--- trunk/tengDissertation/Introduction.tex 2006/04/07 05:03:54 2697 +++ trunk/tengDissertation/Introduction.tex 2006/04/19 19:46:53 2720 @@ -27,11 +27,11 @@ $F_ij$ be the force that particle $i$ exerts on partic \end{equation} A point mass interacting with other bodies moves with the acceleration along the direction of the force acting on it. Let -$F_ij$ be the force that particle $i$ exerts on particle $j$, and -$F_ji$ be the force that particle $j$ exerts on particle $i$. +$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and +$F_{ji}$ be the force that particle $j$ exerts on particle $i$. Newton¡¯s third law states that \begin{equation} -F_ij = -F_ji +F_{ij} = -F_{ji} \label{introEquation:newtonThirdLaw} \end{equation} @@ -117,7 +117,7 @@ for a holonomic system of $f$ degrees of freedom, the \subsubsection{\label{introSection:equationOfMotionLagrangian}The Equations of Motion in Lagrangian Mechanics} -for a holonomic system of $f$ degrees of freedom, the equations of +For a holonomic system of $f$ degrees of freedom, the equations of motion in the Lagrangian form is \begin{equation} \frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - @@ -212,41 +212,252 @@ q_i }}} \right) = 0} }}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - \frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial -q_i }}} \right) = 0} -\label{introEquation:conserveHalmitonian} +q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} \end{equation} -When studying Hamiltonian system, it is more convenient to use -notation -\begin{equation} -r = r(q,p)^T -\end{equation} -and to introduce a $2n \times 2n$ canonical structure matrix $J$, -\begin{equation} -J = \left( {\begin{array}{*{20}c} - 0 & I \\ - { - I} & 0 \\ -\end{array}} \right) -\label{introEquation:canonicalMatrix} -\end{equation} -where $I$ is a $n \times n$ identity matrix and $J$ is a -skew-symmetric matrix ($ J^T = - J $). Thus, Hamiltonian system -can be rewritten as, -\begin{equation} -\frac{d}{{dt}}r = J\nabla _r H(r) -\label{introEquation:compactHamiltonian} -\end{equation} - \section{\label{introSection:statisticalMechanics}Statistical Mechanics} The thermodynamic behaviors and properties of Molecular Dynamics simulation are governed by the principle of Statistical Mechanics. The following section will give a brief introduction to some of the -Statistical Mechanics concepts presented in this dissertation. +Statistical Mechanics concepts and theorem presented in this +dissertation. -\subsection{\label{introSection:ensemble}Ensemble and Phase Space} +\subsection{\label{introSection:ensemble}Phase Space and Ensemble} + +Mathematically, phase space is the space which represents all +possible states. Each possible state of the system corresponds to +one unique point in the phase space. For mechanical systems, the +phase space usually consists of all possible values of position and +momentum variables. Consider a dynamic system in a cartesian space, +where each of the $6f$ coordinates and momenta is assigned to one of +$6f$ mutually orthogonal axes, the phase space of this system is a +$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , +\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and +momenta is a phase space vector. + +A microscopic state or microstate of a classical system is +specification of the complete phase space vector of a system at any +instant in time. An ensemble is defined as a collection of systems +sharing one or more macroscopic characteristics but each being in a +unique microstate. The complete ensemble is specified by giving all +systems or microstates consistent with the common macroscopic +characteristics of the ensemble. Although the state of each +individual system in the ensemble could be precisely described at +any instance in time by a suitable phase space vector, when using +ensembles for statistical purposes, there is no need to maintain +distinctions between individual systems, since the numbers of +systems at any time in the different states which correspond to +different regions of the phase space are more interesting. Moreover, +in the point of view of statistical mechanics, one would prefer to +use ensembles containing a large enough population of separate +members so that the numbers of systems in such different states can +be regarded as changing continuously as we traverse different +regions of the phase space. The condition of an ensemble at any time +can be regarded as appropriately specified by the density $\rho$ +with which representative points are distributed over the phase +space. The density of distribution for an ensemble with $f$ degrees +of freedom is defined as, +\begin{equation} +\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). +\label{introEquation:densityDistribution} +\end{equation} +Governed by the principles of mechanics, the phase points change +their value which would change the density at any time at phase +space. Hence, the density of distribution is also to be taken as a +function of the time. + +The number of systems $\delta N$ at time $t$ can be determined by, +\begin{equation} +\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. +\label{introEquation:deltaN} +\end{equation} +Assuming a large enough population of systems are exploited, we can +sufficiently approximate $\delta N$ without introducing +discontinuity when we go from one region in the phase space to +another. By integrating over the whole phase space, +\begin{equation} +N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f +\label{introEquation:totalNumberSystem} +\end{equation} +gives us an expression for the total number of the systems. Hence, +the probability per unit in the phase space can be obtained by, +\begin{equation} +\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int +{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. +\label{introEquation:unitProbability} +\end{equation} +With the help of Equation(\ref{introEquation:unitProbability}) and +the knowledge of the system, it is possible to calculate the average +value of any desired quantity which depends on the coordinates and +momenta of the system. Even when the dynamics of the real system is +complex, or stochastic, or even discontinuous, the average +properties of the ensemble of possibilities as a whole may still +remain well defined. For a classical system in thermal equilibrium +with its environment, the ensemble average of a mechanical quantity, +$\langle A(q , p) \rangle_t$, takes the form of an integral over the +phase space of the system, +\begin{equation} +\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho +(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho +(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} +\label{introEquation:ensembelAverage} +\end{equation} + +There are several different types of ensembles with different +statistical characteristics. As a function of macroscopic +parameters, such as temperature \textit{etc}, partition function can +be used to describe the statistical properties of a system in +thermodynamic equilibrium. +As an ensemble of systems, each of which is known to be thermally +isolated and conserve energy, Microcanonical ensemble(NVE) has a +partition function like, +\begin{equation} +\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. +\end{equation} +A canonical ensemble(NVT)is an ensemble of systems, each of which +can share its energy with a large heat reservoir. The distribution +of the total energy amongst the possible dynamical states is given +by the partition function, +\begin{equation} +\Omega (N,V,T) = e^{ - \beta A} +\label{introEquation:NVTPartition} +\end{equation} +Here, $A$ is the Helmholtz free energy which is defined as $ A = U - +TS$. Since most experiment are carried out under constant pressure +condition, isothermal-isobaric ensemble(NPT) play a very important +role in molecular simulation. The isothermal-isobaric ensemble allow +the system to exchange energy with a heat bath of temperature $T$ +and to change the volume as well. Its partition function is given as +\begin{equation} +\Delta (N,P,T) = - e^{\beta G}. + \label{introEquation:NPTPartition} +\end{equation} +Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. + +\subsection{\label{introSection:liouville}Liouville's theorem} + +The Liouville's theorem is the foundation on which statistical +mechanics rests. It describes the time evolution of phase space +distribution function. In order to calculate the rate of change of +$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we +consider the two faces perpendicular to the $q_1$ axis, which are +located at $q_1$ and $q_1 + \delta q_1$, the number of phase points +leaving the opposite face is given by the expression, +\begin{equation} +\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } +\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 +}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 +\ldots \delta p_f . +\end{equation} +Summing all over the phase space, we obtain +\begin{equation} +\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho +\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + +\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( +{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial +\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 +\ldots \delta q_f \delta p_1 \ldots \delta p_f . +\end{equation} +Differentiating the equations of motion in Hamiltonian formalism +(\ref{introEquation:motionHamiltonianCoordinate}, +\ref{introEquation:motionHamiltonianMomentum}), we can show, +\begin{equation} +\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} ++ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , +\end{equation} +which cancels the first terms of the right hand side. Furthermore, +divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta +p_f $ in both sides, we can write out Liouville's theorem in a +simple form, +\begin{equation} +\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f +{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + +\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . +\label{introEquation:liouvilleTheorem} +\end{equation} + +Liouville's theorem states that the distribution function is +constant along any trajectory in phase space. In classical +statistical mechanics, since the number of particles in the system +is huge, we may be able to believe the system is stationary, +\begin{equation} +\frac{{\partial \rho }}{{\partial t}} = 0. +\label{introEquation:stationary} +\end{equation} +In such stationary system, the density of distribution $\rho$ can be +connected to the Hamiltonian $H$ through Maxwell-Boltzmann +distribution, +\begin{equation} +\rho \propto e^{ - \beta H} +\label{introEquation:densityAndHamiltonian} +\end{equation} + +\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} +Lets consider a region in the phase space, +\begin{equation} +\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . +\end{equation} +If this region is small enough, the density $\rho$ can be regarded +as uniform over the whole phase space. Thus, the number of phase +points inside this region is given by, +\begin{equation} +\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f +dp_1 } ..dp_f. +\end{equation} + +\begin{equation} +\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho +\frac{d}{{dt}}(\delta v) = 0. +\end{equation} +With the help of stationary assumption +(\ref{introEquation:stationary}), we obtain the principle of the +\emph{conservation of extension in phase space}, +\begin{equation} +\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } +...dq_f dp_1 } ..dp_f = 0. +\label{introEquation:volumePreserving} +\end{equation} + +\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} + +Liouville's theorem can be expresses in a variety of different forms +which are convenient within different contexts. For any two function +$F$ and $G$ of the coordinates and momenta of a system, the Poisson +bracket ${F, G}$ is defined as +\begin{equation} +\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial +F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - +\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial +q_i }}} \right)}. +\label{introEquation:poissonBracket} +\end{equation} +Substituting equations of motion in Hamiltonian formalism( +\ref{introEquation:motionHamiltonianCoordinate} , +\ref{introEquation:motionHamiltonianMomentum} ) into +(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's +theorem using Poisson bracket notion, +\begin{equation} +\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ +{\rho ,H} \right\}. +\label{introEquation:liouvilleTheromInPoissin} +\end{equation} +Moreover, the Liouville operator is defined as +\begin{equation} +iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial +p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial +H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} +\label{introEquation:liouvilleOperator} +\end{equation} +In terms of Liouville operator, Liouville's equation can also be +expressed as +\begin{equation} +\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho +\label{introEquation:liouvilleTheoremInOperator} +\end{equation} + \subsection{\label{introSection:ergodic}The Ergodic Hypothesis} Various thermodynamic properties can be calculated from Molecular @@ -261,22 +472,23 @@ statistical ensemble are identical \cite{Frenkel1996, ensemble average. It states that time average and average over the statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. \begin{equation} -\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty } -\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma -{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq +\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } +\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma +{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp \end{equation} -where $\langle A \rangle_t$ is an equilibrium value of a physical -quantity and $\rho (p(t), q(t))$ is the equilibrium distribution -function. If an observation is averaged over a sufficiently long -time (longer than relaxation time), all accessible microstates in -phase space are assumed to be equally probed, giving a properly -weighted statistical average. This allows the researcher freedom of -choice when deciding how best to measure a given observable. In case -an ensemble averaged approach sounds most reasonable, the Monte -Carlo techniques\cite{metropolis:1949} can be utilized. Or if the -system lends itself to a time averaging approach, the Molecular -Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} -will be the best choice\cite{Frenkel1996}. +where $\langle A(q , p) \rangle_t$ is an equilibrium value of a +physical quantity and $\rho (p(t), q(t))$ is the equilibrium +distribution function. If an observation is averaged over a +sufficiently long time (longer than relaxation time), all accessible +microstates in phase space are assumed to be equally probed, giving +a properly weighted statistical average. This allows the researcher +freedom of choice when deciding how best to measure a given +observable. In case an ensemble averaged approach sounds most +reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be +utilized. Or if the system lends itself to a time averaging +approach, the Molecular Dynamics techniques in +Sec.~\ref{introSection:molecularDynamics} will be the best +choice\cite{Frenkel1996}. \section{\label{introSection:geometricIntegratos}Geometric Integrators} A variety of numerical integrators were proposed to simulate the @@ -324,166 +536,936 @@ classical mechanics. According to Liouville's theorem, is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and the \emph{pullback} of $\eta$ under f is equal to $\omega$. Canonical transformation is an example of symplectomorphism in -classical mechanics. According to Liouville's theorem, the -Hamiltonian \emph{flow} or \emph{symplectomorphism} generated by the -Hamiltonian vector filed preserves the volume form on the phase -space, which is the basis of classical statistical mechanics. +classical mechanics. -\subsection{\label{introSection:exactFlow}The Exact Flow of ODE} +\subsection{\label{introSection:ODE}Ordinary Differential Equations} -\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting} +For a ordinary differential system defined as +\begin{equation} +\dot x = f(x) +\end{equation} +where $x = x(q,p)^T$, this system is canonical Hamiltonian, if +\begin{equation} +f(r) = J\nabla _x H(r). +\end{equation} +$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric +matrix +\begin{equation} +J = \left( {\begin{array}{*{20}c} + 0 & I \\ + { - I} & 0 \\ +\end{array}} \right) +\label{introEquation:canonicalMatrix} +\end{equation} +where $I$ is an identity matrix. Using this notation, Hamiltonian +system can be rewritten as, +\begin{equation} +\frac{d}{{dt}}x = J\nabla _x H(x) +\label{introEquation:compactHamiltonian} +\end{equation}In this case, $f$ is +called a \emph{Hamiltonian vector field}. -\section{\label{introSection:molecularDynamics}Molecular Dynamics} +Another generalization of Hamiltonian dynamics is Poisson Dynamics, +\begin{equation} +\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} +\end{equation} +The most obvious change being that matrix $J$ now depends on $x$. -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{introSection:exactFlow}Exact Flow} -\subsection{\label{introSec:mdInit}Initialization} +Let $x(t)$ be the exact solution of the ODE system, +\begin{equation} +\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} +\end{equation} +The exact flow(solution) $\varphi_\tau$ is defined by +\[ +x(t+\tau) =\varphi_\tau(x(t)) +\] +where $\tau$ is a fixed time step and $\varphi$ is a map from phase +space to itself. The flow has the continuous group property, +\begin{equation} +\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 ++ \tau _2 } . +\end{equation} +In particular, +\begin{equation} +\varphi _\tau \circ \varphi _{ - \tau } = I +\end{equation} +Therefore, the exact flow is self-adjoint, +\begin{equation} +\varphi _\tau = \varphi _{ - \tau }^{ - 1}. +\end{equation} +The exact flow can also be written in terms of the of an operator, +\begin{equation} +\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial +}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). +\label{introEquation:exponentialOperator} +\end{equation} -\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} +In most cases, it is not easy to find the exact flow $\varphi_\tau$. +Instead, we use a approximate map, $\psi_\tau$, which is usually +called integrator. The order of an integrator $\psi_\tau$ is $p$, if +the Taylor series of $\psi_\tau$ agree to order $p$, +\begin{equation} +\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) +\end{equation} -\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} +\subsection{\label{introSection:geometricProperties}Geometric Properties} -A rigid body is a body in which the distance between any two given -points of a rigid body remains constant regardless of external -forces exerted on it. A rigid body therefore conserves its shape -during its motion. +The hidden geometric properties of ODE and its flow play important +roles in numerical studies. Many of them can be found in systems +which occur naturally in applications. -Applications of dynamics of rigid bodies. - -\subsection{\label{introSection:lieAlgebra}Lie Algebra} - -\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} - -\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} - -%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} - -\section{\label{introSection:correlationFunctions}Correlation Functions} - -\section{\label{introSection:langevinDynamics}Langevin Dynamics} - -\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} - -\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} - +Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is +a \emph{symplectic} flow if it satisfies, \begin{equation} -H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) -\label{introEquation:bathGLE} +{\varphi '}^T J \varphi ' = J. \end{equation} -where $H_B$ is harmonic bath Hamiltonian, +According to Liouville's theorem, the symplectic volume is invariant +under a Hamiltonian flow, which is the basis for classical +statistical mechanics. Furthermore, the flow of a Hamiltonian vector +field on a symplectic manifold can be shown to be a +symplectomorphism. As to the Poisson system, +\begin{equation} +{\varphi '}^T J \varphi ' = J \circ \varphi +\end{equation} +is the property must be preserved by the integrator. + +It is possible to construct a \emph{volume-preserving} flow for a +source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ +\det d\varphi = 1$. One can show easily that a symplectic flow will +be volume-preserving. + +Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} +will result in a new system, \[ -H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 -}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} +\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). \] -and $\Delta U$ is bilinear system-bath coupling, +The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. +In other words, the flow of this vector field is reversible if and +only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. + +A \emph{first integral}, or conserved quantity of a general +differential function is a function $ G:R^{2d} \to R^d $ which is +constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , \[ -\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} +\frac{{dG(x(t))}}{{dt}} = 0. +\] +Using chain rule, one may obtain, +\[ +\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, \] -Completing the square, +which is the condition for conserving \emph{first integral}. For a +canonical Hamiltonian system, the time evolution of an arbitrary +smooth function $G$ is given by, +\begin{equation} +\begin{array}{c} + \frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ + = [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ + \end{array} +\label{introEquation:firstIntegral1} +\end{equation} +Using poisson bracket notion, Equation +\ref{introEquation:firstIntegral1} can be rewritten as \[ -H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ -{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha -w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha -w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = -1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 +\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). \] -and putting it back into Eq.~\ref{introEquation:bathGLE}, +Therefore, the sufficient condition for $G$ to be the \emph{first +integral} of a Hamiltonian system is \[ -H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N -{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha -w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha -w_\alpha ^2 }}x} \right)^2 } \right\}} +\left\{ {G,H} \right\} = 0. \] -where +As well known, the Hamiltonian (or energy) H of a Hamiltonian system +is a \emph{first integral}, which is due to the fact $\{ H,H\} = +0$. + + + When designing any numerical methods, one should always try to +preserve the structural properties of the original ODE and its flow. + +\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} +A lot of well established and very effective numerical methods have +been successful precisely because of their symplecticities even +though this fact was not recognized when they were first +constructed. The most famous example is leapfrog methods in +molecular dynamics. In general, symplectic integrators can be +constructed using one of four different methods. +\begin{enumerate} +\item Generating functions +\item Variational methods +\item Runge-Kutta methods +\item Splitting methods +\end{enumerate} + +Generating function tends to lead to methods which are cumbersome +and difficult to use. In dissipative systems, variational methods +can capture the decay of energy accurately. Since their +geometrically unstable nature against non-Hamiltonian perturbations, +ordinary implicit Runge-Kutta methods are not suitable for +Hamiltonian system. Recently, various high-order explicit +Runge--Kutta methods have been developed to overcome this +instability. However, due to computational penalty involved in +implementing the Runge-Kutta methods, they do not attract too much +attention from Molecular Dynamics community. Instead, splitting have +been widely accepted since they exploit natural decompositions of +the system\cite{Tuckerman92}. + +\subsubsection{\label{introSection:splittingMethod}Splitting Method} + +The main idea behind splitting methods is to decompose the discrete +$\varphi_h$ as a composition of simpler flows, +\begin{equation} +\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ +\varphi _{h_n } +\label{introEquation:FlowDecomposition} +\end{equation} +where each of the sub-flow is chosen such that each represent a +simpler integration of the system. + +Suppose that a Hamiltonian system takes the form, \[ -W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 -}}{{2m_\alpha w_\alpha ^2 }}} x^2 +H = H_1 + H_2. \] -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}, +Here, $H_1$ and $H_2$ may represent different physical processes of +the system. For instance, they may relate to kinetic and potential +energy respectively, which is a natural decomposition of the +problem. If $H_1$ and $H_2$ can be integrated using exact flows +$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first +order is then given by the Lie-Trotter formula +\begin{equation} +\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, +\label{introEquation:firstOrderSplitting} +\end{equation} +where $\varphi _h$ is the result of applying the corresponding +continuous $\varphi _i$ over a time $h$. By definition, as +$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it +must follow that each operator $\varphi_i(t)$ is a symplectic map. +It is easy to show that any composition of symplectic flows yields a +symplectic map, +\begin{equation} +(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi +'\phi ' = \phi '^T J\phi ' = J, + \label{introEquation:SymplecticFlowComposition} +\end{equation} +where $\phi$ and $\psi$ both are symplectic maps. Thus operator +splitting in this context automatically generates a symplectic map. + +The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) +introduces local errors proportional to $h^2$, while Strang +splitting gives a second-order decomposition, +\begin{equation} +\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi +_{1,h/2} , \label{introEquation:secondOrderSplitting} +\end{equation} +which has a local error proportional to $h^3$. Sprang splitting's +popularity in molecular simulation community attribute to its +symmetric property, +\begin{equation} +\varphi _h^{ - 1} = \varphi _{ - h}. +\label{introEquation:timeReversible} +\end{equation} + +\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} +The classical equation for a system consisting of interacting +particles can be written in Hamiltonian form, +\[ +H = T + V +\] +where $T$ is the kinetic energy and $V$ is the potential energy. +Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one +obtains the following: \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{introEq:Lp5} +q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + + \frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % +\label{introEquation:Lp10a} \\% +% +\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} + \biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % +\label{introEquation:Lp10b} \end{align} -, and +where $F(t)$ is the force at time $t$. This integration scheme is +known as \emph{velocity verlet} which is +symplectic(\ref{introEquation:SymplecticFlowComposition}), +time-reversible(\ref{introEquation:timeReversible}) and +volume-preserving (\ref{introEquation:volumePreserving}). These +geometric properties attribute to its long-time stability and its +popularity in the community. However, the most commonly used +velocity verlet integration scheme is written as below, \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) +\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= + \dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% +% +q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% + \label{introEquation:Lp9b}\\% +% +\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + + \frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} \end{align} +From the preceding splitting, one can see that the integration of +the equations of motion would follow: +\begin{enumerate} +\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. -\subsection{\label{introSection:laplaceTransform}The Laplace Transform} +\item Use the half step velocities to move positions one whole step, $\Delta t$. -\[ -L(x) = \int_0^\infty {x(t)e^{ - pt} dt} -\] +\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. -\[ -L(x + y) = L(x) + L(y) -\] +\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. +\end{enumerate} +Simply switching the order of splitting and composing, a new +integrator, the \emph{position verlet} integrator, can be generated, +\begin{align} +\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + +\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % +\label{introEquation:positionVerlet1} \\% +% +q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot +q(\Delta t)} \right]. % + \label{introEquation:positionVerlet2} +\end{align} + +\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} + +Baker-Campbell-Hausdorff formula can be used to determine the local +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 +\begin{equation} +\exp (hX + hY) = \exp (hZ) +\end{equation} +where +\begin{equation} +hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( +{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . +\end{equation} +Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by \[ -L(ax) = aL(x) +[X,Y] = XY - YX . \] - +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 ) +\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 +procedure can be applied to general splitting, of the form +\begin{equation} +\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 +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 +method can be constructed by composing, \[ -L(\dot x) = pL(x) - px(0) +\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta +h}^{(2)} \circ \varphi _{\alpha h}^{(2)} \] - +where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta += \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric +integrator $ \varphi _h^{(2n + 2)}$ can be composed by +\begin{equation} +\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi +_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} +\end{equation} +, if the weights are chosen as \[ -L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) +\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = +\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . \] +\section{\label{introSection:molecularDynamics}Molecular Dynamics} + +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 \[ -L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) +T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} \] +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. -Some relatively important transformation, -\[ -L(\cos at) = \frac{p}{{p^2 + a^2 }} +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, including the force evaluation +and the numerical integration schemes of the equations of motion . +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} + +\subsubsection{\label{introSec:forceCalculation}The Force Calculation} + +\subsubsection{\label{introSection:integrationSchemes} Integration +Schemes} + +\subsection{\label{introSection:Analysis} Analysis} + +\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} + +Rigid bodies are frequently involved in the modeling of different +areas, from engineering, physics, to chemistry. For example, +missiles and vehicle are usually modeled by rigid bodies. The +movement of the objects in 3D gaming engine or other physics +simulator is governed by the rigid body dynamics. In molecular +simulation, rigid body is used to simplify the model in +protein-protein docking study{\cite{Gray03}}. + +It is very important to develop stable and efficient methods to +integrate the equations of motion of orientational degrees of +freedom. Euler angles are the nature choice to describe the +rotational degrees of freedom. However, due to its singularity, the +numerical integration of corresponding equations of motion is very +inefficient and inaccurate. Although an alternative integrator using +different sets of Euler angles can overcome this difficulty\cite{}, +the computational penalty and the lost of angular momentum +conservation still remain. A singularity free representation +utilizing quaternions was developed by Evans in 1977. Unfortunately, +this approach suffer from the nonseparable Hamiltonian resulted from +quaternion representation, which prevents the symplectic algorithm +to be utilized. Another different approach is to apply holonomic +constraints to the atoms belonging to the rigid body. Each atom +moves independently under the normal forces deriving from potential +energy and constraint forces which are used to guarantee the +rigidness. However, due to their iterative nature, SHAKE and Rattle +algorithm converge very slowly when the number of constraint +increases. + +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 $Q$ 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 $Q_T Q = 1$. 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{Dullweber1997} in depth. + +\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 ]. +\label{introEquation:RBHamiltonian} +\end{equation} +Here, $q$ and $Q$ are the position and rotation matrix for the +rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and +$J$, a diagonal matrix, is defined by +\[ +I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } \] +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} +\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 . \\ +\label{introEquation:RBFirstOrderConstraint} +\end{equation} +Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, +\ref{introEquation:motionHamiltonianMomentum}), one can write down +the equations of motion, \[ -L(\sin at) = \frac{a}{{p^2 + a^2 }} +\begin{array}{c} + \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ + \frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ + \frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ + \frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ + \end{array} \] +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. The two method are +proved to be equivalent. The holonomic constraint and equations of +motions define a constraint manifold for rigid body \[ -L(1) = \frac{1}{p} +M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} +\right\}. \] -First, the bath coordinates, +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 \[ -p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega -_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha -}}L(x) +\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), \] +the mechanical system subject to a holonomic constraint manifold $M$ +can be re-formulated as a Hamiltonian system on the cotangent space \[ -L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + -px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} +T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = +1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} \] -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, +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 +\[ +V(q,Q) = V(Q X_0 + q). +\] +Hence, the force and torque are given by +\[ +\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, +\] +and +\[ +\nabla _Q V(q,Q) = F(q,Q)X_i^t +\] +respectively. + +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, +\[ +\begin{array}{l} + \Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ + Q^T Q = 1 \\ + \end{array} +\] + +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 +\[ +\hat vu = v \times u +\] + +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 +\[ +\dot Q = Qskew(I^{ - 1} \pi ) +\] + +\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 +\[ +\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = +Q(0)e^{\Delta tR_1 } +\] +with +\[ +e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} + 0 & 0 & 0 \\ + 0 & {\cos \theta _1 } & {\sin \theta _1 } \\ + 0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ +\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. +\] +To reduce the cost of computing expensive functions in $e^{\Delta +tR_1 }$, we can use Cayley transformation, +\[ +e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 +) +\] +The flow maps for $T_2^r$ and $T_3^r$ can be found in the same +manner. + +In order to construct a second-order symplectic method, we split the +angular kinetic Hamiltonian function can into five terms +\[ +T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 +) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r +(\pi _1 ) +\]. +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 +\[ +\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi +) +\] +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 +\[ +\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 +}}{2})\pi +\] +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. + +\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, +\[ +H = T(p,\pi ) + V(q,Q) +\] +The equations of motion corresponding to potential energy and +kinetic energy are listed in the below table, +\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} +A second-order symplectic method is now obtained by the composition +of the flow maps, +\[ +\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi +_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. +\] +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 @@ -503,70 +1485,374 @@ 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. -\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} +\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. +\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} + +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. + +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 +\[ +\begin{array}{l} + \Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ + \Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ + \end{array}. +\] + +\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} + +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} -\xi (t) = \left\langle {R(t)R(0)} \right\rangle -\label{introEquation:secondFluctuationDissipation} +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} -\section{\label{introSection:hydroynamics}Hydrodynamics} +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 -\subsection{\label{introSection:frictionTensor} Friction Tensor} -\subsection{\label{introSection:analyticalApproach}Analytical -Approach} +\[ +C = B^{ - 1} = \left( {\begin{array}{*{20}c} + {C_{11} } & \ldots & {C_{1N} } \\ + \vdots & \ddots & \vdots \\ + {C_{N1} } & \cdots & {C_{NN} } \\ +\end{array}} \right) +\] +, 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} -\subsection{\label{introSection:approximationApproach}Approximation -Approach} +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 $O$. -\subsection{\label{introSection:centersRigidBody}Centers of Rigid -Body} +%\section{\label{introSection:correlationFunctions}Correlation Functions}