--- trunk/tengDissertation/Introduction.tex 2006/04/06 04:07:57 2695 +++ trunk/tengDissertation/Introduction.tex 2006/04/07 22:05:48 2698 @@ -63,10 +63,12 @@ that if all forces are conservative, Energy $E = T + V \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 $E = T + V$ is -conserved. All of these conserved quantities are important factors -to determine the quality of numerical integration scheme for rigid -body \cite{Dullweber1997}. +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} @@ -200,36 +202,19 @@ When studying Hamiltonian system, it is more convenien independent variables and it only works with 1st-order differential equations\cite{Marion90}. -When studying Hamiltonian system, it is more convenient to use -notation +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} -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} +\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} -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} -%\subsection{\label{introSection:canonicalTransformation}Canonical -%Transformation} - -\section{\label{introSection:geometricIntegratos}Geometric Integrators} - -\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods} - -\subsection{\label{Construction of Symplectic Methods}} - \section{\label{introSection:statisticalMechanics}Statistical Mechanics} @@ -238,7 +223,7 @@ Statistical Mechanics concepts presented in this disse The following section will give a brief introduction to some of the Statistical Mechanics concepts presented in this dissertation. -\subsection{\label{introSection::ensemble}Ensemble and Phase Space} +\subsection{\label{introSection:ensemble}Ensemble and Phase Space} \subsection{\label{introSection:ergodic}The Ergodic Hypothesis} @@ -269,8 +254,153 @@ will be the best choice. 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. +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:geometricProperties}Geometric Properties} +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. 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} + +The hidden geometric properties of ODE and its flow play important +roles in numerical studies. The flow of a Hamiltonian vector field +on a symplectic manifold is a symplectomorphism. Let $\varphi$ be +the flow of Hamiltonian vector field, $\varphi$ is a +\emph{symplectic} flow if it satisfies, +\begin{equation} +d \varphi^T J d \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. As to the Poisson system, +\begin{equation} +d\varphi ^T Jd\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, +\[ +\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). +\] +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. + +\subsection{\label{introSection:splittingAndComposition}Splitting and Composition Methods} + \section{\label{introSection:molecularDynamics}Molecular Dynamics} As a special discipline of molecular modeling, Molecular dynamics @@ -303,6 +433,208 @@ Applications of dynamics of rigid bodies. \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{introEq: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}