--- trunk/tengDissertation/Introduction.tex 2006/04/13 04:47:47 2706 +++ trunk/tengDissertation/Introduction.tex 2006/05/26 18:25:41 2779 @@ -570,21 +570,6 @@ The free rigid body is an example of Poisson system (a \dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} \end{equation} The most obvious change being that matrix $J$ now depends on $x$. -The free rigid body is an example of Poisson system (actually a -Lie-Poisson system) with Hamiltonian function of angular kinetic -energy. -\begin{equation} -J(\pi ) = \left( {\begin{array}{*{20}c} - 0 & {\pi _3 } & { - \pi _2 } \\ - { - \pi _3 } & 0 & {\pi _1 } \\ - {\pi _2 } & { - \pi _1 } & 0 \\ -\end{array}} \right) -\end{equation} - -\begin{equation} -H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 -}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) -\end{equation} \subsection{\label{introSection:exactFlow}Exact Flow} @@ -837,7 +822,7 @@ q(\Delta t)} \right]. % % q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot q(\Delta t)} \right]. % - \label{introEquation:positionVerlet1} + \label{introEquation:positionVerlet2} \end{align} \subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} @@ -846,7 +831,7 @@ $\varphi_1(t)$ and $\varphi_2(t$ respectively , we hav error of splitting method in terms of commutator of the operators(\ref{introEquation:exponentialOperator}) associated with the sub-flow. For operators $hX$ and $hY$ which are associate to -$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have +$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have \begin{equation} \exp (hX + hY) = \exp (hZ) \end{equation} @@ -862,10 +847,9 @@ can obtain Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we can obtain \begin{eqnarray*} -\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 -[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ -& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + -\ldots ) +\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ + & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ + & & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) \end{eqnarray*} Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local error of Spring splitting is proportional to $h^3$. The same @@ -874,7 +858,7 @@ Careful choice of coefficient $a_1 ,\ldot , b_m$ will \varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - 1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . \end{equation} -Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher +Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher order method. Yoshida proposed an elegant way to compose higher order methods based on symmetric splitting. Given a symmetric second order base method $ \varphi _h^{(2)} $, a fourth-order symmetric @@ -898,16 +882,299 @@ As a special discipline of molecular modeling, Molecul \section{\label{introSection:molecularDynamics}Molecular Dynamics} -As a special discipline of molecular modeling, Molecular dynamics -has proven to be a powerful tool for studying the functions of -biological systems, providing structural, thermodynamic and -dynamical information. - -\subsection{\label{introSec:mdInit}Initialization} - -\subsection{\label{introSec:forceEvaluation}Force Evaluation} +As one of the principal tools of molecular modeling, Molecular +dynamics has proven to be a powerful tool for studying the functions +of biological systems, providing structural, thermodynamic and +dynamical information. The basic idea of molecular dynamics is that +macroscopic properties are related to microscopic behavior and +microscopic behavior can be calculated from the trajectories in +simulations. For instance, instantaneous temperature of an +Hamiltonian system of $N$ particle can be measured by +\[ +T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} +\] +where $m_i$ and $v_i$ are the mass and velocity of $i$th particle +respectively, $f$ is the number of degrees of freedom, and $k_B$ is +the boltzman constant. -\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} +A typical molecular dynamics run consists of three essential steps: +\begin{enumerate} + \item Initialization + \begin{enumerate} + \item Preliminary preparation + \item Minimization + \item Heating + \item Equilibration + \end{enumerate} + \item Production + \item Analysis +\end{enumerate} +These three individual steps will be covered in the following +sections. Sec.~\ref{introSec:initialSystemSettings} deals with the +initialization of a simulation. Sec.~\ref{introSec:production} will +discusses issues in production run. Sec.~\ref{introSection:Analysis} +provides the theoretical tools for trajectory analysis. + +\subsection{\label{introSec:initialSystemSettings}Initialization} + +\subsubsection{Preliminary preparation} + +When selecting the starting structure of a molecule for molecular +simulation, one may retrieve its Cartesian coordinates from public +databases, such as RCSB Protein Data Bank \textit{etc}. Although +thousands of crystal structures of molecules are discovered every +year, many more remain unknown due to the difficulties of +purification and crystallization. Even for the molecule with known +structure, some important information is missing. For example, the +missing hydrogen atom which acts as donor in hydrogen bonding must +be added. Moreover, in order to include electrostatic interaction, +one may need to specify the partial charges for individual atoms. +Under some circumstances, we may even need to prepare the system in +a special setup. For instance, when studying transport phenomenon in +membrane system, we may prepare the lipids in bilayer structure +instead of placing lipids randomly in solvent, since we are not +interested in self-aggregation and it takes a long time to happen. + +\subsubsection{Minimization} + +It is quite possible that some of molecules in the system from +preliminary preparation may be overlapped with each other. This +close proximity leads to high potential energy which consequently +jeopardizes any molecular dynamics simulations. To remove these +steric overlaps, one typically performs energy minimization to find +a more reasonable conformation. Several energy minimization methods +have been developed to exploit the energy surface and to locate the +local minimum. While converging slowly near the minimum, steepest +descent method is extremely robust when systems are far from +harmonic. Thus, it is often used to refine structure from +crystallographic data. Relied on the gradient or hessian, advanced +methods like conjugate gradient and Newton-Raphson converge rapidly +to a local minimum, while become unstable if the energy surface is +far from quadratic. Another factor must be taken into account, when +choosing energy minimization method, is the size of the system. +Steepest descent and conjugate gradient can deal with models of any +size. Because of the limit of computation power to calculate hessian +matrix and insufficient storage capacity to store them, most +Newton-Raphson methods can not be used with very large models. + +\subsubsection{Heating} + +Typically, Heating is performed by assigning random velocities +according to a Gaussian distribution for a temperature. Beginning at +a lower temperature and gradually increasing the temperature by +assigning greater random velocities, we end up with setting the +temperature of the system to a final temperature at which the +simulation will be conducted. In heating phase, we should also keep +the system from drifting or rotating as a whole. Equivalently, the +net linear momentum and angular momentum of the system should be +shifted to zero. + +\subsubsection{Equilibration} + +The purpose of equilibration is to allow the system to evolve +spontaneously for a period of time and reach equilibrium. The +procedure is continued until various statistical properties, such as +temperature, pressure, energy, volume and other structural +properties \textit{etc}, become independent of time. Strictly +speaking, minimization and heating are not necessary, provided the +equilibration process is long enough. However, these steps can serve +as a means to arrive at an equilibrated structure in an effective +way. + +\subsection{\label{introSection:production}Production} + +Production run is the most important steps of the simulation, in +which the equilibrated structure is used as a starting point and the +motions of the molecules are collected for later analysis. In order +to capture the macroscopic properties of the system, the molecular +dynamics simulation must be performed in correct and efficient way. + +The most expensive part of a molecular dynamics simulation is the +calculation of non-bonded forces, such as van der Waals force and +Coulombic forces \textit{etc}. For a system of $N$ particles, the +complexity of the algorithm for pair-wise interactions is $O(N^2 )$, +which making large simulations prohibitive in the absence of any +computation saving techniques. + +A natural approach to avoid system size issue is to represent the +bulk behavior by a finite number of the particles. However, this +approach will suffer from the surface effect. To offset this, +\textit{Periodic boundary condition} is developed to simulate bulk +properties with a relatively small number of particles. In this +method, the simulation box is replicated throughout space to form an +infinite lattice. During the simulation, when a particle moves in +the primary cell, its image in other cells move in exactly the same +direction with exactly the same orientation. Thus, as a particle +leaves the primary cell, one of its images will enter through the +opposite face. +%\begin{figure} +%\centering +%\includegraphics[width=\linewidth]{pbcFig.eps} +%\caption[An illustration of periodic boundary conditions]{A 2-D +%illustration of periodic boundary conditions. As one particle leaves +%the right of the simulation box, an image of it enters the left.} +%\label{introFig:pbc} +%\end{figure} + +%cutoff and minimum image convention +Another important technique to improve the efficiency of force +evaluation is to apply cutoff where particles farther than a +predetermined distance, are not included in the calculation +\cite{Frenkel1996}. The use of a cutoff radius will cause a +discontinuity in the potential energy curve. Fortunately, one can +shift the potential to ensure the potential curve go smoothly to +zero at the cutoff radius. Cutoff strategy works pretty well for +Lennard-Jones interaction because of its short range nature. +However, simply truncating the electrostatic interaction with the +use of cutoff has been shown to lead to severe artifacts in +simulations. Ewald summation, in which the slowly conditionally +convergent Coulomb potential is transformed into direct and +reciprocal sums with rapid and absolute convergence, has proved to +minimize the periodicity artifacts in liquid simulations. Taking the +advantages of the fast Fourier transform (FFT) for calculating +discrete Fourier transforms, the particle mesh-based methods are +accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative +approach is \emph{fast multipole method}, which treats Coulombic +interaction exactly at short range, and approximate the potential at +long range through multipolar expansion. In spite of their wide +acceptances at the molecular simulation community, these two methods +are hard to be implemented correctly and efficiently. Instead, we +use a damped and charge-neutralized Coulomb potential method +developed by Wolf and his coworkers. The shifted Coulomb potential +for particle $i$ and particle $j$ at distance $r_{rj}$ is given by: +\begin{equation} +V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha +r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow +R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha +r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} +\end{equation} +where $\alpha$ is the convergence parameter. Due to the lack of +inherent periodicity and rapid convergence,this method is extremely +efficient and easy to implement. +%\begin{figure} +%\centering +%\includegraphics[width=\linewidth]{pbcFig.eps} +%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} +%\label{introFigure:shiftedCoulomb} +%\end{figure} + +%multiple time step + +\subsection{\label{introSection:Analysis} Analysis} + +Recently, advanced visualization technique are widely applied to +monitor the motions of molecules. Although the dynamics of the +system can be described qualitatively from animation, quantitative +trajectory analysis are more appreciable. According to the +principles of Statistical Mechanics, +Sec.~\ref{introSection:statisticalMechanics}, one can compute +thermodynamics properties, analyze fluctuations of structural +parameters, and investigate time-dependent processes of the molecule +from the trajectories. + +\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} + +Thermodynamics properties, which can be expressed in terms of some +function of the coordinates and momenta of all particles in the +system, can be directly computed from molecular dynamics. The usual +way to measure the pressure is based on virial theorem of Clausius +which states that the virial is equal to $-3Nk_BT$. For a system +with forces between particles, the total virial, $W$, contains the +contribution from external pressure and interaction between the +particles: +\[ +W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot +f_{ij} } } \right\rangle +\] +where $f_{ij}$ is the force between particle $i$ and $j$ at a +distance $r_{ij}$. Thus, the expression for the pressure is given +by: +\begin{equation} +P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i +< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle +\end{equation} + +\subsubsection{\label{introSection:structuralProperties}Structural Properties} + +Structural Properties of a simple fluid can be described by a set of +distribution functions. Among these functions,\emph{pair +distribution function}, also known as \emph{radial distribution +function}, is of most fundamental importance to liquid-state theory. +Pair distribution function can be gathered by Fourier transforming +raw data from a series of neutron diffraction experiments and +integrating over the surface factor \cite{Powles73}. The experiment +result can serve as a criterion to justify the correctness of the +theory. Moreover, various equilibrium thermodynamic and structural +properties can also be expressed in terms of radial distribution +function \cite{allen87:csl}. + +A pair distribution functions $g(r)$ gives the probability that a +particle $i$ will be located at a distance $r$ from a another +particle $j$ in the system +\[ +g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j +\ne i} {\delta (r - r_{ij} )} } } \right\rangle. +\] +Note that the delta function can be replaced by a histogram in +computer simulation. Figure +\ref{introFigure:pairDistributionFunction} shows a typical pair +distribution function for the liquid argon system. The occurrence of +several peaks in the plot of $g(r)$ suggests that it is more likely +to find particles at certain radial values than at others. This is a +result of the attractive interaction at such distances. Because of +the strong repulsive forces at short distance, the probability of +locating particles at distances less than about 2.5{\AA} from each +other is essentially zero. + +%\begin{figure} +%\centering +%\includegraphics[width=\linewidth]{pdf.eps} +%\caption[Pair distribution function for the liquid argon +%]{Pair distribution function for the liquid argon} +%\label{introFigure:pairDistributionFunction} +%\end{figure} + +\subsubsection{\label{introSection:timeDependentProperties}Time-dependent +Properties} + +Time-dependent properties are usually calculated using \emph{time +correlation function}, which correlates random variables $A$ and $B$ +at two different time +\begin{equation} +C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. +\label{introEquation:timeCorrelationFunction} +\end{equation} +If $A$ and $B$ refer to same variable, this kind of correlation +function is called \emph{auto correlation function}. One example of +auto correlation function is velocity auto-correlation function +which is directly related to transport properties of molecular +liquids: +\[ +D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} +\right\rangle } dt +\] +where $D$ is diffusion constant. Unlike velocity autocorrelation +function which is averaging over time origins and over all the +atoms, dipole autocorrelation are calculated for the entire system. +The dipole autocorrelation function is given by: +\[ +c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} +\right\rangle +\] +Here $u_{tot}$ is the net dipole of the entire system and is given +by +\[ +u_{tot} (t) = \sum\limits_i {u_i (t)} +\] +In principle, many time correlation functions can be related with +Fourier transforms of the infrared, Raman, and inelastic neutron +scattering spectra of molecular liquids. In practice, one can +extract the IR spectrum from the intensity of dipole fluctuation at +each frequency using the following relationship: +\[ +\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - +i2\pi vt} dt} +\] \section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} @@ -942,19 +1209,19 @@ rotation matrix $A$ and re-formulating Hamiltonian's e The break through in geometric literature suggests that, in order to develop a long-term integration scheme, one should preserve the symplectic structure of the flow. Introducing conjugate momentum to -rotation matrix $A$ and re-formulating Hamiltonian's equation, a +rotation matrix $Q$ and re-formulating Hamiltonian's equation, a symplectic integrator, RSHAKE, was proposed to evolve the Hamiltonian system in a constraint manifold by iteratively -satisfying the orthogonality constraint $A_t A = 1$. An alternative +satisfying the orthogonality constraint $Q_T Q = 1$. An alternative method using quaternion representation was developed by Omelyan. However, both of these methods are iterative and inefficient. In this section, we will present a symplectic Lie-Poisson integrator -for rigid body developed by Dullweber and his coworkers\cite{}. +for rigid body developed by Dullweber and his +coworkers\cite{Dullweber1997} in depth. -\subsection{\label{introSection:lieAlgebra}Lie Algebra} - \subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} - +The motion of the rigid body is Hamiltonian with the Hamiltonian +function \begin{equation} H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. @@ -969,13 +1236,13 @@ Q^T Q = 1$, \label{introEquation:orthogonalConstraint} where $I_{ii}$ is the diagonal element of the inertia tensor. This constrained Hamiltonian equation subjects to a holonomic constraint, \begin{equation} -Q^T Q = 1$, \label{introEquation:orthogonalConstraint} +Q^T Q = 1, \label{introEquation:orthogonalConstraint} \end{equation} which is used to ensure rotation matrix's orthogonality. Differentiating \ref{introEquation:orthogonalConstraint} and using Equation \ref{introEquation:RBMotionMomentum}, one may obtain, \begin{equation} -Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0 . \\ +Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ \label{introEquation:RBFirstOrderConstraint} \end{equation} @@ -987,142 +1254,404 @@ the equations of motion, \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ \frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ \frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ - \frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ + \frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ \end{array} \] - +In general, there are two ways to satisfy the holonomic constraints. +We can use constraint force provided by lagrange multiplier on the +normal manifold to keep the motion on constraint space. Or we can +simply evolve the system in constraint manifold. These two methods +are proved to be equivalent. The holonomic constraint and equations +of motions define a constraint manifold for rigid body \[ -M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} -\right\} . +M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} +\right\}. \] -\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} - -\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Discretization of Euler Equations} - - -\section{\label{introSection:langevinDynamics}Langevin Dynamics} - -\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} - -\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} - -\begin{equation} -H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) -\label{introEquation:bathGLE} -\end{equation} -where $H_B$ is harmonic bath Hamiltonian, +Unfortunately, this constraint manifold is not the cotangent bundle +$T_{\star}SO(3)$. However, it turns out that under symplectic +transformation, the cotangent space and the phase space are +diffeomorphic. Introducing \[ -H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 -}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} +\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), \] -and $\Delta U$ is bilinear system-bath coupling, +the mechanical system subject to a holonomic constraint manifold $M$ +can be re-formulated as a Hamiltonian system on the cotangent space \[ -\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} +T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = +1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} \] -Completing the square, + +For a body fixed vector $X_i$ with respect to the center of mass of +the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is +given as +\begin{equation} +X_i^{lab} = Q X_i + q. +\end{equation} +Therefore, potential energy $V(q,Q)$ is defined by \[ -H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ -{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha -w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha -w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = -1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 +V(q,Q) = V(Q X_0 + q). \] -and putting it back into Eq.~\ref{introEquation:bathGLE}, +Hence, the force and torque are given by \[ -H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N -{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha -w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha -w_\alpha ^2 }}x} \right)^2 } \right\}} +\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, \] -where +and \[ -W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 -}}{{2m_\alpha w_\alpha ^2 }}} x^2 +\nabla _Q V(q,Q) = F(q,Q)X_i^t \] -Since the first two terms of the new Hamiltonian depend only on the -system coordinates, we can get the equations of motion for -Generalized Langevin Dynamics by Hamilton's equations -\ref{introEquation:motionHamiltonianCoordinate, -introEquation:motionHamiltonianMomentum}, -\begin{align} -\dot p &= - \frac{{\partial H}}{{\partial x}} - &= m\ddot x - &= - \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)} -\label{introEquation:Lp5} -\end{align} -, and -\begin{align} -\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} - &= m\ddot x_\alpha - &= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) -\end{align} +respectively. -\subsection{\label{introSection:laplaceTransform}The Laplace Transform} - +As a common choice to describe the rotation dynamics of the rigid +body, angular momentum on body frame $\Pi = Q^t P$ is introduced to +rewrite the equations of motion, +\begin{equation} + \begin{array}{l} + \mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ + \mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ + \end{array} + \label{introEqaution:RBMotionPI} +\end{equation} +, as well as holonomic constraints, \[ -L(x) = \int_0^\infty {x(t)e^{ - pt} dt} +\begin{array}{l} + \Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ + Q^T Q = 1 \\ + \end{array} \] +For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in +so(3)^ \star$, the hat-map isomorphism, +\begin{equation} +v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( +{\begin{array}{*{20}c} + 0 & { - v_3 } & {v_2 } \\ + {v_3 } & 0 & { - v_1 } \\ + { - v_2 } & {v_1 } & 0 \\ +\end{array}} \right), +\label{introEquation:hatmapIsomorphism} +\end{equation} +will let us associate the matrix products with traditional vector +operations \[ -L(x + y) = L(x) + L(y) +\hat vu = v \times u \] +Using \ref{introEqaution:RBMotionPI}, one can construct a skew +matrix, +\begin{equation} +(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T +){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ +- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - +(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} +\end{equation} +Since $\Lambda$ is symmetric, the last term of Equation +\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange +multiplier $\Lambda$ is absent from the equations of motion. This +unique property eliminate the requirement of iterations which can +not be avoided in other methods\cite{}. + +Applying hat-map isomorphism, we obtain the equation of motion for +angular momentum on body frame +\begin{equation} +\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T +F_i (r,Q)} \right) \times X_i }. +\label{introEquation:bodyAngularMotion} +\end{equation} +In the same manner, the equation of motion for rotation matrix is +given by \[ -L(ax) = aL(x) +\dot Q = Qskew(I^{ - 1} \pi ) \] +\subsection{\label{introSection:SymplecticFreeRB}Symplectic +Lie-Poisson Integrator for Free Rigid Body} + +If there is not external forces exerted on the rigid body, the only +contribution to the rotational is from the kinetic potential (the +first term of \ref{ introEquation:bodyAngularMotion}). The free +rigid body is an example of Lie-Poisson system with Hamiltonian +function +\begin{equation} +T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) +\label{introEquation:rotationalKineticRB} +\end{equation} +where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and +Lie-Poisson structure matrix, +\begin{equation} +J(\pi ) = \left( {\begin{array}{*{20}c} + 0 & {\pi _3 } & { - \pi _2 } \\ + { - \pi _3 } & 0 & {\pi _1 } \\ + {\pi _2 } & { - \pi _1 } & 0 \\ +\end{array}} \right) +\end{equation} +Thus, the dynamics of free rigid body is governed by +\begin{equation} +\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) +\end{equation} + +One may notice that each $T_i^r$ in Equation +\ref{introEquation:rotationalKineticRB} can be solved exactly. For +instance, the equations of motion due to $T_1^r$ are given by +\begin{equation} +\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 +\label{introEqaution:RBMotionSingleTerm} +\end{equation} +where +\[ R_1 = \left( {\begin{array}{*{20}c} + 0 & 0 & 0 \\ + 0 & 0 & {\pi _1 } \\ + 0 & { - \pi _1 } & 0 \\ +\end{array}} \right). +\] +The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is \[ -L(\dot x) = pL(x) - px(0) +\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = +Q(0)e^{\Delta tR_1 } \] - +with \[ -L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) +e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} + 0 & 0 & 0 \\ + 0 & {\cos \theta _1 } & {\sin \theta _1 } \\ + 0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ +\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. \] - +To reduce the cost of computing expensive functions in $e^{\Delta +tR_1 }$, we can use Cayley transformation, \[ -L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) +e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 +) \] +The flow maps for $T_2^r$ and $T_3^r$ can be found in the same +manner. -Some relatively important transformation, +In order to construct a second-order symplectic method, we split the +angular kinetic Hamiltonian function can into five terms \[ -L(\cos at) = \frac{p}{{p^2 + a^2 }} +T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 +) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r +(\pi _1 ) +\]. +Concatenating flows corresponding to these five terms, we can obtain +an symplectic integrator, +\[ +\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ +\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } +\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi +_1 }. \] +The non-canonical Lie-Poisson bracket ${F, G}$ of two function +$F(\pi )$ and $G(\pi )$ is defined by \[ -L(\sin at) = \frac{a}{{p^2 + a^2 }} +\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi +) \] - +If the Poisson bracket of a function $F$ with an arbitrary smooth +function $G$ is zero, $F$ is a \emph{Casimir}, which is the +conserved quantity in Poisson system. We can easily verify that the +norm of the angular momentum, $\parallel \pi +\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel +\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , +then by the chain rule \[ -L(1) = \frac{1}{p} +\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 +}}{2})\pi \] +Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi +\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit +Lie-Poisson integrator is found to be extremely efficient and stable +which can be explained by the fact the small angle approximation is +used and the norm of the angular momentum is conserved. -First, the bath coordinates, +\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian +Splitting for Rigid Body} + +The Hamiltonian of rigid body can be separated in terms of kinetic +energy and potential energy, \[ -p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega -_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha -}}L(x) +H = T(p,\pi ) + V(q,Q) \] -\[ -L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + -px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} +The equations of motion corresponding to potential energy and +kinetic energy are listed in the below table, +\begin{table} +\caption{Equations of motion due to Potential and Kinetic Energies} +\begin{center} +\begin{tabular}{|l|l|} + \hline + % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... + Potential & Kinetic \\ + $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ + $\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ + $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ + $ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ + \hline +\end{tabular} +\end{center} +\end{table} +A second-order symplectic method is now obtained by the +composition of the flow maps, +\[ +\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi +_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. \] -Then, the system coordinates, -\begin{align} -mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - -\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha -}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha -(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha -}}\omega _\alpha ^2 L(x)} \right\}} -% - &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - - \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\}} -\end{align} -Then, the inverse transform, +Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two +sub-flows which corresponding to force and torque respectively, +\[ +\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi +_{\Delta t/2,\tau }. +\] +Since the associated operators of $\varphi _{\Delta t/2,F} $ and +$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition +order inside $\varphi _{\Delta t/2,V}$ does not matter. +Furthermore, kinetic potential can be separated to translational +kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, +\begin{equation} +T(p,\pi ) =T^t (p) + T^r (\pi ). +\end{equation} +where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is +defined by \ref{introEquation:rotationalKineticRB}. Therefore, the +corresponding flow maps are given by +\[ +\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi +_{\Delta t,T^r }. +\] +Finally, we obtain the overall symplectic flow maps for free moving +rigid body +\begin{equation} +\begin{array}{c} + \varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ + \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 } \\ + \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ + \end{array} +\label{introEquation:overallRBFlowMaps} +\end{equation} + +\section{\label{introSection:langevinDynamics}Langevin Dynamics} +As an alternative to newtonian dynamics, Langevin dynamics, which +mimics a simple heat bath with stochastic and dissipative forces, +has been applied in a variety of studies. This section will review +the theory of Langevin dynamics simulation. A brief derivation of +generalized Langevin equation will be given first. Follow that, we +will discuss the physical meaning of the terms appearing in the +equation as well as the calculation of friction tensor from +hydrodynamics theory. + +\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} + +Harmonic bath model, in which an effective set of harmonic +oscillators are used to mimic the effect of a linearly responding +environment, has been widely used in quantum chemistry and +statistical mechanics. One of the successful applications of +Harmonic bath model is the derivation of Deriving Generalized +Langevin Dynamics. Lets consider a system, in which the degree of +freedom $x$ is assumed to couple to the bath linearly, giving a +Hamiltonian of the form +\begin{equation} +H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) +\label{introEquation:bathGLE}. +\end{equation} +Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated +with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, +\[ +H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 +}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } +\right\}} +\] +where the index $\alpha$ runs over all the bath degrees of freedom, +$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are +the harmonic bath masses, and $\Delta U$ is bilinear system-bath +coupling, +\[ +\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} +\] +where $g_\alpha$ are the coupling constants between the bath and the +coordinate $x$. Introducing +\[ +W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 +}}{{2m_\alpha w_\alpha ^2 }}} x^2 +\] and combining the last two terms in Equation +\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath +Hamiltonian as +\[ +H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N +{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha +w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha +w_\alpha ^2 }}x} \right)^2 } \right\}} +\] +Since the first two terms of the new Hamiltonian depend only on the +system coordinates, we can get the equations of motion for +Generalized Langevin Dynamics by Hamilton's equations +\ref{introEquation:motionHamiltonianCoordinate, +introEquation:motionHamiltonianMomentum}, +\begin{equation} +m\ddot x = - \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)}, +\label{introEquation:coorMotionGLE} +\end{equation} +and +\begin{equation} +m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - +\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). +\label{introEquation:bathMotionGLE} +\end{equation} + +In order to derive an equation for $x$, the dynamics of the bath +variables $x_\alpha$ must be solved exactly first. As an integral +transform which is particularly useful in solving linear ordinary +differential equations, Laplace transform is the appropriate tool to +solve this problem. The basic idea is to transform the difficult +differential equations into simple algebra problems which can be +solved easily. Then applying inverse Laplace transform, also known +as the Bromwich integral, we can retrieve the solutions of the +original problems. + +Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace +transform of f(t) is a new function defined as +\[ +L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} +\] +where $p$ is real and $L$ is called the Laplace Transform +Operator. Below are some important properties of Laplace transform +\begin{equation} +\begin{array}{c} + L(x + y) = L(x) + L(y) \\ + L(ax) = aL(x) \\ + L(\dot x) = pL(x) - px(0) \\ + L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ + L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ + \end{array} +\end{equation} + +Applying Laplace transform to the bath coordinates, we obtain +\[ +\begin{array}{c} + p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ + L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ + \end{array} +\] +By the same way, the system coordinates become +\[ +\begin{array}{c} + mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ + - \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ + \end{array} +\] + +With the help of some relatively important inverse Laplace +transformations: +\[ +\begin{array}{c} + L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ + L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ + L(1) = \frac{1}{p} \\ + \end{array} +\] +, we obtain \begin{align} m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 @@ -1142,72 +1671,372 @@ t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = (\omega _\alpha t)} \right\}} \end{align} +Introducing a \emph{dynamic friction kernel} \begin{equation} +\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 +}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} +\label{introEquation:dynamicFrictionKernelDefinition} +\end{equation} +and \emph{a random force} +\begin{equation} +R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) +- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} +\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha +(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), +\label{introEquation:randomForceDefinition} +\end{equation} +the equation of motion can be rewritten as +\begin{equation} m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi (t)\dot x(t - \tau )d\tau } + R(t) \label{introEuqation:GeneralizedLangevinDynamics} \end{equation} -%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and -%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ +which is known as the \emph{generalized Langevin equation}. + +\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel} + +One may notice that $R(t)$ depends only on initial conditions, which +implies it is completely deterministic within the context of a +harmonic bath. However, it is easy to verify that $R(t)$ is totally +uncorrelated to $x$ and $\dot x$, \[ -\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 -}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} +\begin{array}{l} + \left\langle {x(t)R(t)} \right\rangle = 0, \\ + \left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ + \end{array} \] -For an infinite harmonic bath, we can use the spectral density and -an integral over frequencies. +This property is what we expect from a truly random process. As long +as the model, which is gaussian distribution in general, chosen for +$R(t)$ is a truly random process, the stochastic nature of the GLE +still remains. +%dynamic friction kernel +The convolution integral \[ -R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) -- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} -\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha -(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) +\int_0^t {\xi (t)\dot x(t - \tau )d\tau } \] -The random forces depend only on initial conditions. +depends on the entire history of the evolution of $x$, which implies +that the bath retains memory of previous motions. In other words, +the bath requires a finite time to respond to change in the motion +of the system. For a sluggish bath which responds slowly to changes +in the system coordinate, we may regard $\xi(t)$ as a constant +$\xi(t) = \Xi_0$. Hence, the convolution integral becomes +\[ +\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) +\] +and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes +\[ +m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + +\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), +\] +which can be used to describe dynamic caging effect. The other +extreme is the bath that responds infinitely quickly to motions in +the system. Thus, $\xi (t)$ can be taken as a $delta$ function in +time: +\[ +\xi (t) = 2\xi _0 \delta (t) +\] +Hence, the convolution integral becomes +\[ +\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t +{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), +\] +and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes +\begin{equation} +m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot +x(t) + R(t) \label{introEquation:LangevinEquation} +\end{equation} +which is known as the Langevin equation. The static friction +coefficient $\xi _0$ can either be calculated from spectral density +or be determined by Stokes' law for regular shaped particles.A +briefly review on calculating friction tensor for arbitrary shaped +particles is given in Sec.~\ref{introSection:frictionTensor}. \subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} -So we can define a new set of coordinates, + +Defining a new set of coordinates, \[ q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha ^2 }}x(0) -\] -This makes +\], +we can rewrite $R(T)$ as \[ -R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} +R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. \] And since the $q$ coordinates are harmonic oscillators, \[ -\begin{array}{l} +\begin{array}{c} + \left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ + \left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ + = \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ + = kT\xi (t) \\ \end{array} +\] +Thus, we recover the \emph{second fluctuation dissipation theorem} +\begin{equation} +\xi (t) = \left\langle {R(t)R(0)} \right\rangle +\label{introEquation:secondFluctuationDissipation}. +\end{equation} +In effect, it acts as a constraint on the possible ways in which one +can model the random force and friction kernel. + +\subsection{\label{introSection:frictionTensor} Friction Tensor} +Theoretically, the friction kernel can be determined using velocity +autocorrelation function. However, this approach become impractical +when the system become more and more complicate. Instead, various +approaches based on hydrodynamics have been developed to calculate +the friction coefficients. The friction effect is isotropic in +Equation, $\zeta$ can be taken as a scalar. In general, friction +tensor $\Xi$ is a $6\times 6$ matrix given by +\[ +\Xi = \left( {\begin{array}{*{20}c} + {\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ + {\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ +\end{array}} \right). \] +Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction +tensor and rotational resistance (friction) tensor respectively, +while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ +{\Xi^{rt} }$ is rotation-translation coupling tensor. When a +particle moves in a fluid, it may experience friction force or +torque along the opposite direction of the velocity or angular +velocity, +\[ +\left( \begin{array}{l} + F_R \\ + \tau _R \\ + \end{array} \right) = - \left( {\begin{array}{*{20}c} + {\Xi ^{tt} } & {\Xi ^{rt} } \\ + {\Xi ^{tr} } & {\Xi ^{rr} } \\ +\end{array}} \right)\left( \begin{array}{l} + v \\ + w \\ + \end{array} \right) +\] +where $F_r$ is the friction force and $\tau _R$ is the friction +toque. -\begin{align} -\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 } } -% -&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} -\right\rangle \cos (\omega _\alpha t)} -% -&= kT\xi (t) -\end{align} +\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} -\begin{equation} -\xi (t) = \left\langle {R(t)R(0)} \right\rangle -\label{introEquation:secondFluctuationDissipation} -\end{equation} +For a spherical particle, the translational and rotational friction +constant can be calculated from Stoke's law, +\[ +\Xi ^{tt} = \left( {\begin{array}{*{20}c} + {6\pi \eta R} & 0 & 0 \\ + 0 & {6\pi \eta R} & 0 \\ + 0 & 0 & {6\pi \eta R} \\ +\end{array}} \right) +\] +and +\[ +\Xi ^{rr} = \left( {\begin{array}{*{20}c} + {8\pi \eta R^3 } & 0 & 0 \\ + 0 & {8\pi \eta R^3 } & 0 \\ + 0 & 0 & {8\pi \eta R^3 } \\ +\end{array}} \right) +\] +where $\eta$ is the viscosity of the solvent and $R$ is the +hydrodynamics radius. -\section{\label{introSection:hydroynamics}Hydrodynamics} +Other non-spherical shape, such as cylinder and ellipsoid +\textit{etc}, are widely used as reference for developing new +hydrodynamics theory, because their properties can be calculated +exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, +also called a triaxial ellipsoid, which is given in Cartesian +coordinates by +\[ +\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 +}} = 1 +\] +where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, +due to the complexity of the elliptic integral, only the ellipsoid +with the restriction of two axes having to be equal, \textit{i.e.} +prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved +exactly. Introducing an elliptic integral parameter $S$ for prolate, +\[ +S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 +} }}{b}, +\] +and oblate, +\[ +S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } +}}{a} +\], +one can write down the translational and rotational resistance +tensors +\[ +\begin{array}{l} + \Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ + \Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ + \end{array}, +\] +and +\[ +\begin{array}{l} + \Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ + \Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ + \end{array}. +\] -\subsection{\label{introSection:frictionTensor} Friction Tensor} -\subsection{\label{introSection:analyticalApproach}Analytical -Approach} +\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} -\subsection{\label{introSection:approximationApproach}Approximation -Approach} +Unlike spherical and other regular shaped molecules, there is not +analytical solution for friction tensor of any arbitrary shaped +rigid molecules. The ellipsoid of revolution model and general +triaxial ellipsoid model have been used to approximate the +hydrodynamic properties of rigid bodies. However, since the mapping +from all possible ellipsoidal space, $r$-space, to all possible +combination of rotational diffusion coefficients, $D$-space is not +unique\cite{Wegener79} as well as the intrinsic coupling between +translational and rotational motion of rigid body\cite{}, general +ellipsoid is not always suitable for modeling arbitrarily shaped +rigid molecule. A number of studies have been devoted to determine +the friction tensor for irregularly shaped rigid bodies using more +advanced method\cite{} where the molecule of interest was modeled by +combinations of spheres(beads)\cite{} and the hydrodynamics +properties of the molecule can be calculated using the hydrodynamic +interaction tensor. Let us consider a rigid assembly of $N$ beads +immersed in a continuous medium. Due to hydrodynamics interaction, +the ``net'' velocity of $i$th bead, $v'_i$ is different than its +unperturbed velocity $v_i$, +\[ +v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } +\] +where $F_i$ is the frictional force, and $T_{ij}$ is the +hydrodynamic interaction tensor. The friction force of $i$th bead is +proportional to its ``net'' velocity +\begin{equation} +F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. +\label{introEquation:tensorExpression} +\end{equation} +This equation is the basis for deriving the hydrodynamic tensor. In +1930, Oseen and Burgers gave a simple solution to Equation +\ref{introEquation:tensorExpression} +\begin{equation} +T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} +R_{ij}^T }}{{R_{ij}^2 }}} \right). +\label{introEquation:oseenTensor} +\end{equation} +Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. +A second order expression for element of different size was +introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de +la Torre and Bloomfield, +\begin{equation} +T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + +\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma +_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - +\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. +\label{introEquation:RPTensorNonOverlapped} +\end{equation} +Both of the Equation \ref{introEquation:oseenTensor} and Equation +\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} +\ge \sigma _i + \sigma _j$. An alternative expression for +overlapping beads with the same radius, $\sigma$, is given by +\begin{equation} +T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - +\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + +\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] +\label{introEquation:RPTensorOverlapped} +\end{equation} -\subsection{\label{introSection:centersRigidBody}Centers of Rigid -Body} +To calculate the resistance tensor at an arbitrary origin $O$, we +construct a $3N \times 3N$ matrix consisting of $N \times N$ +$B_{ij}$ blocks +\begin{equation} +B = \left( {\begin{array}{*{20}c} + {B_{11} } & \ldots & {B_{1N} } \\ + \vdots & \ddots & \vdots \\ + {B_{N1} } & \cdots & {B_{NN} } \\ +\end{array}} \right), +\end{equation} +where $B_{ij}$ is given by +\[ +B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} +)T_{ij} +\] +where $\delta _{ij}$ is Kronecker delta function. Inverting matrix +$B$, we obtain -\section{\label{introSection:correlationFunctions}Correlation Functions} +\[ +C = B^{ - 1} = \left( {\begin{array}{*{20}c} + {C_{11} } & \ldots & {C_{1N} } \\ + \vdots & \ddots & \vdots \\ + {C_{N1} } & \cdots & {C_{NN} } \\ +\end{array}} \right) +\] +, which can be partitioned into $N \times N$ $3 \times 3$ block +$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ +\[ +U_i = \left( {\begin{array}{*{20}c} + 0 & { - z_i } & {y_i } \\ + {z_i } & 0 & { - x_i } \\ + { - y_i } & {x_i } & 0 \\ +\end{array}} \right) +\] +where $x_i$, $y_i$, $z_i$ are the components of the vector joining +bead $i$ and origin $O$. Hence, the elements of resistance tensor at +arbitrary origin $O$ can be written as +\begin{equation} +\begin{array}{l} + \Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ + \Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ + \Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ + \end{array} +\label{introEquation:ResistanceTensorArbitraryOrigin} +\end{equation} + +The resistance tensor depends on the origin to which they refer. The +proper location for applying friction force is the center of +resistance (reaction), at which the trace of rotational resistance +tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of +resistance is defined as an unique point of the rigid body at which +the translation-rotation coupling tensor are symmetric, +\begin{equation} +\Xi^{tr} = \left( {\Xi^{tr} } \right)^T +\label{introEquation:definitionCR} +\end{equation} +Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, +we can easily find out that the translational resistance tensor is +origin independent, while the rotational resistance tensor and +translation-rotation coupling resistance tensor depend on the +origin. Given resistance tensor at an arbitrary origin $O$, and a +vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can +obtain the resistance tensor at $P$ by +\begin{equation} +\begin{array}{l} + \Xi _P^{tt} = \Xi _O^{tt} \\ + \Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ + \Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ + \end{array} + \label{introEquation:resistanceTensorTransformation} +\end{equation} +where +\[ +U_{OP} = \left( {\begin{array}{*{20}c} + 0 & { - z_{OP} } & {y_{OP} } \\ + {z_i } & 0 & { - x_{OP} } \\ + { - y_{OP} } & {x_{OP} } & 0 \\ +\end{array}} \right) +\] +Using Equations \ref{introEquation:definitionCR} and +\ref{introEquation:resistanceTensorTransformation}, one can locate +the position of center of resistance, +\[ +\left( \begin{array}{l} + x_{OR} \\ + y_{OR} \\ + z_{OR} \\ + \end{array} \right) = \left( {\begin{array}{*{20}c} + {(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ + { - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ + { - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ +\end{array}} \right)^{ - 1} \left( \begin{array}{l} + (\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ + (\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ + (\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ + \end{array} \right). +\] +where $x_OR$, $y_OR$, $z_OR$ are the components of the vector +joining center of resistance $R$ and origin $O$.