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\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
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\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
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As a special discipline of molecular modeling, Molecular dynamics |
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has proven to be a powerful tool for studying the functions of |
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biological systems, providing structural, thermodynamic and |
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dynamical information. |
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\section{\label{introSection:classicalMechanics}Classical |
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Mechanics} |
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sufficient to predict the future behavior of the system. |
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\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
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The discovery of Newton's three laws of mechanics which govern the |
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motion of particles is the foundation of the classical mechanics. |
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Newton¡¯s first law defines a class of inertial frames. Inertial |
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frames are reference frames where a particle not interacting with |
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other bodies will move with constant speed in the same direction. |
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With respect to inertial frames Newton¡¯s second law has the form |
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\begin{equation} |
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F = \frac {dp}{dt} = \frac {mv}{dt} |
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\label{introEquation:newtonSecondLaw} |
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\end{equation} |
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A point mass interacting with other bodies moves with the |
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acceleration along the direction of the force acting on it. Let |
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$F_ij$ be the force that particle $i$ exerts on particle $j$, and |
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$F_ji$ be the force that particle $j$ exerts on particle $i$. |
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Newton¡¯s third law states that |
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\begin{equation} |
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F_ij = -F_ji |
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\label{introEquation:newtonThirdLaw} |
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\end{equation} |
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Conservation laws of Newtonian Mechanics play very important roles |
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in solving mechanics problems. The linear momentum of a particle is |
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conserved if it is free or it experiences no force. The second |
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conservation theorem concerns the angular momentum of a particle. |
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The angular momentum $L$ of a particle with respect to an origin |
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from which $r$ is measured is defined to be |
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\begin{equation} |
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L \equiv r \times p \label{introEquation:angularMomentumDefinition} |
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\end{equation} |
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The torque $\tau$ with respect to the same origin is defined to be |
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\begin{equation} |
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N \equiv r \times F \label{introEquation:torqueDefinition} |
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\end{equation} |
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Differentiating Eq.~\ref{introEquation:angularMomentumDefinition}, |
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\[ |
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\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times |
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\dot p) |
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\] |
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since |
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\[ |
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\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0 |
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\] |
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thus, |
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\begin{equation} |
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\dot L = r \times \dot p = N |
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\end{equation} |
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If there are no external torques acting on a body, the angular |
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momentum of it is conserved. The last conservation theorem state |
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that if all forces are conservative, Energy $E = T + V$ is |
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conserved. All of these conserved quantities are important factors |
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to determine the quality of numerical integration scheme for rigid |
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body \cite{Dullweber1997}. |
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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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Newtonian Mechanics suffers from two important limitations: it |
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which arise in attempts to apply Newton's equation to complex |
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system, alternative procedures may be developed. |
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\subsection{\label{introSection:halmiltonPrinciple}Hamilton's |
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\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
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Principle} |
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Hamilton introduced the dynamical principle upon which it is |
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The actual trajectory, along which a dynamical system may move from |
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one point to another within a specified time, is derived by finding |
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the path which minimizes the time integral of the difference between |
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the kinetic, $K$, and potential energies, $U$. |
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the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
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\begin{equation} |
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\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
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\label{introEquation:halmitonianPrinciple1} |
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\label{introEquation:halmitonianPrinciple2} |
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\end{equation} |
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\subsection{\label{introSection:equationOfMotionLagrangian}The |
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\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
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Equations of Motion in Lagrangian Mechanics} |
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for a holonomic system of $f$ degrees of freedom, the equations of |
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Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
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Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
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equation of motion. Due to their symmetrical formula, they are also |
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known as the canonical equations of motions. |
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known as the canonical equations of motions \cite{Goldstein01}. |
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An important difference between Lagrangian approach and the |
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Hamiltonian approach is that the Lagrangian is considered to be a |
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appropriate for application to statistical mechanics and quantum |
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mechanics, since it treats the coordinate and its time derivative as |
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independent variables and it only works with 1st-order differential |
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equations. |
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equations\cite{Marion90}. |
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\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
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When studying Hamiltonian system, it is more convenient to use |
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notation |
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\begin{equation} |
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r = r(q,p)^T |
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\end{equation} |
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and to introduce a $2n \times 2n$ canonical structure matrix $J$, |
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\begin{equation} |
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J = \left( {\begin{array}{*{20}c} |
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0 & I \\ |
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{ - I} & 0 \\ |
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\end{array}} \right) |
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\label{introEquation:canonicalMatrix} |
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\end{equation} |
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Thus, Hamiltonian system can be rewritten as, |
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\begin{equation} |
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\frac{d}{{dt}}r = J\nabla _r H(r) |
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\label{introEquation:compactHamiltonian} |
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\end{equation} |
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where $I$ is an identity matrix and $J$ is a skew-symmetrix matrix |
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($ J^T = - J $). |
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\subsection{\label{introSection:canonicalTransformation}Canonical |
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%\subsection{\label{introSection:canonicalTransformation}Canonical |
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Transformation} |
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\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
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\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods} |
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\subsection{\label{Construction of Symplectic Methods}} |
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\section{\label{introSection:statisticalMechanics}Statistical |
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Mechanics} |
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The thermodynamic behaviors and properties of Molecular Dynamics |
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The thermodynamic behaviors and properties of Molecular Dynamics |
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simulation are governed by the principle of Statistical Mechanics. |
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The following section will give a brief introduction to some of the |
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Statistical Mechanics concepts presented in this dissertation. |
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\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
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\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
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As a special discipline of molecular modeling, Molecular dynamics |
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has proven to be a powerful tool for studying the functions of |
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biological systems, providing structural, thermodynamic and |
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dynamical information. |
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\subsection{\label{introSec:mdInit}Initialization} |
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\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
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\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
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A rigid body is a body in which the distance between any two given |
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points of a rigid body remains constant regardless of external |
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forces exerted on it. A rigid body therefore conserves its shape |
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during its motion. |
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Applications of dynamics of rigid bodies. |
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%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
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\section{\label{introSection:correlationFunctions}Correlation Functions} |
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\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
