamic friction kernel |
| 1444 |
+ |
The convolution integral |
| 1445 |
|
\[ |
| 1446 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 1282 |
< |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 1283 |
< |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 1284 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
| 1446 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } |
| 1447 |
|
\] |
| 1448 |
< |
The random forces depend only on initial conditions. |
| 1448 |
> |
depends on the entire history of the evolution of $x$, which implies |
| 1449 |
> |
that the bath retains memory of previous motions. In other words, |
| 1450 |
> |
the bath requires a finite time to respond to change in the motion |
| 1451 |
> |
of the system. For a sluggish bath which responds slowly to changes |
| 1452 |
> |
in the system coordinate, we may regard $\xi(t)$ as a constant |
| 1453 |
> |
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes |
| 1454 |
> |
\[ |
| 1455 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
| 1456 |
> |
\] |
| 1457 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1458 |
> |
\[ |
| 1459 |
> |
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1460 |
> |
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1461 |
> |
\] |
| 1462 |
> |
which can be used to describe dynamic caging effect. The other |
| 1463 |
> |
extreme is the bath that responds infinitely quickly to motions in |
| 1464 |
> |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
| 1465 |
> |
time: |
| 1466 |
> |
\[ |
| 1467 |
> |
\xi (t) = 2\xi _0 \delta (t) |
| 1468 |
> |
\] |
| 1469 |
> |
Hence, the convolution integral becomes |
| 1470 |
> |
\[ |
| 1471 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1472 |
> |
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1473 |
> |
\] |
| 1474 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1475 |
> |
\begin{equation} |
| 1476 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1477 |
> |
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1478 |
> |
\end{equation} |
| 1479 |
> |
which is known as the Langevin equation. The static friction |
| 1480 |
> |
coefficient $\xi _0$ can either be calculated from spectral density |
| 1481 |
> |
or be determined by Stokes' law for regular shaped particles.A |
| 1482 |
> |
briefly review on calculating friction tensor for arbitrary shaped |
| 1483 |
> |
particles is given in section \ref{introSection:frictionTensor}. |
| 1484 |
|
|
| 1485 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 1486 |
< |
So we can define a new set of coordinates, |
| 1486 |
> |
|
| 1487 |
> |
Defining a new set of coordinates, |
| 1488 |
|
\[ |
| 1489 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1490 |
|
^2 }}x(0) |
| 1491 |
< |
\] |
| 1492 |
< |
This makes |
| 1491 |
> |
\], |
| 1492 |
> |
we can rewrite $R(T)$ as |
| 1493 |
|
\[ |
| 1494 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
| 1494 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1495 |
|
\] |
| 1496 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1497 |
|
\[ |
| 1498 |
< |
\begin{array}{l} |
| 1498 |
> |
\begin{array}{c} |
| 1499 |
> |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1500 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1501 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1502 |
+ |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
| 1503 |
+ |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1504 |
+ |
= kT\xi (t) \\ |
| 1505 |
|
\end{array} |
| 1506 |
|
\] |
| 1507 |
< |
|
| 1306 |
< |
\begin{align} |
| 1307 |
< |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
| 1308 |
< |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
| 1309 |
< |
(t)q_\beta (0)} \right\rangle } } |
| 1310 |
< |
% |
| 1311 |
< |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
| 1312 |
< |
\right\rangle \cos (\omega _\alpha t)} |
| 1313 |
< |
% |
| 1314 |
< |
&= kT\xi (t) |
| 1315 |
< |
\end{align} |
| 1316 |
< |
|
| 1507 |
> |
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1508 |
|
\begin{equation} |
| 1509 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 1510 |
< |
\label{introEquation:secondFluctuationDissipation} |
| 1510 |
> |
\label{introEquation:secondFluctuationDissipation}. |
| 1511 |
|
\end{equation} |
| 1512 |
+ |
In effect, it acts as a constraint on the possible ways in which one |
| 1513 |
+ |
can model the random force and friction kernel. |
| 1514 |
|
|
| 1322 |
– |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
| 1323 |
– |
|
| 1515 |
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1516 |
< |
\subsection{\label{introSection:analyticalApproach}Analytical |
| 1517 |
< |
Approach} |
| 1516 |
> |
Theoretically, the friction kernel can be determined using velocity |
| 1517 |
> |
autocorrelation function. However, this approach become impractical |
| 1518 |
> |
when the system become more and more complicate. Instead, various |
| 1519 |
> |
approaches based on hydrodynamics have been developed to calculate |
| 1520 |
> |
the friction coefficients. The friction effect is isotropic in |
| 1521 |
> |
Equation, \zeta can be taken as a scalar. In general, friction |
| 1522 |
> |
tensor \Xi is a $6\times 6$ matrix given by |
| 1523 |
> |
\[ |
| 1524 |
> |
\Xi = \left( {\begin{array}{*{20}c} |
| 1525 |
> |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1526 |
> |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 1527 |
> |
\end{array}} \right). |
| 1528 |
> |
\] |
| 1529 |
> |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 1530 |
> |
tensor and rotational resistance (friction) tensor respectively, |
| 1531 |
> |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 1532 |
> |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 1533 |
> |
particle moves in a fluid, it may experience friction force or |
| 1534 |
> |
torque along the opposite direction of the velocity or angular |
| 1535 |
> |
velocity, |
| 1536 |
> |
\[ |
| 1537 |
> |
\left( \begin{array}{l} |
| 1538 |
> |
F_R \\ |
| 1539 |
> |
\tau _R \\ |
| 1540 |
> |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1541 |
> |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1542 |
> |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1543 |
> |
\end{array}} \right)\left( \begin{array}{l} |
| 1544 |
> |
v \\ |
| 1545 |
> |
w \\ |
| 1546 |
> |
\end{array} \right) |
| 1547 |
> |
\] |
| 1548 |
> |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1549 |
> |
toque. |
| 1550 |
|
|
| 1551 |
< |
\subsection{\label{introSection:approximationApproach}Approximation |
| 1329 |
< |
Approach} |
| 1551 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
| 1552 |
|
|
| 1553 |
< |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1554 |
< |
Body} |
| 1553 |
> |
For a spherical particle, the translational and rotational friction |
| 1554 |
> |
constant can be calculated from Stoke's law, |
| 1555 |
> |
\[ |
| 1556 |
> |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1557 |
> |
{6\pi \eta R} & 0 & 0 \\ |
| 1558 |
> |
0 & {6\pi \eta R} & 0 \\ |
| 1559 |
> |
0 & 0 & {6\pi \eta R} \\ |
| 1560 |
> |
\end{array}} \right) |
| 1561 |
> |
\] |
| 1562 |
> |
and |
| 1563 |
> |
\[ |
| 1564 |
> |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1565 |
> |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1566 |
> |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1567 |
> |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1568 |
> |
\end{array}} \right) |
| 1569 |
> |
\] |
| 1570 |
> |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1571 |
> |
hydrodynamics radius. |
| 1572 |
|
|
| 1573 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 1573 |
> |
Other non-spherical shape, such as cylinder and ellipsoid |
| 1574 |
> |
\textit{etc}, are widely used as reference for developing new |
| 1575 |
> |
hydrodynamics theory, because their properties can be calculated |
| 1576 |
> |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 1577 |
> |
also called a triaxial ellipsoid, which is given in Cartesian |
| 1578 |
> |
coordinates by |
| 1579 |
> |
\[ |
| 1580 |
> |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 1581 |
> |
}} = 1 |
| 1582 |
> |
\] |
| 1583 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 1584 |
> |
due to the complexity of the elliptic integral, only the ellipsoid |
| 1585 |
> |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 1586 |
> |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 1587 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 1588 |
> |
\[ |
| 1589 |
> |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1590 |
> |
} }}{b}, |
| 1591 |
> |
\] |
| 1592 |
> |
and oblate, |
| 1593 |
> |
\[ |
| 1594 |
> |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1595 |
> |
}}{a} |
| 1596 |
> |
\], |
| 1597 |
> |
one can write down the translational and rotational resistance |
| 1598 |
> |
tensors |
| 1599 |
> |
\[ |
| 1600 |
> |
\begin{array}{l} |
| 1601 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1602 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1603 |
> |
\end{array}, |
| 1604 |
> |
\] |
| 1605 |
> |
and |
| 1606 |
> |
\[ |
| 1607 |
> |
\begin{array}{l} |
| 1608 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1609 |
> |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1610 |
> |
\end{array}. |
| 1611 |
> |
\] |
| 1612 |
> |
|
| 1613 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
| 1614 |
> |
|
| 1615 |
> |
Unlike spherical and other regular shaped molecules, there is not |
| 1616 |
> |
analytical solution for friction tensor of any arbitrary shaped |
| 1617 |
> |
rigid molecules. The ellipsoid of revolution model and general |
| 1618 |
> |
triaxial ellipsoid model have been used to approximate the |
| 1619 |
> |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1620 |
> |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1621 |
> |
combination of rotational diffusion coefficients, $D$-space is not |
| 1622 |
> |
unique\cite{Wegener79} as well as the intrinsic coupling between |
| 1623 |
> |
translational and rotational motion of rigid body\cite{}, general |
| 1624 |
> |
ellipsoid is not always suitable for modeling arbitrarily shaped |
| 1625 |
> |
rigid molecule. A number of studies have been devoted to determine |
| 1626 |
> |
the friction tensor for irregularly shaped rigid bodies using more |
| 1627 |
> |
advanced method\cite{} where the molecule of interest was modeled by |
| 1628 |
> |
combinations of spheres(beads)\cite{} and the hydrodynamics |
| 1629 |
> |
properties of the molecule can be calculated using the hydrodynamic |
| 1630 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 1631 |
> |
immersed in a continuous medium. Due to hydrodynamics interaction, |
| 1632 |
> |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 1633 |
> |
unperturbed velocity $v_i$, |
| 1634 |
> |
\[ |
| 1635 |
> |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1636 |
> |
\] |
| 1637 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1638 |
> |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1639 |
> |
proportional to its ``net'' velocity |
| 1640 |
> |
\begin{equation} |
| 1641 |
> |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1642 |
> |
\label{introEquation:tensorExpression} |
| 1643 |
> |
\end{equation} |
| 1644 |
> |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1645 |
> |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1646 |
> |
\ref{introEquation:tensorExpression} |
| 1647 |
> |
\begin{equation} |
| 1648 |
> |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1649 |
> |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1650 |
> |
\label{introEquation:oseenTensor} |
| 1651 |
> |
\end{equation} |
| 1652 |
> |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1653 |
> |
A second order expression for element of different size was |
| 1654 |
> |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
| 1655 |
> |
la Torre and Bloomfield, |
| 1656 |
> |
\begin{equation} |
| 1657 |
> |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1658 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1659 |
> |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1660 |
> |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1661 |
> |
\label{introEquation:RPTensorNonOverlapped} |
| 1662 |
> |
\end{equation} |
| 1663 |
> |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1664 |
> |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1665 |
> |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1666 |
> |
overlapping beads with the same radius, $\sigma$, is given by |
| 1667 |
> |
\begin{equation} |
| 1668 |
> |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1669 |
> |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1670 |
> |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1671 |
> |
\label{introEquation:RPTensorOverlapped} |
| 1672 |
> |
\end{equation} |
| 1673 |
> |
|
| 1674 |
> |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1675 |
> |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1676 |
> |
$B_{ij}$ blocks |
| 1677 |
> |
\begin{equation} |
| 1678 |
> |
B = \left( {\begin{array}{*{20}c} |
| 1679 |
> |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1680 |
> |
\vdots & \ddots & \vdots \\ |
| 1681 |
> |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1682 |
> |
\end{array}} \right), |
| 1683 |
> |
\end{equation} |
| 1684 |
> |
where $B_{ij}$ is given by |
| 1685 |
> |
\[ |
| 1686 |
> |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1687 |
> |
)T_{ij} |
| 1688 |
> |
\] |
| 1689 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 1690 |
> |
$B$, we obtain |
| 1691 |
> |
|
| 1692 |
> |
\[ |
| 1693 |
> |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1694 |
> |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1695 |
> |
\vdots & \ddots & \vdots \\ |
| 1696 |
> |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1697 |
> |
\end{array}} \right) |
| 1698 |
> |
\] |
| 1699 |
> |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1700 |
> |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1701 |
> |
\[ |
| 1702 |
> |
U_i = \left( {\begin{array}{*{20}c} |
| 1703 |
> |
0 & { - z_i } & {y_i } \\ |
| 1704 |
> |
{z_i } & 0 & { - x_i } \\ |
| 1705 |
> |
{ - y_i } & {x_i } & 0 \\ |
| 1706 |
> |
\end{array}} \right) |
| 1707 |
> |
\] |
| 1708 |
> |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1709 |
> |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1710 |
> |
arbitrary origin $O$ can be written as |
| 1711 |
> |
\begin{equation} |
| 1712 |
> |
\begin{array}{l} |
| 1713 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1714 |
> |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1715 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1716 |
> |
\end{array} |
| 1717 |
> |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1718 |
> |
\end{equation} |
| 1719 |
> |
|
| 1720 |
> |
The resistance tensor depends on the origin to which they refer. The |
| 1721 |
> |
proper location for applying friction force is the center of |
| 1722 |
> |
resistance (reaction), at which the trace of rotational resistance |
| 1723 |
> |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1724 |
> |
resistance is defined as an unique point of the rigid body at which |
| 1725 |
> |
the translation-rotation coupling tensor are symmetric, |
| 1726 |
> |
\begin{equation} |
| 1727 |
> |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1728 |
> |
\label{introEquation:definitionCR} |
| 1729 |
> |
\end{equation} |
| 1730 |
> |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1731 |
> |
we can easily find out that the translational resistance tensor is |
| 1732 |
> |
origin independent, while the rotational resistance tensor and |
| 1733 |
> |
translation-rotation coupling resistance tensor depend on the |
| 1734 |
> |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1735 |
> |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1736 |
> |
obtain the resistance tensor at $P$ by |
| 1737 |
> |
\begin{equation} |
| 1738 |
> |
\begin{array}{l} |
| 1739 |
> |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 1740 |
> |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 1741 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
| 1742 |
> |
\end{array} |
| 1743 |
> |
\label{introEquation:resistanceTensorTransformation} |
| 1744 |
> |
\end{equation} |
| 1745 |
> |
where |
| 1746 |
> |
\[ |
| 1747 |
> |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 1748 |
> |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 1749 |
> |
{z_i } & 0 & { - x_{OP} } \\ |
| 1750 |
> |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 1751 |
> |
\end{array}} \right) |
| 1752 |
> |
\] |
| 1753 |
> |
Using Equations \ref{introEquation:definitionCR} and |
| 1754 |
> |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 1755 |
> |
the position of center of resistance, |
| 1756 |
> |
\[ |
| 1757 |
> |
\left( \begin{array}{l} |
| 1758 |
> |
x_{OR} \\ |
| 1759 |
> |
y_{OR} \\ |
| 1760 |
> |
z_{OR} \\ |
| 1761 |
> |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
| 1762 |
> |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 1763 |
> |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 1764 |
> |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 1765 |
> |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
| 1766 |
> |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 1767 |
> |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 1768 |
> |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 1769 |
> |
\end{array} \right). |
| 1770 |
> |
\] |
| 1771 |
> |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1772 |
> |
joining center of resistance $R$ and origin $O$. |
| 1773 |
> |
|
| 1774 |
> |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
