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\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
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|
3 |
\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 |
7 |
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 |
24 |
\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 |
33 |
\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 |
44 |
\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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\[ |
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\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times |
54 |
\dot p) |
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\] |
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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, |
61 |
\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 |
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\begin{equation}E = T + V \label{introEquation:energyConservation} |
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\end{equation} |
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is conserved. All of these conserved quantities are |
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important factors to determine the quality of numerical integration |
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scheme for rigid 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 |
87 |
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 }} - |
124 |
\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} |
127 |
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 |
142 |
\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 |
180 |
\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 |
185 |
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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In Newtonian Mechanics, a system described by conservative forces |
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conserves the total energy \ref{introEquation:energyConservation}. |
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It follows that Hamilton's equations of motion conserve the total |
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Hamiltonian. |
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\begin{equation} |
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\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
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H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
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}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
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H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
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\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
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q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
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\end{equation} |
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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 and theorem presented in this |
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dissertation. |
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|
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\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
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|
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Mathematically, phase space is the space which represents all |
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possible states. Each possible state of the system corresponds to |
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one unique point in the phase space. For mechanical systems, the |
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phase space usually consists of all possible values of position and |
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momentum variables. Consider a dynamic system in a cartesian space, |
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where each of the $6f$ coordinates and momenta is assigned to one of |
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$6f$ mutually orthogonal axes, the phase space of this system is a |
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$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , |
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\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and |
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momenta is a phase space vector. |
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|
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A microscopic state or microstate of a classical system is |
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specification of the complete phase space vector of a system at any |
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instant in time. An ensemble is defined as a collection of systems |
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sharing one or more macroscopic characteristics but each being in a |
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unique microstate. The complete ensemble is specified by giving all |
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systems or microstates consistent with the common macroscopic |
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characteristics of the ensemble. Although the state of each |
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individual system in the ensemble could be precisely described at |
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any instance in time by a suitable phase space vector, when using |
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ensembles for statistical purposes, there is no need to maintain |
250 |
distinctions between individual systems, since the numbers of |
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systems at any time in the different states which correspond to |
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different regions of the phase space are more interesting. Moreover, |
253 |
in the point of view of statistical mechanics, one would prefer to |
254 |
use ensembles containing a large enough population of separate |
255 |
members so that the numbers of systems in such different states can |
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be regarded as changing continuously as we traverse different |
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regions of the phase space. The condition of an ensemble at any time |
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can be regarded as appropriately specified by the density $\rho$ |
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with which representative points are distributed over the phase |
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space. The density of distribution for an ensemble with $f$ degrees |
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of freedom is defined as, |
262 |
\begin{equation} |
263 |
\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). |
264 |
\label{introEquation:densityDistribution} |
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\end{equation} |
266 |
Governed by the principles of mechanics, the phase points change |
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their value which would change the density at any time at phase |
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space. Hence, the density of distribution is also to be taken as a |
269 |
function of the time. |
270 |
|
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The number of systems $\delta N$ at time $t$ can be determined by, |
272 |
\begin{equation} |
273 |
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
274 |
\label{introEquation:deltaN} |
275 |
\end{equation} |
276 |
Assuming a large enough population of systems are exploited, we can |
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sufficiently approximate $\delta N$ without introducing |
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discontinuity when we go from one region in the phase space to |
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another. By integrating over the whole phase space, |
280 |
\begin{equation} |
281 |
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
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\label{introEquation:totalNumberSystem} |
283 |
\end{equation} |
284 |
gives us an expression for the total number of the systems. Hence, |
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the probability per unit in the phase space can be obtained by, |
286 |
\begin{equation} |
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\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int |
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{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
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\label{introEquation:unitProbability} |
290 |
\end{equation} |
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With the help of Equation(\ref{introEquation:unitProbability}) and |
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the knowledge of the system, it is possible to calculate the average |
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value of any desired quantity which depends on the coordinates and |
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momenta of the system. Even when the dynamics of the real system is |
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complex, or stochastic, or even discontinuous, the average |
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properties of the ensemble of possibilities as a whole may still |
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remain well defined. For a classical system in thermal equilibrium |
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with its environment, the ensemble average of a mechanical quantity, |
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$\langle A(q , p) \rangle_t$, takes the form of an integral over the |
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phase space of the system, |
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\begin{equation} |
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\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
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(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
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(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
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\label{introEquation:ensembelAverage} |
306 |
\end{equation} |
307 |
|
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There are several different types of ensembles with different |
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statistical characteristics. As a function of macroscopic |
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parameters, such as temperature \textit{etc}, partition function can |
311 |
be used to describe the statistical properties of a system in |
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thermodynamic equilibrium. |
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|
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As an ensemble of systems, each of which is known to be thermally |
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isolated and conserve energy, Microcanonical ensemble(NVE) has a |
316 |
partition function like, |
317 |
\begin{equation} |
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\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
319 |
\end{equation} |
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A canonical ensemble(NVT)is an ensemble of systems, each of which |
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can share its energy with a large heat reservoir. The distribution |
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of the total energy amongst the possible dynamical states is given |
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by the partition function, |
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\begin{equation} |
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\Omega (N,V,T) = e^{ - \beta A} |
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\label{introEquation:NVTPartition} |
327 |
\end{equation} |
328 |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
329 |
TS$. Since most experiment are carried out under constant pressure |
330 |
condition, isothermal-isobaric ensemble(NPT) play a very important |
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role in molecular simulation. The isothermal-isobaric ensemble allow |
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the system to exchange energy with a heat bath of temperature $T$ |
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and to change the volume as well. Its partition function is given as |
334 |
\begin{equation} |
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\Delta (N,P,T) = - e^{\beta G}. |
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\label{introEquation:NPTPartition} |
337 |
\end{equation} |
338 |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
339 |
|
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\subsection{\label{introSection:liouville}Liouville's theorem} |
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|
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The Liouville's theorem is the foundation on which statistical |
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mechanics rests. It describes the time evolution of phase space |
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distribution function. In order to calculate the rate of change of |
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$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
346 |
consider the two faces perpendicular to the $q_1$ axis, which are |
347 |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
348 |
leaving the opposite face is given by the expression, |
349 |
\begin{equation} |
350 |
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
351 |
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
352 |
}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 |
353 |
\ldots \delta p_f . |
354 |
\end{equation} |
355 |
Summing all over the phase space, we obtain |
356 |
\begin{equation} |
357 |
\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho |
358 |
\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + |
359 |
\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( |
360 |
{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial |
361 |
\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 |
362 |
\ldots \delta q_f \delta p_1 \ldots \delta p_f . |
363 |
\end{equation} |
364 |
Differentiating the equations of motion in Hamiltonian formalism |
365 |
(\ref{introEquation:motionHamiltonianCoordinate}, |
366 |
\ref{introEquation:motionHamiltonianMomentum}), we can show, |
367 |
\begin{equation} |
368 |
\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} |
369 |
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
370 |
\end{equation} |
371 |
which cancels the first terms of the right hand side. Furthermore, |
372 |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
373 |
p_f $ in both sides, we can write out Liouville's theorem in a |
374 |
simple form, |
375 |
\begin{equation} |
376 |
\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f |
377 |
{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + |
378 |
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
379 |
\label{introEquation:liouvilleTheorem} |
380 |
\end{equation} |
381 |
|
382 |
Liouville's theorem states that the distribution function is |
383 |
constant along any trajectory in phase space. In classical |
384 |
statistical mechanics, since the number of particles in the system |
385 |
is huge, we may be able to believe the system is stationary, |
386 |
\begin{equation} |
387 |
\frac{{\partial \rho }}{{\partial t}} = 0. |
388 |
\label{introEquation:stationary} |
389 |
\end{equation} |
390 |
In such stationary system, the density of distribution $\rho$ can be |
391 |
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
392 |
distribution, |
393 |
\begin{equation} |
394 |
\rho \propto e^{ - \beta H} |
395 |
\label{introEquation:densityAndHamiltonian} |
396 |
\end{equation} |
397 |
|
398 |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
399 |
Lets consider a region in the phase space, |
400 |
\begin{equation} |
401 |
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
402 |
\end{equation} |
403 |
If this region is small enough, the density $\rho$ can be regarded |
404 |
as uniform over the whole phase space. Thus, the number of phase |
405 |
points inside this region is given by, |
406 |
\begin{equation} |
407 |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
408 |
dp_1 } ..dp_f. |
409 |
\end{equation} |
410 |
|
411 |
\begin{equation} |
412 |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
413 |
\frac{d}{{dt}}(\delta v) = 0. |
414 |
\end{equation} |
415 |
With the help of stationary assumption |
416 |
(\ref{introEquation:stationary}), we obtain the principle of the |
417 |
\emph{conservation of extension in phase space}, |
418 |
\begin{equation} |
419 |
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
420 |
...dq_f dp_1 } ..dp_f = 0. |
421 |
\label{introEquation:volumePreserving} |
422 |
\end{equation} |
423 |
|
424 |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
425 |
|
426 |
Liouville's theorem can be expresses in a variety of different forms |
427 |
which are convenient within different contexts. For any two function |
428 |
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
429 |
bracket ${F, G}$ is defined as |
430 |
\begin{equation} |
431 |
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
432 |
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
433 |
\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial |
434 |
q_i }}} \right)}. |
435 |
\label{introEquation:poissonBracket} |
436 |
\end{equation} |
437 |
Substituting equations of motion in Hamiltonian formalism( |
438 |
\ref{introEquation:motionHamiltonianCoordinate} , |
439 |
\ref{introEquation:motionHamiltonianMomentum} ) into |
440 |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
441 |
theorem using Poisson bracket notion, |
442 |
\begin{equation} |
443 |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
444 |
{\rho ,H} \right\}. |
445 |
\label{introEquation:liouvilleTheromInPoissin} |
446 |
\end{equation} |
447 |
Moreover, the Liouville operator is defined as |
448 |
\begin{equation} |
449 |
iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial |
450 |
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial |
451 |
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} |
452 |
\label{introEquation:liouvilleOperator} |
453 |
\end{equation} |
454 |
In terms of Liouville operator, Liouville's equation can also be |
455 |
expressed as |
456 |
\begin{equation} |
457 |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
458 |
\label{introEquation:liouvilleTheoremInOperator} |
459 |
\end{equation} |
460 |
|
461 |
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
462 |
|
463 |
Various thermodynamic properties can be calculated from Molecular |
464 |
Dynamics simulation. By comparing experimental values with the |
465 |
calculated properties, one can determine the accuracy of the |
466 |
simulation and the quality of the underlying model. However, both of |
467 |
experiment and computer simulation are usually performed during a |
468 |
certain time interval and the measurements are averaged over a |
469 |
period of them which is different from the average behavior of |
470 |
many-body system in Statistical Mechanics. Fortunately, Ergodic |
471 |
Hypothesis is proposed to make a connection between time average and |
472 |
ensemble average. It states that time average and average over the |
473 |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
474 |
\begin{equation} |
475 |
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
476 |
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
477 |
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp |
478 |
\end{equation} |
479 |
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
480 |
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
481 |
distribution function. If an observation is averaged over a |
482 |
sufficiently long time (longer than relaxation time), all accessible |
483 |
microstates in phase space are assumed to be equally probed, giving |
484 |
a properly weighted statistical average. This allows the researcher |
485 |
freedom of choice when deciding how best to measure a given |
486 |
observable. In case an ensemble averaged approach sounds most |
487 |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
488 |
utilized. Or if the system lends itself to a time averaging |
489 |
approach, the Molecular Dynamics techniques in |
490 |
Sec.~\ref{introSection:molecularDynamics} will be the best |
491 |
choice\cite{Frenkel1996}. |
492 |
|
493 |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
494 |
A variety of numerical integrators were proposed to simulate the |
495 |
motions. They usually begin with an initial conditionals and move |
496 |
the objects in the direction governed by the differential equations. |
497 |
However, most of them ignore the hidden physical law contained |
498 |
within the equations. Since 1990, geometric integrators, which |
499 |
preserve various phase-flow invariants such as symplectic structure, |
500 |
volume and time reversal symmetry, are developed to address this |
501 |
issue. The velocity verlet method, which happens to be a simple |
502 |
example of symplectic integrator, continues to gain its popularity |
503 |
in molecular dynamics community. This fact can be partly explained |
504 |
by its geometric nature. |
505 |
|
506 |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
507 |
A \emph{manifold} is an abstract mathematical space. It locally |
508 |
looks like Euclidean space, but when viewed globally, it may have |
509 |
more complicate structure. A good example of manifold is the surface |
510 |
of Earth. It seems to be flat locally, but it is round if viewed as |
511 |
a whole. A \emph{differentiable manifold} (also known as |
512 |
\emph{smooth manifold}) is a manifold with an open cover in which |
513 |
the covering neighborhoods are all smoothly isomorphic to one |
514 |
another. In other words,it is possible to apply calculus on |
515 |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
516 |
defined as a pair $(M, \omega)$ which consisting of a |
517 |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
518 |
bilinear symplectic form, $\omega$. A symplectic form on a vector |
519 |
space $V$ is a function $\omega(x, y)$ which satisfies |
520 |
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
521 |
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
522 |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
523 |
example of symplectic form. |
524 |
|
525 |
One of the motivations to study \emph{symplectic manifold} in |
526 |
Hamiltonian Mechanics is that a symplectic manifold can represent |
527 |
all possible configurations of the system and the phase space of the |
528 |
system can be described by it's cotangent bundle. Every symplectic |
529 |
manifold is even dimensional. For instance, in Hamilton equations, |
530 |
coordinate and momentum always appear in pairs. |
531 |
|
532 |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
533 |
\[ |
534 |
f : M \rightarrow N |
535 |
\] |
536 |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
537 |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
538 |
Canonical transformation is an example of symplectomorphism in |
539 |
classical mechanics. |
540 |
|
541 |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
542 |
|
543 |
For a ordinary differential system defined as |
544 |
\begin{equation} |
545 |
\dot x = f(x) |
546 |
\end{equation} |
547 |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
548 |
\begin{equation} |
549 |
f(r) = J\nabla _x H(r). |
550 |
\end{equation} |
551 |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
552 |
matrix |
553 |
\begin{equation} |
554 |
J = \left( {\begin{array}{*{20}c} |
555 |
0 & I \\ |
556 |
{ - I} & 0 \\ |
557 |
\end{array}} \right) |
558 |
\label{introEquation:canonicalMatrix} |
559 |
\end{equation} |
560 |
where $I$ is an identity matrix. Using this notation, Hamiltonian |
561 |
system can be rewritten as, |
562 |
\begin{equation} |
563 |
\frac{d}{{dt}}x = J\nabla _x H(x) |
564 |
\label{introEquation:compactHamiltonian} |
565 |
\end{equation}In this case, $f$ is |
566 |
called a \emph{Hamiltonian vector field}. |
567 |
|
568 |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
569 |
\begin{equation} |
570 |
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
571 |
\end{equation} |
572 |
The most obvious change being that matrix $J$ now depends on $x$. |
573 |
The free rigid body is an example of Poisson system (actually a |
574 |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
575 |
energy. |
576 |
\begin{equation} |
577 |
J(\pi ) = \left( {\begin{array}{*{20}c} |
578 |
0 & {\pi _3 } & { - \pi _2 } \\ |
579 |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
580 |
{\pi _2 } & { - \pi _1 } & 0 \\ |
581 |
\end{array}} \right) |
582 |
\end{equation} |
583 |
|
584 |
\begin{equation} |
585 |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
586 |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
587 |
\end{equation} |
588 |
|
589 |
\subsection{\label{introSection:exactFlow}Exact Flow} |
590 |
|
591 |
Let $x(t)$ be the exact solution of the ODE system, |
592 |
\begin{equation} |
593 |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
594 |
\end{equation} |
595 |
The exact flow(solution) $\varphi_\tau$ is defined by |
596 |
\[ |
597 |
x(t+\tau) =\varphi_\tau(x(t)) |
598 |
\] |
599 |
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
600 |
space to itself. The flow has the continuous group property, |
601 |
\begin{equation} |
602 |
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
603 |
+ \tau _2 } . |
604 |
\end{equation} |
605 |
In particular, |
606 |
\begin{equation} |
607 |
\varphi _\tau \circ \varphi _{ - \tau } = I |
608 |
\end{equation} |
609 |
Therefore, the exact flow is self-adjoint, |
610 |
\begin{equation} |
611 |
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
612 |
\end{equation} |
613 |
The exact flow can also be written in terms of the of an operator, |
614 |
\begin{equation} |
615 |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
616 |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
617 |
\label{introEquation:exponentialOperator} |
618 |
\end{equation} |
619 |
|
620 |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
621 |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
622 |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
623 |
the Taylor series of $\psi_\tau$ agree to order $p$, |
624 |
\begin{equation} |
625 |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
626 |
\end{equation} |
627 |
|
628 |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
629 |
|
630 |
The hidden geometric properties of ODE and its flow play important |
631 |
roles in numerical studies. Many of them can be found in systems |
632 |
which occur naturally in applications. |
633 |
|
634 |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
635 |
a \emph{symplectic} flow if it satisfies, |
636 |
\begin{equation} |
637 |
{\varphi '}^T J \varphi ' = J. |
638 |
\end{equation} |
639 |
According to Liouville's theorem, the symplectic volume is invariant |
640 |
under a Hamiltonian flow, which is the basis for classical |
641 |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
642 |
field on a symplectic manifold can be shown to be a |
643 |
symplectomorphism. As to the Poisson system, |
644 |
\begin{equation} |
645 |
{\varphi '}^T J \varphi ' = J \circ \varphi |
646 |
\end{equation} |
647 |
is the property must be preserved by the integrator. |
648 |
|
649 |
It is possible to construct a \emph{volume-preserving} flow for a |
650 |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
651 |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
652 |
be volume-preserving. |
653 |
|
654 |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
655 |
will result in a new system, |
656 |
\[ |
657 |
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
658 |
\] |
659 |
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
660 |
In other words, the flow of this vector field is reversible if and |
661 |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
662 |
|
663 |
A \emph{first integral}, or conserved quantity of a general |
664 |
differential function is a function $ G:R^{2d} \to R^d $ which is |
665 |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
666 |
\[ |
667 |
\frac{{dG(x(t))}}{{dt}} = 0. |
668 |
\] |
669 |
Using chain rule, one may obtain, |
670 |
\[ |
671 |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
672 |
\] |
673 |
which is the condition for conserving \emph{first integral}. For a |
674 |
canonical Hamiltonian system, the time evolution of an arbitrary |
675 |
smooth function $G$ is given by, |
676 |
\begin{equation} |
677 |
\begin{array}{c} |
678 |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
679 |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
680 |
\end{array} |
681 |
\label{introEquation:firstIntegral1} |
682 |
\end{equation} |
683 |
Using poisson bracket notion, Equation |
684 |
\ref{introEquation:firstIntegral1} can be rewritten as |
685 |
\[ |
686 |
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
687 |
\] |
688 |
Therefore, the sufficient condition for $G$ to be the \emph{first |
689 |
integral} of a Hamiltonian system is |
690 |
\[ |
691 |
\left\{ {G,H} \right\} = 0. |
692 |
\] |
693 |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
694 |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
695 |
0$. |
696 |
|
697 |
|
698 |
When designing any numerical methods, one should always try to |
699 |
preserve the structural properties of the original ODE and its flow. |
700 |
|
701 |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
702 |
A lot of well established and very effective numerical methods have |
703 |
been successful precisely because of their symplecticities even |
704 |
though this fact was not recognized when they were first |
705 |
constructed. The most famous example is leapfrog methods in |
706 |
molecular dynamics. In general, symplectic integrators can be |
707 |
constructed using one of four different methods. |
708 |
\begin{enumerate} |
709 |
\item Generating functions |
710 |
\item Variational methods |
711 |
\item Runge-Kutta methods |
712 |
\item Splitting methods |
713 |
\end{enumerate} |
714 |
|
715 |
Generating function tends to lead to methods which are cumbersome |
716 |
and difficult to use. In dissipative systems, variational methods |
717 |
can capture the decay of energy accurately. Since their |
718 |
geometrically unstable nature against non-Hamiltonian perturbations, |
719 |
ordinary implicit Runge-Kutta methods are not suitable for |
720 |
Hamiltonian system. Recently, various high-order explicit |
721 |
Runge--Kutta methods have been developed to overcome this |
722 |
instability. However, due to computational penalty involved in |
723 |
implementing the Runge-Kutta methods, they do not attract too much |
724 |
attention from Molecular Dynamics community. Instead, splitting have |
725 |
been widely accepted since they exploit natural decompositions of |
726 |
the system\cite{Tuckerman92}. |
727 |
|
728 |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
729 |
|
730 |
The main idea behind splitting methods is to decompose the discrete |
731 |
$\varphi_h$ as a composition of simpler flows, |
732 |
\begin{equation} |
733 |
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
734 |
\varphi _{h_n } |
735 |
\label{introEquation:FlowDecomposition} |
736 |
\end{equation} |
737 |
where each of the sub-flow is chosen such that each represent a |
738 |
simpler integration of the system. |
739 |
|
740 |
Suppose that a Hamiltonian system takes the form, |
741 |
\[ |
742 |
H = H_1 + H_2. |
743 |
\] |
744 |
Here, $H_1$ and $H_2$ may represent different physical processes of |
745 |
the system. For instance, they may relate to kinetic and potential |
746 |
energy respectively, which is a natural decomposition of the |
747 |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
748 |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
749 |
order is then given by the Lie-Trotter formula |
750 |
\begin{equation} |
751 |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
752 |
\label{introEquation:firstOrderSplitting} |
753 |
\end{equation} |
754 |
where $\varphi _h$ is the result of applying the corresponding |
755 |
continuous $\varphi _i$ over a time $h$. By definition, as |
756 |
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
757 |
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
758 |
It is easy to show that any composition of symplectic flows yields a |
759 |
symplectic map, |
760 |
\begin{equation} |
761 |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
762 |
'\phi ' = \phi '^T J\phi ' = J, |
763 |
\label{introEquation:SymplecticFlowComposition} |
764 |
\end{equation} |
765 |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
766 |
splitting in this context automatically generates a symplectic map. |
767 |
|
768 |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
769 |
introduces local errors proportional to $h^2$, while Strang |
770 |
splitting gives a second-order decomposition, |
771 |
\begin{equation} |
772 |
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
773 |
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
774 |
\end{equation} |
775 |
which has a local error proportional to $h^3$. Sprang splitting's |
776 |
popularity in molecular simulation community attribute to its |
777 |
symmetric property, |
778 |
\begin{equation} |
779 |
\varphi _h^{ - 1} = \varphi _{ - h}. |
780 |
\label{introEquation:timeReversible} |
781 |
\end{equation} |
782 |
|
783 |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
784 |
The classical equation for a system consisting of interacting |
785 |
particles can be written in Hamiltonian form, |
786 |
\[ |
787 |
H = T + V |
788 |
\] |
789 |
where $T$ is the kinetic energy and $V$ is the potential energy. |
790 |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
791 |
obtains the following: |
792 |
\begin{align} |
793 |
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
794 |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
795 |
\label{introEquation:Lp10a} \\% |
796 |
% |
797 |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
798 |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
799 |
\label{introEquation:Lp10b} |
800 |
\end{align} |
801 |
where $F(t)$ is the force at time $t$. This integration scheme is |
802 |
known as \emph{velocity verlet} which is |
803 |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
804 |
time-reversible(\ref{introEquation:timeReversible}) and |
805 |
volume-preserving (\ref{introEquation:volumePreserving}). These |
806 |
geometric properties attribute to its long-time stability and its |
807 |
popularity in the community. However, the most commonly used |
808 |
velocity verlet integration scheme is written as below, |
809 |
\begin{align} |
810 |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
811 |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
812 |
% |
813 |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
814 |
\label{introEquation:Lp9b}\\% |
815 |
% |
816 |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
817 |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
818 |
\end{align} |
819 |
From the preceding splitting, one can see that the integration of |
820 |
the equations of motion would follow: |
821 |
\begin{enumerate} |
822 |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
823 |
|
824 |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
825 |
|
826 |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
827 |
|
828 |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
829 |
\end{enumerate} |
830 |
|
831 |
Simply switching the order of splitting and composing, a new |
832 |
integrator, the \emph{position verlet} integrator, can be generated, |
833 |
\begin{align} |
834 |
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
835 |
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
836 |
\label{introEquation:positionVerlet1} \\% |
837 |
% |
838 |
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
839 |
q(\Delta t)} \right]. % |
840 |
\label{introEquation:positionVerlet1} |
841 |
\end{align} |
842 |
|
843 |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
844 |
|
845 |
Baker-Campbell-Hausdorff formula can be used to determine the local |
846 |
error of splitting method in terms of commutator of the |
847 |
operators(\ref{introEquation:exponentialOperator}) associated with |
848 |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
849 |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
850 |
\begin{equation} |
851 |
\exp (hX + hY) = \exp (hZ) |
852 |
\end{equation} |
853 |
where |
854 |
\begin{equation} |
855 |
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
856 |
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
857 |
\end{equation} |
858 |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
859 |
\[ |
860 |
[X,Y] = XY - YX . |
861 |
\] |
862 |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
863 |
can obtain |
864 |
\begin{eqnarray*} |
865 |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
866 |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
867 |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
868 |
\ldots ) |
869 |
\end{eqnarray*} |
870 |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
871 |
error of Spring splitting is proportional to $h^3$. The same |
872 |
procedure can be applied to general splitting, of the form |
873 |
\begin{equation} |
874 |
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
875 |
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
876 |
\end{equation} |
877 |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
878 |
order method. Yoshida proposed an elegant way to compose higher |
879 |
order methods based on symmetric splitting. Given a symmetric second |
880 |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
881 |
method can be constructed by composing, |
882 |
\[ |
883 |
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
884 |
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
885 |
\] |
886 |
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
887 |
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
888 |
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
889 |
\begin{equation} |
890 |
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
891 |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
892 |
\end{equation} |
893 |
, if the weights are chosen as |
894 |
\[ |
895 |
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
896 |
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
897 |
\] |
898 |
|
899 |
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
900 |
|
901 |
As a special discipline of molecular modeling, Molecular dynamics |
902 |
has proven to be a powerful tool for studying the functions of |
903 |
biological systems, providing structural, thermodynamic and |
904 |
dynamical information. |
905 |
|
906 |
\subsection{\label{introSec:mdInit}Initialization} |
907 |
|
908 |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
909 |
|
910 |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
911 |
|
912 |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
913 |
|
914 |
Rigid bodies are frequently involved in the modeling of different |
915 |
areas, from engineering, physics, to chemistry. For example, |
916 |
missiles and vehicle are usually modeled by rigid bodies. The |
917 |
movement of the objects in 3D gaming engine or other physics |
918 |
simulator is governed by the rigid body dynamics. In molecular |
919 |
simulation, rigid body is used to simplify the model in |
920 |
protein-protein docking study{\cite{Gray03}}. |
921 |
|
922 |
It is very important to develop stable and efficient methods to |
923 |
integrate the equations of motion of orientational degrees of |
924 |
freedom. Euler angles are the nature choice to describe the |
925 |
rotational degrees of freedom. However, due to its singularity, the |
926 |
numerical integration of corresponding equations of motion is very |
927 |
inefficient and inaccurate. Although an alternative integrator using |
928 |
different sets of Euler angles can overcome this difficulty\cite{}, |
929 |
the computational penalty and the lost of angular momentum |
930 |
conservation still remain. A singularity free representation |
931 |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
932 |
this approach suffer from the nonseparable Hamiltonian resulted from |
933 |
quaternion representation, which prevents the symplectic algorithm |
934 |
to be utilized. Another different approach is to apply holonomic |
935 |
constraints to the atoms belonging to the rigid body. Each atom |
936 |
moves independently under the normal forces deriving from potential |
937 |
energy and constraint forces which are used to guarantee the |
938 |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
939 |
algorithm converge very slowly when the number of constraint |
940 |
increases. |
941 |
|
942 |
The break through in geometric literature suggests that, in order to |
943 |
develop a long-term integration scheme, one should preserve the |
944 |
symplectic structure of the flow. Introducing conjugate momentum to |
945 |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
946 |
symplectic integrator, RSHAKE, was proposed to evolve the |
947 |
Hamiltonian system in a constraint manifold by iteratively |
948 |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
949 |
method using quaternion representation was developed by Omelyan. |
950 |
However, both of these methods are iterative and inefficient. In |
951 |
this section, we will present a symplectic Lie-Poisson integrator |
952 |
for rigid body developed by Dullweber and his |
953 |
coworkers\cite{Dullweber1997}. |
954 |
|
955 |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
956 |
|
957 |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
958 |
|
959 |
\begin{equation} |
960 |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
961 |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
962 |
\label{introEquation:RBHamiltonian} |
963 |
\end{equation} |
964 |
Here, $q$ and $Q$ are the position and rotation matrix for the |
965 |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
966 |
$J$, a diagonal matrix, is defined by |
967 |
\[ |
968 |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
969 |
\] |
970 |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
971 |
constrained Hamiltonian equation subjects to a holonomic constraint, |
972 |
\begin{equation} |
973 |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
974 |
\end{equation} |
975 |
which is used to ensure rotation matrix's orthogonality. |
976 |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
977 |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
978 |
\begin{equation} |
979 |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
980 |
\label{introEquation:RBFirstOrderConstraint} |
981 |
\end{equation} |
982 |
|
983 |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
984 |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
985 |
the equations of motion, |
986 |
\[ |
987 |
\begin{array}{c} |
988 |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
989 |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
990 |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
991 |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
992 |
\end{array} |
993 |
\] |
994 |
|
995 |
In general, there are two ways to satisfy the holonomic constraints. |
996 |
We can use constraint force provided by lagrange multiplier on the |
997 |
normal manifold to keep the motion on constraint space. Or we can |
998 |
simply evolve the system in constraint manifold. The two method are |
999 |
proved to be equivalent. The holonomic constraint and equations of |
1000 |
motions define a constraint manifold for rigid body |
1001 |
\[ |
1002 |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1003 |
\right\}. |
1004 |
\] |
1005 |
|
1006 |
Unfortunately, this constraint manifold is not the cotangent bundle |
1007 |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
1008 |
transformation, the cotangent space and the phase space are |
1009 |
diffeomorphic. Introducing |
1010 |
\[ |
1011 |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1012 |
\] |
1013 |
the mechanical system subject to a holonomic constraint manifold $M$ |
1014 |
can be re-formulated as a Hamiltonian system on the cotangent space |
1015 |
\[ |
1016 |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
1017 |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
1018 |
\] |
1019 |
|
1020 |
For a body fixed vector $X_i$ with respect to the center of mass of |
1021 |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
1022 |
given as |
1023 |
\begin{equation} |
1024 |
X_i^{lab} = Q X_i + q. |
1025 |
\end{equation} |
1026 |
Therefore, potential energy $V(q,Q)$ is defined by |
1027 |
\[ |
1028 |
V(q,Q) = V(Q X_0 + q). |
1029 |
\] |
1030 |
Hence, |
1031 |
\[ |
1032 |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)} |
1033 |
\] |
1034 |
|
1035 |
\[ |
1036 |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
1037 |
\] |
1038 |
|
1039 |
As a common choice to describe the rotation dynamics of the rigid |
1040 |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
1041 |
rewrite the equations of motion, |
1042 |
\begin{equation} |
1043 |
\begin{array}{l} |
1044 |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1045 |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
1046 |
\end{array} |
1047 |
\label{introEqaution:RBMotionPI} |
1048 |
\end{equation} |
1049 |
, as well as holonomic constraints, |
1050 |
\[ |
1051 |
\begin{array}{l} |
1052 |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
1053 |
Q^T Q = 1 \\ |
1054 |
\end{array} |
1055 |
\] |
1056 |
|
1057 |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1058 |
so(3)^ \star$, the hat-map isomorphism, |
1059 |
\begin{equation} |
1060 |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1061 |
{\begin{array}{*{20}c} |
1062 |
0 & { - v_3 } & {v_2 } \\ |
1063 |
{v_3 } & 0 & { - v_1 } \\ |
1064 |
{ - v_2 } & {v_1 } & 0 \\ |
1065 |
\end{array}} \right), |
1066 |
\label{introEquation:hatmapIsomorphism} |
1067 |
\end{equation} |
1068 |
will let us associate the matrix products with traditional vector |
1069 |
operations |
1070 |
\[ |
1071 |
\hat vu = v \times u |
1072 |
\] |
1073 |
|
1074 |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1075 |
matrix, |
1076 |
\begin{equation} |
1077 |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
1078 |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
1079 |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1080 |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
1081 |
\end{equation} |
1082 |
Since $\Lambda$ is symmetric, the last term of Equation |
1083 |
\ref{introEquation:skewMatrixPI}, which implies the Lagrange |
1084 |
multiplier $\Lambda$ is ignored in the integration. |
1085 |
|
1086 |
Hence, applying hat-map isomorphism, we obtain the equation of |
1087 |
motion for angular momentum on body frame |
1088 |
\[ |
1089 |
\dot \pi = \pi \times I^{ - 1} \pi + Q^T \sum\limits_i {F_i (r,Q) |
1090 |
\times X_i } |
1091 |
\] |
1092 |
In the same manner, the equation of motion for rotation matrix is |
1093 |
given by |
1094 |
\[ |
1095 |
\dot Q = Qskew(M^{ - 1} \pi ) |
1096 |
\] |
1097 |
|
1098 |
The free rigid body equation is an example of a non-canonical |
1099 |
Hamiltonian system. |
1100 |
|
1101 |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Integration of Euler Equations} |
1102 |
|
1103 |
\[ |
1104 |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1105 |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V} |
1106 |
\] |
1107 |
|
1108 |
\[ |
1109 |
\varphi _{\Delta t,T} = \varphi _{\Delta t,R} \circ \varphi |
1110 |
_{\Delta t,\pi } |
1111 |
\] |
1112 |
|
1113 |
\[ |
1114 |
\varphi _{\Delta t,\pi } = \varphi _{\Delta t/2,\pi _1 } \circ |
1115 |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1116 |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1117 |
_1 } |
1118 |
\] |
1119 |
|
1120 |
\[ |
1121 |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1122 |
_{\Delta t/2,\tau } |
1123 |
\] |
1124 |
|
1125 |
|
1126 |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1127 |
|
1128 |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
1129 |
|
1130 |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
1131 |
|
1132 |
\begin{equation} |
1133 |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1134 |
\label{introEquation:bathGLE} |
1135 |
\end{equation} |
1136 |
where $H_B$ is harmonic bath Hamiltonian, |
1137 |
\[ |
1138 |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1139 |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
1140 |
\] |
1141 |
and $\Delta U$ is bilinear system-bath coupling, |
1142 |
\[ |
1143 |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1144 |
\] |
1145 |
Completing the square, |
1146 |
\[ |
1147 |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
1148 |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1149 |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1150 |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
1151 |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1152 |
\] |
1153 |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
1154 |
\[ |
1155 |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1156 |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1157 |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1158 |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1159 |
\] |
1160 |
where |
1161 |
\[ |
1162 |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1163 |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1164 |
\] |
1165 |
Since the first two terms of the new Hamiltonian depend only on the |
1166 |
system coordinates, we can get the equations of motion for |
1167 |
Generalized Langevin Dynamics by Hamilton's equations |
1168 |
\ref{introEquation:motionHamiltonianCoordinate, |
1169 |
introEquation:motionHamiltonianMomentum}, |
1170 |
\begin{align} |
1171 |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
1172 |
&= m\ddot x |
1173 |
&= - \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)} |
1174 |
\label{introEquation:Lp5} |
1175 |
\end{align} |
1176 |
, and |
1177 |
\begin{align} |
1178 |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
1179 |
&= m\ddot x_\alpha |
1180 |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
1181 |
\end{align} |
1182 |
|
1183 |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
1184 |
|
1185 |
\[ |
1186 |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
1187 |
\] |
1188 |
|
1189 |
\[ |
1190 |
L(x + y) = L(x) + L(y) |
1191 |
\] |
1192 |
|
1193 |
\[ |
1194 |
L(ax) = aL(x) |
1195 |
\] |
1196 |
|
1197 |
\[ |
1198 |
L(\dot x) = pL(x) - px(0) |
1199 |
\] |
1200 |
|
1201 |
\[ |
1202 |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
1203 |
\] |
1204 |
|
1205 |
\[ |
1206 |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
1207 |
\] |
1208 |
|
1209 |
Some relatively important transformation, |
1210 |
\[ |
1211 |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
1212 |
\] |
1213 |
|
1214 |
\[ |
1215 |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
1216 |
\] |
1217 |
|
1218 |
\[ |
1219 |
L(1) = \frac{1}{p} |
1220 |
\] |
1221 |
|
1222 |
First, the bath coordinates, |
1223 |
\[ |
1224 |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
1225 |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
1226 |
}}L(x) |
1227 |
\] |
1228 |
\[ |
1229 |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
1230 |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
1231 |
\] |
1232 |
Then, the system coordinates, |
1233 |
\begin{align} |
1234 |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1235 |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
1236 |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
1237 |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
1238 |
}}\omega _\alpha ^2 L(x)} \right\}} |
1239 |
% |
1240 |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1241 |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
1242 |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
1243 |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
1244 |
\end{align} |
1245 |
Then, the inverse transform, |
1246 |
|
1247 |
\begin{align} |
1248 |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
1249 |
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1250 |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
1251 |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
1252 |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
1253 |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
1254 |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
1255 |
% |
1256 |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1257 |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1258 |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1259 |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
1260 |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
1261 |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
1262 |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
1263 |
(\omega _\alpha t)} \right\}} |
1264 |
\end{align} |
1265 |
|
1266 |
\begin{equation} |
1267 |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
1268 |
(t)\dot x(t - \tau )d\tau } + R(t) |
1269 |
\label{introEuqation:GeneralizedLangevinDynamics} |
1270 |
\end{equation} |
1271 |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
1272 |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
1273 |
\[ |
1274 |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1275 |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
1276 |
\] |
1277 |
For an infinite harmonic bath, we can use the spectral density and |
1278 |
an integral over frequencies. |
1279 |
|
1280 |
\[ |
1281 |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
1282 |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
1283 |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
1284 |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
1285 |
\] |
1286 |
The random forces depend only on initial conditions. |
1287 |
|
1288 |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
1289 |
So we can define a new set of coordinates, |
1290 |
\[ |
1291 |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1292 |
^2 }}x(0) |
1293 |
\] |
1294 |
This makes |
1295 |
\[ |
1296 |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
1297 |
\] |
1298 |
And since the $q$ coordinates are harmonic oscillators, |
1299 |
\[ |
1300 |
\begin{array}{l} |
1301 |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1302 |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1303 |
\end{array} |
1304 |
\] |
1305 |
|
1306 |
\begin{align} |
1307 |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
1308 |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
1309 |
(t)q_\beta (0)} \right\rangle } } |
1310 |
% |
1311 |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
1312 |
\right\rangle \cos (\omega _\alpha t)} |
1313 |
% |
1314 |
&= kT\xi (t) |
1315 |
\end{align} |
1316 |
|
1317 |
\begin{equation} |
1318 |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1319 |
\label{introEquation:secondFluctuationDissipation} |
1320 |
\end{equation} |
1321 |
|
1322 |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
1323 |
|
1324 |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1325 |
\subsection{\label{introSection:analyticalApproach}Analytical |
1326 |
Approach} |
1327 |
|
1328 |
\subsection{\label{introSection:approximationApproach}Approximation |
1329 |
Approach} |
1330 |
|
1331 |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
1332 |
Body} |
1333 |
|
1334 |
\section{\label{introSection:correlationFunctions}Correlation Functions} |