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\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
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|
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\section{\label{introSection:classicalMechanics}Classical |
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Mechanics} |
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|
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Closely related to Classical Mechanics, Molecular Dynamics |
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simulations are carried out by integrating the equations of motion |
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for a given system of particles. There are three fundamental ideas |
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behind classical mechanics. Firstly, One can determine the state of |
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a mechanical system at any time of interest; Secondly, all the |
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mechanical properties of the system at that time can be determined |
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by combining the knowledge of the properties of the system with the |
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specification of this state; Finally, the specification of the state |
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when further combine with the laws of mechanics will also be |
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sufficient to predict the future behavior of the system. |
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|
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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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|
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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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|
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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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|
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Newtonian Mechanics suffers from two important limitations: it |
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describes their motion in special cartesian coordinate systems. |
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Another limitation of Newtonian mechanics becomes obvious when we |
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try to describe systems with large numbers of particles. It becomes |
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very difficult to predict the properties of the system by carrying |
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out calculations involving the each individual interaction between |
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all the particles, even if we know all of the details of the |
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interaction. In order to overcome some of the practical difficulties |
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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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|
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\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
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Principle} |
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|
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Hamilton introduced the dynamical principle upon which it is |
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possible to base all of mechanics and, indeed, most of classical |
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physics. Hamilton's Principle may be stated as follow, |
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|
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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, $U$ \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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\end{equation} |
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|
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For simple mechanical systems, where the forces acting on the |
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different part are derivable from a potential and the velocities are |
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small compared with that of light, the Lagrangian function $L$ can |
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be define as the difference between the kinetic energy of the system |
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and its potential energy, |
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\begin{equation} |
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L \equiv K - U = L(q_i ,\dot q_i ) , |
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\label{introEquation:lagrangianDef} |
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\end{equation} |
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then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
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\begin{equation} |
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\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
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\label{introEquation:halmitonianPrinciple2} |
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\end{equation} |
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|
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\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
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Equations of Motion in Lagrangian Mechanics} |
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|
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for a holonomic system of $f$ degrees of freedom, the equations of |
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motion in the Lagrangian form is |
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\begin{equation} |
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\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - |
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\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f |
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\label{introEquation:eqMotionLagrangian} |
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\end{equation} |
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where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is |
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generalized velocity. |
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|
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\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} |
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|
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Arising from Lagrangian Mechanics, Hamiltonian Mechanics was |
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introduced by William Rowan Hamilton in 1833 as a re-formulation of |
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classical mechanics. If the potential energy of a system is |
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independent of generalized velocities, the generalized momenta can |
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be defined as |
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\begin{equation} |
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p_i = \frac{\partial L}{\partial \dot q_i} |
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\label{introEquation:generalizedMomenta} |
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\end{equation} |
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The Lagrange equations of motion are then expressed by |
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\begin{equation} |
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p_i = \frac{{\partial L}}{{\partial q_i }} |
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\label{introEquation:generalizedMomentaDot} |
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\end{equation} |
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|
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With the help of the generalized momenta, we may now define a new |
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quantity $H$ by the equation |
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\begin{equation} |
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H = \sum\limits_k {p_k \dot q_k } - L , |
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\label{introEquation:hamiltonianDefByLagrangian} |
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\end{equation} |
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where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
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$L$ is the Lagrangian function for the system. |
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|
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Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
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one can obtain |
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\begin{equation} |
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dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
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\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
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L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
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L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
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\end{equation} |
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Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
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second and fourth terms in the parentheses cancel. Therefore, |
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Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
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\begin{equation} |
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dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
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\right)} - \frac{{\partial L}}{{\partial t}}dt |
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\label{introEquation:diffHamiltonian2} |
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\end{equation} |
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By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
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find |
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\begin{equation} |
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\frac{{\partial H}}{{\partial p_k }} = q_k |
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\label{introEquation:motionHamiltonianCoordinate} |
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\end{equation} |
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\begin{equation} |
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\frac{{\partial H}}{{\partial q_k }} = - p_k |
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\label{introEquation:motionHamiltonianMomentum} |
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\end{equation} |
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and |
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\begin{equation} |
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\frac{{\partial H}}{{\partial t}} = - \frac{{\partial L}}{{\partial |
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t}} |
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\label{introEquation:motionHamiltonianTime} |
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\end{equation} |
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|
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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 \cite{Goldstein01}. |
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|
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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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function of the generalized velocities $\dot q_i$ and the |
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generalized coordinates $q_i$, while the Hamiltonian is considered |
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to be a function of the generalized momenta $p_i$ and the conjugate |
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generalized coordinate $q_i$. Hamiltonian Mechanics is more |
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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\cite{Marion90}. |
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|
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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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where $I$ is a $n \times n$ identity matrix and $J$ is a |
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skew-symmetric matrix ($ J^T = - J $). Thus, Hamiltonian system |
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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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|
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%\subsection{\label{introSection:canonicalTransformation}Canonical |
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%Transformation} |
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|
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\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
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|
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\subsection{\label{introSection:symplecticMaps}Symplectic Maps and Methods} |
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|
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\subsection{\label{Construction of Symplectic Methods}} |
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|
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\section{\label{introSection:statisticalMechanics}Statistical |
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Mechanics} |
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|
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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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|
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\subsection{\label{introSection::ensemble}Ensemble and Phase Space} |
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|
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\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
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|
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Various thermodynamic properties can be calculated from Molecular |
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Dynamics simulation. By comparing experimental values with the |
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calculated properties, one can determine the accuracy of the |
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simulation and the quality of the underlying model. However, both of |
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experiment and computer simulation are usually performed during a |
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certain time interval and the measurements are averaged over a |
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period of them which is different from the average behavior of |
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many-body system in Statistical Mechanics. Fortunately, Ergodic |
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Hypothesis is proposed to make a connection between time average and |
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ensemble average. It states that time average and average over the |
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statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
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\begin{equation} |
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\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
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\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma |
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{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq |
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\end{equation} |
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where $\langle A \rangle_t$ is an equilibrium value of a physical |
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quantity and $\rho (p(t), q(t))$ is the equilibrium distribution |
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function. If an observation is averaged over a sufficiently long |
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time (longer than relaxation time), all accessible microstates in |
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phase space are assumed to be equally probed, giving a properly |
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weighted statistical average. This allows the researcher freedom of |
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choice when deciding how best to measure a given observable. In case |
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an ensemble averaged approach sounds most reasonable, the Monte |
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Carlo techniques\cite{metropolis:1949} can be utilized. Or if the |
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system lends itself to a time averaging approach, the Molecular |
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Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
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will be the best choice. |
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|
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\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
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|
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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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|
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\subsection{\label{introSec:mdInit}Initialization} |
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|
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\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
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|
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\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
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|
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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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|
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Applications of dynamics of rigid bodies. |
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|
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\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
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|
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\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
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|
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\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
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|
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%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
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|
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\section{\label{introSection:correlationFunctions}Correlation Functions} |
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|
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\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
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|
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\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
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|
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\subsection{\label{introSection:hydroynamics}Hydrodynamics} |
