--- trunk/tengDissertation/Introduction.tex 2006/04/10 05:35:55 2699 +++ trunk/tengDissertation/Introduction.tex 2006/04/13 04:47:47 2706 @@ -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 }} - @@ -221,10 +221,243 @@ Statistical Mechanics concepts presented in this disse 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 @@ -239,22 +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 @@ -352,7 +586,8 @@ H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \f }}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) \end{equation} -\subsection{\label{introSection:geometricProperties}Geometric Properties} +\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} @@ -362,20 +597,44 @@ space to itself. In most cases, it is not easy to find x(t+\tau) =\varphi_\tau(x(t)) \] where $\tau$ is a fixed time step and $\varphi$ is a map from phase -space to itself. 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 $p$, +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. Let $\varphi$ be the flow of Hamiltonian -vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies, +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. +{\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 @@ -383,22 +642,62 @@ symplectomorphism. As to the Poisson system, 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 +{\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$. Changing the variables $y = h(x)$ in a -ODE\ref{introEquation:ODE} will result in a new system, +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 $. When -designing any numerical methods, one should always try to preserve -the structural properties of the original ODE and its flow. +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 @@ -414,36 +713,188 @@ and difficult to use\cite{}. In dissipative systems, v \end{enumerate} Generating function tends to lead to methods which are cumbersome -and difficult to use\cite{}. In dissipative systems, variational -methods can capture the decay of energy accurately\cite{}. 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 \cite{}. 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}. The main idea behind splitting -methods is to decompose the discrete $\varphi_h$ as a composition of -simpler flows, +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. Let $\phi$ and $\psi$ both be -symplectic maps, it is easy to show that any composition of -symplectic flows yields a symplectic map, +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. +'\phi ' = \phi '^T J\phi ' = J, \label{introEquation:SymplecticFlowComposition} \end{equation} -Suppose that a Hamiltonian system has a form with $H = T + V$ +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} @@ -454,27 +905,103 @@ 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} -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. +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}}. -Applications of dynamics of rigid bodies. +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:DLMMotionEquation}The Euler Equations of Rigid Body Motion} +\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} -\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} +\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} -%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} +Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, +\ref{introEquation:motionHamiltonianMomentum}), one can write down +the equations of motion, +\[ +\begin{array}{c} + \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ + \frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ + \frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ + \frac{{dP}}{{dt}} = - \nabla _q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ + \end{array} +\] -\section{\label{introSection:correlationFunctions}Correlation Functions} +\[ +M = \left\{ {(Q,P):Q^T Q = 1,Q^t PJ^{ - 1} + J^{ - 1} P^t Q = 0} +\right\} . +\] + +\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} @@ -523,7 +1050,7 @@ introEquation:motionHamiltonianMomentum}, \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} +\label{introEquation:Lp5} \end{align} , and \begin{align} @@ -682,3 +1209,5 @@ Body} \subsection{\label{introSection:centersRigidBody}Centers of Rigid Body} + +\section{\label{introSection:correlationFunctions}Correlation Functions}