--- trunk/tengDissertation/Introduction.tex 2006/04/04 21:32:58 2692 +++ trunk/tengDissertation/Introduction.tex 2006/04/13 04:47:47 2706 @@ -1,14 +1,8 @@ \chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} -\section{\label{introSection:molecularDynamics}Molecular Dynamics} +\section{\label{introSection:classicalMechanics}Classical +Mechanics} -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:classicalMechanics}Classical Mechanics} - Closely related to Classical Mechanics, Molecular Dynamics simulations are carried out by integrating the equations of motion for a given system of particles. There are three fundamental ideas @@ -20,10 +14,64 @@ sufficient to predict the future behavior of the syste when further combine with the laws of mechanics will also be sufficient to predict the future behavior of the system. -\subsubsection{\label{introSection:newtonian}Newtonian Mechanics} +\subsection{\label{introSection:newtonian}Newtonian Mechanics} +The discovery of Newton's three laws of mechanics which govern the +motion of particles is the foundation of the classical mechanics. +Newton¡¯s first law defines a class of inertial frames. Inertial +frames are reference frames where a particle not interacting with +other bodies will move with constant speed in the same direction. +With respect to inertial frames Newton¡¯s second law has the form +\begin{equation} +F = \frac {dp}{dt} = \frac {mv}{dt} +\label{introEquation:newtonSecondLaw} +\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 $i$. +Newton¡¯s third law states that +\begin{equation} +F_{ij} = -F_{ji} + \label{introEquation:newtonThirdLaw} +\end{equation} -\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics} +Conservation laws of Newtonian Mechanics play very important roles +in solving mechanics problems. The linear momentum of a particle is +conserved if it is free or it experiences no force. The second +conservation theorem concerns the angular momentum of a particle. +The angular momentum $L$ of a particle with respect to an origin +from which $r$ is measured is defined to be +\begin{equation} +L \equiv r \times p \label{introEquation:angularMomentumDefinition} +\end{equation} +The torque $\tau$ with respect to the same origin is defined to be +\begin{equation} +N \equiv r \times F \label{introEquation:torqueDefinition} +\end{equation} +Differentiating Eq.~\ref{introEquation:angularMomentumDefinition}, +\[ +\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times +\dot p) +\] +since +\[ +\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0 +\] +thus, +\begin{equation} +\dot L = r \times \dot p = N +\end{equation} +If there are no external torques acting on a body, the angular +momentum of it is conserved. The last conservation theorem state +that if all forces are conservative, Energy +\begin{equation}E = T + V \label{introEquation:energyConservation} +\end{equation} + is conserved. All of these conserved quantities are +important factors to determine the quality of numerical integration +scheme for rigid body \cite{Dullweber1997}. +\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} + Newtonian Mechanics suffers from two important limitations: it describes their motion in special cartesian coordinate systems. Another limitation of Newtonian mechanics becomes obvious when we @@ -35,7 +83,7 @@ system, alternative procedures may be developed. which arise in attempts to apply Newton's equation to complex system, alternative procedures may be developed. -\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's +\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's Principle} Hamilton introduced the dynamical principle upon which it is @@ -45,10 +93,10 @@ the kinetic, $K$, and potential energies, $U$. The actual trajectory, along which a dynamical system may move from one point to another within a specified time, is derived by finding the path which minimizes the time integral of the difference between -the kinetic, $K$, and potential energies, $U$. +the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. \begin{equation} \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , -\lable{introEquation:halmitonianPrinciple1} +\label{introEquation:halmitonianPrinciple1} \end{equation} For simple mechanical systems, where the forces acting on the @@ -62,24 +110,24 @@ then Eq.~\ref{introEquation:halmitonianPrinciple1} bec \end{equation} then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes \begin{equation} -\delta \int_{t_1 }^{t_2 } {K dt = 0} , -\lable{introEquation:halmitonianPrinciple2} +\delta \int_{t_1 }^{t_2 } {L dt = 0} , +\label{introEquation:halmitonianPrinciple2} \end{equation} -\subsubsubsection{\label{introSection:equationOfMotionLagrangian}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 }} - \frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f -\lable{introEquation:eqMotionLagrangian} +\label{introEquation:eqMotionLagrangian} \end{equation} where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is generalized velocity. -\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} +\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} Arising from Lagrangian Mechanics, Hamiltonian Mechanics was introduced by William Rowan Hamilton in 1833 as a re-formulation of @@ -90,15 +138,59 @@ With the help of these momenta, we may now define a ne p_i = \frac{\partial L}{\partial \dot q_i} \label{introEquation:generalizedMomenta} \end{equation} -With the help of these momenta, we may now define a new quantity $H$ -by the equation +The Lagrange equations of motion are then expressed by \begin{equation} -H = p_1 \dot q_1 + \ldots + p_f \dot q_f - L, +p_i = \frac{{\partial L}}{{\partial q_i }} +\label{introEquation:generalizedMomentaDot} +\end{equation} + +With the help of the generalized momenta, we may now define a new +quantity $H$ by the equation +\begin{equation} +H = \sum\limits_k {p_k \dot q_k } - L , \label{introEquation:hamiltonianDefByLagrangian} \end{equation} where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and $L$ is the Lagrangian function for the system. +Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, +one can obtain +\begin{equation} +dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - +\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial +L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial +L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} +\end{equation} +Making use of Eq.~\ref{introEquation:generalizedMomenta}, the +second and fourth terms in the parentheses cancel. Therefore, +Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as +\begin{equation} +dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } +\right)} - \frac{{\partial L}}{{\partial t}}dt +\label{introEquation:diffHamiltonian2} +\end{equation} +By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can +find +\begin{equation} +\frac{{\partial H}}{{\partial p_k }} = q_k +\label{introEquation:motionHamiltonianCoordinate} +\end{equation} +\begin{equation} +\frac{{\partial H}}{{\partial q_k }} = - p_k +\label{introEquation:motionHamiltonianMomentum} +\end{equation} +and +\begin{equation} +\frac{{\partial H}}{{\partial t}} = - \frac{{\partial L}}{{\partial +t}} +\label{introEquation:motionHamiltonianTime} +\end{equation} + +Eq.~\ref{introEquation:motionHamiltonianCoordinate} and +Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's +equation of motion. Due to their symmetrical formula, they are also +known as the canonical equations of motions \cite{Goldstein01}. + An important difference between Lagrangian approach and the Hamiltonian approach is that the Lagrangian is considered to be a function of the generalized velocities $\dot q_i$ and the @@ -108,28 +200,1014 @@ equations. appropriate for application to statistical mechanics and quantum mechanics, since it treats the coordinate and its time derivative as independent variables and it only works with 1st-order differential -equations. +equations\cite{Marion90}. +In Newtonian Mechanics, a system described by conservative forces +conserves the total energy \ref{introEquation:energyConservation}. +It follows that Hamilton's equations of motion conserve the total +Hamiltonian. +\begin{equation} +\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial +H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i +}}\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} +\end{equation} -\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation} +\section{\label{introSection:statisticalMechanics}Statistical +Mechanics} -\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics} - -The thermodynamic behaviors and properties of Molecular Dynamics +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. -\subsubsection{\label{introSection::ensemble}Ensemble} +\subsection{\label{introSection:ensemble}Phase Space and Ensemble} -\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis} +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. -\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies} +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. -\subsection{\label{introSection:correlationFunctions}Correlation Functions} +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 +Dynamics simulation. By comparing experimental values with the +calculated properties, one can determine the accuracy of the +simulation and the quality of the underlying model. However, both of +experiment and computer simulation are usually performed during a +certain time interval and the measurements are averaged over a +period of them which is different from the average behavior of +many-body system in Statistical Mechanics. Fortunately, Ergodic +Hypothesis is proposed to make a connection between time average and +ensemble average. It states that time average and average over the +statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. +\begin{equation} +\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 $\langle A(q , p) \rangle_t$ 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 +motions. They usually begin with an initial conditionals and move +the objects in the direction governed by the differential equations. +However, most of them ignore the hidden physical law contained +within the equations. Since 1990, geometric integrators, which +preserve various phase-flow invariants such as symplectic structure, +volume and time reversal symmetry, are developed to address this +issue. The velocity verlet method, which happens to be a simple +example of symplectic integrator, continues to gain its popularity +in molecular dynamics community. This fact can be partly explained +by its geometric nature. + +\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} +A \emph{manifold} is an abstract mathematical space. It locally +looks like Euclidean space, but when viewed globally, it may have +more complicate structure. A good example of manifold is the surface +of Earth. It seems to be flat locally, but it is round if viewed as +a whole. A \emph{differentiable manifold} (also known as +\emph{smooth manifold}) is a manifold with an open cover in which +the covering neighborhoods are all smoothly isomorphic to one +another. In other words,it is possible to apply calculus on +\emph{differentiable manifold}. A \emph{symplectic manifold} is +defined as a pair $(M, \omega)$ which consisting of a +\emph{differentiable manifold} $M$ and a close, non-degenerated, +bilinear symplectic form, $\omega$. A symplectic form on a vector +space $V$ is a function $\omega(x, y)$ which satisfies +$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ +\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and +$\omega(x, x) = 0$. Cross product operation in vector field is an +example of symplectic form. + +One of the motivations to study \emph{symplectic manifold} in +Hamiltonian Mechanics is that a symplectic manifold can represent +all possible configurations of the system and the phase space of the +system can be described by it's cotangent bundle. Every symplectic +manifold is even dimensional. For instance, in Hamilton equations, +coordinate and momentum always appear in pairs. + +Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map +\[ +f : M \rightarrow N +\] +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. + +\subsection{\label{introSection:ODE}Ordinary Differential Equations} + +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}. + +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 $x$. +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} + +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} + +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} + +\subsection{\label{introSection:geometricProperties}Geometric Properties} + +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. + +Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is +a \emph{symplectic} flow if it satisfies, +\begin{equation} +{\varphi '}^T J \varphi ' = J. +\end{equation} +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, +\[ +\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). +\] +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 $\frac{{dx}}{{dt}} = f(x)$ , +\[ +\frac{{dG(x(t))}}{{dt}} = 0. +\] +Using chain rule, one may obtain, +\[ +\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, +\] +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 +\[ +\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). +\] +Therefore, the sufficient condition for $G$ to be the \emph{first +integral} of a Hamiltonian system is +\[ +\left\{ {G,H} \right\} = 0. +\] +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, +\[ +H = H_1 + H_2. +\] +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} +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} +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{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. + +\item Use the half step velocities to move positions one whole step, $\Delta t$. + +\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. + +\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:positionVerlet1} +\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 +\[ +[X,Y] = XY - YX . +\] +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, +\[ +\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta +h}^{(2)} \circ \varphi _{\alpha h}^{(2)} +\] +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 +\[ +\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = +\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . +\] + +\section{\label{introSection:molecularDynamics}Molecular Dynamics} + +As 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} + +\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} + +\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 $A$ 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 +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{}. + +\subsection{\label{introSection:lieAlgebra}Lie Algebra} + +\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} + +\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, +\[ +\begin{array}{c} + \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ + \frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ + \frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ + \frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ + \end{array} +\] + + +\[ +M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} +\right\} . +\] + +\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} -\subsection{\label{introSection:hydroynamics}Hydrodynamics} +\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, +\[ +H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 +}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} +\] +and $\Delta U$ is bilinear system-bath coupling, +\[ +\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} +\] +Completing the square, +\[ +H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ +{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha +w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha +w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = +1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 +\] +and putting it back into Eq.~\ref{introEquation:bathGLE}, +\[ +H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N +{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha +w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha +w_\alpha ^2 }}x} \right)^2 } \right\}} +\] +where +\[ +W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 +}}{{2m_\alpha w_\alpha ^2 }}} x^2 +\] +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}, +\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} + +\subsection{\label{introSection:laplaceTransform}The Laplace Transform} + +\[ +L(x) = \int_0^\infty {x(t)e^{ - pt} dt} +\] + +\[ +L(x + y) = L(x) + L(y) +\] + +\[ +L(ax) = aL(x) +\] + +\[ +L(\dot x) = pL(x) - px(0) +\] + +\[ +L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) +\] + +\[ +L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) +\] + +Some relatively important transformation, +\[ +L(\cos at) = \frac{p}{{p^2 + a^2 }} +\] + +\[ +L(\sin at) = \frac{a}{{p^2 + a^2 }} +\] + +\[ +L(1) = \frac{1}{p} +\] + +First, the bath coordinates, +\[ +p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega +_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha +}}L(x) +\] +\[ +L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + +px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} +\] +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, + +\begin{align} +m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - +\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 +}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega +_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) +- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos +(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega +_\alpha }}\sin (\omega _\alpha t)} } \right\}} +% +&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t +{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 +}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha +t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ +{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha +\omega _\alpha }}} \right]\cos (\omega _\alpha t) + +\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin +(\omega _\alpha t)} \right\}} +\end{align} + +\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. $W(x) = - kT\ln p(x)$ +\[ +\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 +}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} +\] +For an infinite harmonic bath, we can use the spectral density and +an integral over frequencies. + +\[ +R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) +- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} +\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha +(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) +\] +The random forces depend only on initial conditions. + +\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} +So we can define a new set of coordinates, +\[ +q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha +^2 }}x(0) +\] +This makes +\[ +R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} +\] +And since the $q$ coordinates are harmonic oscillators, +\[ +\begin{array}{l} + \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ + \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ + \end{array} +\] + +\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} + +\begin{equation} +\xi (t) = \left\langle {R(t)R(0)} \right\rangle +\label{introEquation:secondFluctuationDissipation} +\end{equation} + +\section{\label{introSection:hydroynamics}Hydrodynamics} + +\subsection{\label{introSection:frictionTensor} Friction Tensor} +\subsection{\label{introSection:analyticalApproach}Analytical +Approach} + +\subsection{\label{introSection:approximationApproach}Approximation +Approach} + +\subsection{\label{introSection:centersRigidBody}Centers of Rigid +Body} + +\section{\label{introSection:correlationFunctions}Correlation Functions}