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\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
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\section{\label{introSection:classicalMechanics}Classical |
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Mechanics} |
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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 {mdv}{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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\tau \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 = \tau |
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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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schemes for rigid bodies \cite{Dullweber1997}. |
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|
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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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Newtonian Mechanics suffers from two important limitations: motions |
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can only be described in cartesian coordinate systems. Moreover, it |
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becomes impossible to predict analytically the properties of the |
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system even if we know all of the details of the interaction. In |
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order to overcome some of the practical difficulties which arise in |
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attempts to apply Newton's equation to complex system, approximate |
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numerical procedures may be developed. |
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|
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\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
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Principle}} |
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Hamilton introduced the dynamical principle upon which it is |
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possible to base all of mechanics and most of classical physics. |
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Hamilton's Principle may be stated as follows, |
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The actual trajectory, along which a dynamical system may move from |
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one point to another within a specified time, is derived by finding |
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the path which minimizes the time integral of the difference between |
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the kinetic, $K$, and potential energies, $U$. |
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\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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For simple mechanical systems, where the forces acting on the |
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different parts are derivable from a potential, the Lagrangian |
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function $L$ can be defined as the difference between the kinetic |
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energy of the system 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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\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The |
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Equations of Motion in Lagrangian Mechanics}} |
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For a system of $f$ degrees of freedom, the equations of motion in |
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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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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 velocities, the momenta can 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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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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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 }} = \dot {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 }} = - \dot {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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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{Goldstein2001}. |
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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 coordinates |
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$q_i$, while the Hamiltonian is considered to be a function of the |
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generalized momenta $p_i$ and the conjugate coordinates $q_i$. |
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Hamiltonian Mechanics is more appropriate for application to |
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statistical mechanics and quantum mechanics, since it treats the |
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coordinate and its time derivative as independent variables and it |
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only works with 1st-order differential equations\cite{Marion1990}. |
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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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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 of $f$ particles in a |
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cartesian space, where each of the $6f$ coordinates and momenta is |
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assigned to one of $6f$ mutually orthogonal axes, the phase space of |
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this system is a $6f$ dimensional space. A point, $x = (\rightarrow |
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q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow |
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p_f )$, with a unique set of values of $6f$ coordinates and momenta |
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is a phase space vector. |
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%%%fix me |
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|
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In statistical mechanics, 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 distribution for an ensemble with $f$ degrees of |
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freedom is defined as, |
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\begin{equation} |
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\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). |
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\label{introEquation:densityDistribution} |
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\end{equation} |
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Governed by the principles of mechanics, the phase points change |
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their locations which would change the density at any time at phase |
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space. Hence, the density distribution is also to be taken as a |
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function of the time. |
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The number of systems $\delta N$ at time $t$ can be determined by, |
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\begin{equation} |
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\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
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\label{introEquation:deltaN} |
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\end{equation} |
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Assuming a large enough population of systems, we can sufficiently |
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approximate $\delta N$ without introducing discontinuity when we go |
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from one region in the phase space to another. By integrating over |
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the whole phase space, |
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\begin{equation} |
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N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
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\label{introEquation:totalNumberSystem} |
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\end{equation} |
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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, |
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\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} |
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\end{equation} |
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With the help of Eq.~\ref{introEquation:unitProbability} and the |
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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 remaining |
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well defined. For a classical system in thermal equilibrium with its |
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environment, the ensemble average of a mechanical quantity, $\langle |
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A(q , p) \rangle_t$, takes the form of an integral over the phase |
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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} |
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\end{equation} |
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|
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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}, the partition function |
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can 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, the Microcanonical ensemble (NVE) has |
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a partition function like, |
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\begin{equation} |
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\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
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\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} |
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\end{equation} |
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Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
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TS$. Since most experiments are carried out under constant pressure |
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condition, the isothermal-isobaric ensemble (NPT) plays a very |
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important role in molecular simulations. The isothermal-isobaric |
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ensemble allow the system to exchange energy with a heat bath of |
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temperature $T$ and to change the volume as well. Its partition |
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function is given as |
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\begin{equation} |
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\Delta (N,P,T) = - e^{\beta G}. |
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\label{introEquation:NPTPartition} |
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\end{equation} |
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Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
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\subsection{\label{introSection:liouville}Liouville's theorem} |
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|
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Liouville's theorem is the foundation on which statistical mechanics |
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rests. It describes the time evolution of the 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 Eq.~\ref{introEquation:deltaN}. If we consider |
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the two faces perpendicular to the $q_1$ axis, which are located at |
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$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the |
328 |
|
|
opposite face is given by the expression, |
329 |
tim |
2700 |
\begin{equation} |
330 |
|
|
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
331 |
|
|
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
332 |
|
|
}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 |
333 |
|
|
\ldots \delta p_f . |
334 |
|
|
\end{equation} |
335 |
|
|
Summing all over the phase space, we obtain |
336 |
|
|
\begin{equation} |
337 |
|
|
\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho |
338 |
|
|
\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + |
339 |
|
|
\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( |
340 |
|
|
{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial |
341 |
|
|
\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 |
342 |
|
|
\ldots \delta q_f \delta p_1 \ldots \delta p_f . |
343 |
|
|
\end{equation} |
344 |
|
|
Differentiating the equations of motion in Hamiltonian formalism |
345 |
|
|
(\ref{introEquation:motionHamiltonianCoordinate}, |
346 |
|
|
\ref{introEquation:motionHamiltonianMomentum}), we can show, |
347 |
|
|
\begin{equation} |
348 |
|
|
\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} |
349 |
|
|
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
350 |
|
|
\end{equation} |
351 |
|
|
which cancels the first terms of the right hand side. Furthermore, |
352 |
tim |
2819 |
dividing $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
353 |
tim |
2700 |
p_f $ in both sides, we can write out Liouville's theorem in a |
354 |
|
|
simple form, |
355 |
|
|
\begin{equation} |
356 |
|
|
\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f |
357 |
|
|
{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + |
358 |
|
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
359 |
|
|
\label{introEquation:liouvilleTheorem} |
360 |
|
|
\end{equation} |
361 |
|
|
|
362 |
|
|
Liouville's theorem states that the distribution function is |
363 |
|
|
constant along any trajectory in phase space. In classical |
364 |
tim |
2850 |
statistical mechanics, since the number of members in an ensemble is |
365 |
|
|
huge and constant, we can assume the local density has no reason |
366 |
|
|
(other than classical mechanics) to change, |
367 |
tim |
2700 |
\begin{equation} |
368 |
|
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
369 |
|
|
\label{introEquation:stationary} |
370 |
|
|
\end{equation} |
371 |
|
|
In such stationary system, the density of distribution $\rho$ can be |
372 |
|
|
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
373 |
|
|
distribution, |
374 |
|
|
\begin{equation} |
375 |
|
|
\rho \propto e^{ - \beta H} |
376 |
|
|
\label{introEquation:densityAndHamiltonian} |
377 |
|
|
\end{equation} |
378 |
|
|
|
379 |
tim |
2819 |
\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}} |
380 |
tim |
2702 |
Lets consider a region in the phase space, |
381 |
|
|
\begin{equation} |
382 |
|
|
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
383 |
|
|
\end{equation} |
384 |
|
|
If this region is small enough, the density $\rho$ can be regarded |
385 |
tim |
2819 |
as uniform over the whole integral. Thus, the number of phase points |
386 |
|
|
inside this region is given by, |
387 |
tim |
2702 |
\begin{equation} |
388 |
|
|
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
389 |
|
|
dp_1 } ..dp_f. |
390 |
|
|
\end{equation} |
391 |
|
|
|
392 |
|
|
\begin{equation} |
393 |
|
|
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
394 |
|
|
\frac{d}{{dt}}(\delta v) = 0. |
395 |
|
|
\end{equation} |
396 |
|
|
With the help of stationary assumption |
397 |
|
|
(\ref{introEquation:stationary}), we obtain the principle of the |
398 |
tim |
2819 |
\emph{conservation of volume in phase space}, |
399 |
tim |
2702 |
\begin{equation} |
400 |
|
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
401 |
|
|
...dq_f dp_1 } ..dp_f = 0. |
402 |
|
|
\label{introEquation:volumePreserving} |
403 |
|
|
\end{equation} |
404 |
|
|
|
405 |
tim |
2819 |
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
406 |
tim |
2702 |
|
407 |
tim |
2700 |
Liouville's theorem can be expresses in a variety of different forms |
408 |
|
|
which are convenient within different contexts. For any two function |
409 |
|
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
410 |
|
|
bracket ${F, G}$ is defined as |
411 |
|
|
\begin{equation} |
412 |
|
|
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
413 |
|
|
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
414 |
|
|
\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial |
415 |
|
|
q_i }}} \right)}. |
416 |
|
|
\label{introEquation:poissonBracket} |
417 |
|
|
\end{equation} |
418 |
|
|
Substituting equations of motion in Hamiltonian formalism( |
419 |
tim |
2850 |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
420 |
|
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into |
421 |
|
|
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
422 |
|
|
Liouville's theorem using Poisson bracket notion, |
423 |
tim |
2700 |
\begin{equation} |
424 |
|
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
425 |
|
|
{\rho ,H} \right\}. |
426 |
|
|
\label{introEquation:liouvilleTheromInPoissin} |
427 |
|
|
\end{equation} |
428 |
|
|
Moreover, the Liouville operator is defined as |
429 |
|
|
\begin{equation} |
430 |
|
|
iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial |
431 |
|
|
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial |
432 |
|
|
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} |
433 |
|
|
\label{introEquation:liouvilleOperator} |
434 |
|
|
\end{equation} |
435 |
|
|
In terms of Liouville operator, Liouville's equation can also be |
436 |
|
|
expressed as |
437 |
|
|
\begin{equation} |
438 |
|
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
439 |
|
|
\label{introEquation:liouvilleTheoremInOperator} |
440 |
|
|
\end{equation} |
441 |
|
|
|
442 |
tim |
2693 |
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
443 |
tim |
2692 |
|
444 |
tim |
2695 |
Various thermodynamic properties can be calculated from Molecular |
445 |
|
|
Dynamics simulation. By comparing experimental values with the |
446 |
|
|
calculated properties, one can determine the accuracy of the |
447 |
tim |
2819 |
simulation and the quality of the underlying model. However, both |
448 |
|
|
experiments and computer simulations are usually performed during a |
449 |
tim |
2695 |
certain time interval and the measurements are averaged over a |
450 |
|
|
period of them which is different from the average behavior of |
451 |
tim |
2819 |
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
452 |
|
|
Hypothesis makes a connection between time average and the ensemble |
453 |
|
|
average. It states that the time average and average over the |
454 |
tim |
2786 |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
455 |
tim |
2695 |
\begin{equation} |
456 |
tim |
2700 |
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
457 |
|
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
458 |
|
|
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp |
459 |
tim |
2695 |
\end{equation} |
460 |
tim |
2700 |
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
461 |
|
|
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
462 |
|
|
distribution function. If an observation is averaged over a |
463 |
|
|
sufficiently long time (longer than relaxation time), all accessible |
464 |
|
|
microstates in phase space are assumed to be equally probed, giving |
465 |
|
|
a properly weighted statistical average. This allows the researcher |
466 |
|
|
freedom of choice when deciding how best to measure a given |
467 |
|
|
observable. In case an ensemble averaged approach sounds most |
468 |
tim |
2786 |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
469 |
tim |
2700 |
utilized. Or if the system lends itself to a time averaging |
470 |
|
|
approach, the Molecular Dynamics techniques in |
471 |
|
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
472 |
|
|
choice\cite{Frenkel1996}. |
473 |
tim |
2694 |
|
474 |
tim |
2697 |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
475 |
tim |
2819 |
A variety of numerical integrators have been proposed to simulate |
476 |
|
|
the motions of atoms in MD simulation. They usually begin with |
477 |
|
|
initial conditionals and move the objects in the direction governed |
478 |
|
|
by the differential equations. However, most of them ignore the |
479 |
|
|
hidden physical laws contained within the equations. Since 1990, |
480 |
|
|
geometric integrators, which preserve various phase-flow invariants |
481 |
|
|
such as symplectic structure, volume and time reversal symmetry, are |
482 |
|
|
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
483 |
tim |
2872 |
Leimkuhler1999}. The velocity Verlet method, which happens to be a |
484 |
tim |
2819 |
simple example of symplectic integrator, continues to gain |
485 |
|
|
popularity in the molecular dynamics community. This fact can be |
486 |
|
|
partly explained by its geometric nature. |
487 |
tim |
2697 |
|
488 |
tim |
2819 |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
489 |
|
|
A \emph{manifold} is an abstract mathematical space. It looks |
490 |
|
|
locally like Euclidean space, but when viewed globally, it may have |
491 |
|
|
more complicated structure. A good example of manifold is the |
492 |
|
|
surface of Earth. It seems to be flat locally, but it is round if |
493 |
|
|
viewed as a whole. A \emph{differentiable manifold} (also known as |
494 |
|
|
\emph{smooth manifold}) is a manifold on which it is possible to |
495 |
|
|
apply calculus on \emph{differentiable manifold}. A \emph{symplectic |
496 |
|
|
manifold} is defined as a pair $(M, \omega)$ which consists of a |
497 |
tim |
2697 |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
498 |
|
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
499 |
|
|
space $V$ is a function $\omega(x, y)$ which satisfies |
500 |
|
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
501 |
|
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
502 |
tim |
2819 |
$\omega(x, x) = 0$. The cross product operation in vector field is |
503 |
|
|
an example of symplectic form. |
504 |
tim |
2697 |
|
505 |
tim |
2819 |
One of the motivations to study \emph{symplectic manifolds} in |
506 |
tim |
2697 |
Hamiltonian Mechanics is that a symplectic manifold can represent |
507 |
|
|
all possible configurations of the system and the phase space of the |
508 |
|
|
system can be described by it's cotangent bundle. Every symplectic |
509 |
|
|
manifold is even dimensional. For instance, in Hamilton equations, |
510 |
|
|
coordinate and momentum always appear in pairs. |
511 |
|
|
|
512 |
tim |
2698 |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
513 |
tim |
2697 |
|
514 |
tim |
2819 |
For an ordinary differential system defined as |
515 |
tim |
2698 |
\begin{equation} |
516 |
|
|
\dot x = f(x) |
517 |
|
|
\end{equation} |
518 |
tim |
2819 |
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
519 |
tim |
2698 |
\begin{equation} |
520 |
tim |
2699 |
f(r) = J\nabla _x H(r). |
521 |
tim |
2698 |
\end{equation} |
522 |
|
|
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
523 |
|
|
matrix |
524 |
|
|
\begin{equation} |
525 |
|
|
J = \left( {\begin{array}{*{20}c} |
526 |
|
|
0 & I \\ |
527 |
|
|
{ - I} & 0 \\ |
528 |
|
|
\end{array}} \right) |
529 |
|
|
\label{introEquation:canonicalMatrix} |
530 |
|
|
\end{equation} |
531 |
|
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
532 |
|
|
system can be rewritten as, |
533 |
|
|
\begin{equation} |
534 |
|
|
\frac{d}{{dt}}x = J\nabla _x H(x) |
535 |
|
|
\label{introEquation:compactHamiltonian} |
536 |
|
|
\end{equation}In this case, $f$ is |
537 |
|
|
called a \emph{Hamiltonian vector field}. |
538 |
tim |
2697 |
|
539 |
tim |
2789 |
Another generalization of Hamiltonian dynamics is Poisson |
540 |
|
|
Dynamics\cite{Olver1986}, |
541 |
tim |
2698 |
\begin{equation} |
542 |
|
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
543 |
|
|
\end{equation} |
544 |
|
|
The most obvious change being that matrix $J$ now depends on $x$. |
545 |
|
|
|
546 |
tim |
2702 |
\subsection{\label{introSection:exactFlow}Exact Flow} |
547 |
|
|
|
548 |
tim |
2698 |
Let $x(t)$ be the exact solution of the ODE system, |
549 |
|
|
\begin{equation} |
550 |
|
|
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
551 |
|
|
\end{equation} |
552 |
|
|
The exact flow(solution) $\varphi_\tau$ is defined by |
553 |
|
|
\[ |
554 |
|
|
x(t+\tau) =\varphi_\tau(x(t)) |
555 |
|
|
\] |
556 |
|
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
557 |
tim |
2702 |
space to itself. The flow has the continuous group property, |
558 |
tim |
2698 |
\begin{equation} |
559 |
tim |
2702 |
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
560 |
|
|
+ \tau _2 } . |
561 |
|
|
\end{equation} |
562 |
|
|
In particular, |
563 |
|
|
\begin{equation} |
564 |
|
|
\varphi _\tau \circ \varphi _{ - \tau } = I |
565 |
|
|
\end{equation} |
566 |
|
|
Therefore, the exact flow is self-adjoint, |
567 |
|
|
\begin{equation} |
568 |
|
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
569 |
|
|
\end{equation} |
570 |
|
|
The exact flow can also be written in terms of the of an operator, |
571 |
|
|
\begin{equation} |
572 |
|
|
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
573 |
|
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
574 |
|
|
\label{introEquation:exponentialOperator} |
575 |
|
|
\end{equation} |
576 |
|
|
|
577 |
|
|
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
578 |
tim |
2872 |
Instead, we use an approximate map, $\psi_\tau$, which is usually |
579 |
tim |
2702 |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
580 |
|
|
the Taylor series of $\psi_\tau$ agree to order $p$, |
581 |
|
|
\begin{equation} |
582 |
tim |
2872 |
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
583 |
tim |
2698 |
\end{equation} |
584 |
|
|
|
585 |
tim |
2702 |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
586 |
|
|
|
587 |
tim |
2872 |
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
588 |
|
|
ODE and its flow play important roles in numerical studies. Many of |
589 |
|
|
them can be found in systems which occur naturally in applications. |
590 |
tim |
2702 |
|
591 |
|
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
592 |
|
|
a \emph{symplectic} flow if it satisfies, |
593 |
tim |
2698 |
\begin{equation} |
594 |
tim |
2703 |
{\varphi '}^T J \varphi ' = J. |
595 |
tim |
2698 |
\end{equation} |
596 |
|
|
According to Liouville's theorem, the symplectic volume is invariant |
597 |
|
|
under a Hamiltonian flow, which is the basis for classical |
598 |
tim |
2699 |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
599 |
|
|
field on a symplectic manifold can be shown to be a |
600 |
|
|
symplectomorphism. As to the Poisson system, |
601 |
tim |
2698 |
\begin{equation} |
602 |
tim |
2703 |
{\varphi '}^T J \varphi ' = J \circ \varphi |
603 |
tim |
2698 |
\end{equation} |
604 |
tim |
2872 |
is the property that must be preserved by the integrator. |
605 |
tim |
2702 |
|
606 |
|
|
It is possible to construct a \emph{volume-preserving} flow for a |
607 |
tim |
2872 |
source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ |
608 |
tim |
2702 |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
609 |
|
|
be volume-preserving. |
610 |
|
|
|
611 |
tim |
2872 |
Changing the variables $y = h(x)$ in an ODE |
612 |
|
|
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
613 |
tim |
2698 |
\[ |
614 |
|
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
615 |
|
|
\] |
616 |
|
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
617 |
|
|
In other words, the flow of this vector field is reversible if and |
618 |
tim |
2702 |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
619 |
tim |
2698 |
|
620 |
tim |
2705 |
A \emph{first integral}, or conserved quantity of a general |
621 |
|
|
differential function is a function $ G:R^{2d} \to R^d $ which is |
622 |
|
|
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
623 |
|
|
\[ |
624 |
|
|
\frac{{dG(x(t))}}{{dt}} = 0. |
625 |
|
|
\] |
626 |
|
|
Using chain rule, one may obtain, |
627 |
|
|
\[ |
628 |
|
|
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
629 |
|
|
\] |
630 |
|
|
which is the condition for conserving \emph{first integral}. For a |
631 |
|
|
canonical Hamiltonian system, the time evolution of an arbitrary |
632 |
|
|
smooth function $G$ is given by, |
633 |
tim |
2789 |
|
634 |
|
|
\begin{eqnarray} |
635 |
|
|
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
636 |
|
|
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
637 |
tim |
2705 |
\label{introEquation:firstIntegral1} |
638 |
tim |
2789 |
\end{eqnarray} |
639 |
|
|
|
640 |
|
|
|
641 |
tim |
2705 |
Using poisson bracket notion, Equation |
642 |
|
|
\ref{introEquation:firstIntegral1} can be rewritten as |
643 |
|
|
\[ |
644 |
|
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
645 |
|
|
\] |
646 |
|
|
Therefore, the sufficient condition for $G$ to be the \emph{first |
647 |
|
|
integral} of a Hamiltonian system is |
648 |
|
|
\[ |
649 |
|
|
\left\{ {G,H} \right\} = 0. |
650 |
|
|
\] |
651 |
|
|
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
652 |
|
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
653 |
|
|
0$. |
654 |
|
|
|
655 |
tim |
2789 |
When designing any numerical methods, one should always try to |
656 |
tim |
2702 |
preserve the structural properties of the original ODE and its flow. |
657 |
|
|
|
658 |
tim |
2699 |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
659 |
|
|
A lot of well established and very effective numerical methods have |
660 |
|
|
been successful precisely because of their symplecticities even |
661 |
|
|
though this fact was not recognized when they were first |
662 |
tim |
2872 |
constructed. The most famous example is the Verlet-leapfrog method |
663 |
tim |
2819 |
in molecular dynamics. In general, symplectic integrators can be |
664 |
tim |
2699 |
constructed using one of four different methods. |
665 |
|
|
\begin{enumerate} |
666 |
|
|
\item Generating functions |
667 |
|
|
\item Variational methods |
668 |
|
|
\item Runge-Kutta methods |
669 |
|
|
\item Splitting methods |
670 |
|
|
\end{enumerate} |
671 |
tim |
2698 |
|
672 |
tim |
2789 |
Generating function\cite{Channell1990} tends to lead to methods |
673 |
|
|
which are cumbersome and difficult to use. In dissipative systems, |
674 |
|
|
variational methods can capture the decay of energy |
675 |
|
|
accurately\cite{Kane2000}. Since their geometrically unstable nature |
676 |
|
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
677 |
|
|
methods are not suitable for Hamiltonian system. Recently, various |
678 |
|
|
high-order explicit Runge-Kutta methods |
679 |
|
|
\cite{Owren1992,Chen2003}have been developed to overcome this |
680 |
tim |
2703 |
instability. However, due to computational penalty involved in |
681 |
tim |
2819 |
implementing the Runge-Kutta methods, they have not attracted much |
682 |
|
|
attention from the Molecular Dynamics community. Instead, splitting |
683 |
|
|
methods have been widely accepted since they exploit natural |
684 |
|
|
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
685 |
tim |
2702 |
|
686 |
tim |
2819 |
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
687 |
tim |
2702 |
|
688 |
|
|
The main idea behind splitting methods is to decompose the discrete |
689 |
|
|
$\varphi_h$ as a composition of simpler flows, |
690 |
tim |
2699 |
\begin{equation} |
691 |
|
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
692 |
|
|
\varphi _{h_n } |
693 |
|
|
\label{introEquation:FlowDecomposition} |
694 |
|
|
\end{equation} |
695 |
|
|
where each of the sub-flow is chosen such that each represent a |
696 |
tim |
2702 |
simpler integration of the system. |
697 |
|
|
|
698 |
|
|
Suppose that a Hamiltonian system takes the form, |
699 |
|
|
\[ |
700 |
|
|
H = H_1 + H_2. |
701 |
|
|
\] |
702 |
|
|
Here, $H_1$ and $H_2$ may represent different physical processes of |
703 |
|
|
the system. For instance, they may relate to kinetic and potential |
704 |
|
|
energy respectively, which is a natural decomposition of the |
705 |
|
|
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
706 |
|
|
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
707 |
tim |
2819 |
order expression is then given by the Lie-Trotter formula |
708 |
tim |
2699 |
\begin{equation} |
709 |
tim |
2702 |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
710 |
|
|
\label{introEquation:firstOrderSplitting} |
711 |
|
|
\end{equation} |
712 |
|
|
where $\varphi _h$ is the result of applying the corresponding |
713 |
|
|
continuous $\varphi _i$ over a time $h$. By definition, as |
714 |
|
|
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
715 |
|
|
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
716 |
|
|
It is easy to show that any composition of symplectic flows yields a |
717 |
|
|
symplectic map, |
718 |
|
|
\begin{equation} |
719 |
tim |
2699 |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
720 |
tim |
2702 |
'\phi ' = \phi '^T J\phi ' = J, |
721 |
tim |
2699 |
\label{introEquation:SymplecticFlowComposition} |
722 |
|
|
\end{equation} |
723 |
tim |
2702 |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
724 |
|
|
splitting in this context automatically generates a symplectic map. |
725 |
tim |
2699 |
|
726 |
tim |
2702 |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
727 |
|
|
introduces local errors proportional to $h^2$, while Strang |
728 |
|
|
splitting gives a second-order decomposition, |
729 |
|
|
\begin{equation} |
730 |
|
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
731 |
tim |
2706 |
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
732 |
tim |
2702 |
\end{equation} |
733 |
tim |
2819 |
which has a local error proportional to $h^3$. The Sprang |
734 |
|
|
splitting's popularity in molecular simulation community attribute |
735 |
|
|
to its symmetric property, |
736 |
tim |
2702 |
\begin{equation} |
737 |
|
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
738 |
tim |
2703 |
\label{introEquation:timeReversible} |
739 |
tim |
2882 |
\end{equation} |
740 |
tim |
2702 |
|
741 |
tim |
2872 |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
742 |
tim |
2702 |
The classical equation for a system consisting of interacting |
743 |
|
|
particles can be written in Hamiltonian form, |
744 |
|
|
\[ |
745 |
|
|
H = T + V |
746 |
|
|
\] |
747 |
|
|
where $T$ is the kinetic energy and $V$ is the potential energy. |
748 |
tim |
2872 |
Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one |
749 |
tim |
2702 |
obtains the following: |
750 |
|
|
\begin{align} |
751 |
|
|
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
752 |
|
|
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
753 |
|
|
\label{introEquation:Lp10a} \\% |
754 |
|
|
% |
755 |
|
|
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
756 |
|
|
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
757 |
|
|
\label{introEquation:Lp10b} |
758 |
|
|
\end{align} |
759 |
|
|
where $F(t)$ is the force at time $t$. This integration scheme is |
760 |
|
|
known as \emph{velocity verlet} which is |
761 |
|
|
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
762 |
|
|
time-reversible(\ref{introEquation:timeReversible}) and |
763 |
|
|
volume-preserving (\ref{introEquation:volumePreserving}). These |
764 |
|
|
geometric properties attribute to its long-time stability and its |
765 |
|
|
popularity in the community. However, the most commonly used |
766 |
|
|
velocity verlet integration scheme is written as below, |
767 |
|
|
\begin{align} |
768 |
|
|
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
769 |
|
|
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
770 |
|
|
% |
771 |
|
|
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
772 |
|
|
\label{introEquation:Lp9b}\\% |
773 |
|
|
% |
774 |
|
|
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
775 |
tim |
2872 |
\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c} |
776 |
tim |
2702 |
\end{align} |
777 |
|
|
From the preceding splitting, one can see that the integration of |
778 |
|
|
the equations of motion would follow: |
779 |
|
|
\begin{enumerate} |
780 |
|
|
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
781 |
|
|
|
782 |
|
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
783 |
|
|
|
784 |
tim |
2872 |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
785 |
tim |
2702 |
|
786 |
|
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
787 |
|
|
\end{enumerate} |
788 |
|
|
|
789 |
tim |
2872 |
By simply switching the order of the propagators in the splitting |
790 |
|
|
and composing a new integrator, the \emph{position verlet} |
791 |
|
|
integrator, can be generated, |
792 |
tim |
2702 |
\begin{align} |
793 |
|
|
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
794 |
|
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
795 |
|
|
\label{introEquation:positionVerlet1} \\% |
796 |
|
|
% |
797 |
tim |
2703 |
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
798 |
tim |
2702 |
q(\Delta t)} \right]. % |
799 |
tim |
2719 |
\label{introEquation:positionVerlet2} |
800 |
tim |
2702 |
\end{align} |
801 |
|
|
|
802 |
tim |
2819 |
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
803 |
tim |
2702 |
|
804 |
tim |
2872 |
The Baker-Campbell-Hausdorff formula can be used to determine the |
805 |
|
|
local error of splitting method in terms of the commutator of the |
806 |
tim |
2702 |
operators(\ref{introEquation:exponentialOperator}) associated with |
807 |
tim |
2872 |
the sub-flow. For operators $hX$ and $hY$ which are associated with |
808 |
tim |
2726 |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
809 |
tim |
2702 |
\begin{equation} |
810 |
|
|
\exp (hX + hY) = \exp (hZ) |
811 |
|
|
\end{equation} |
812 |
|
|
where |
813 |
|
|
\begin{equation} |
814 |
|
|
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
815 |
|
|
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
816 |
|
|
\end{equation} |
817 |
|
|
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
818 |
|
|
\[ |
819 |
|
|
[X,Y] = XY - YX . |
820 |
|
|
\] |
821 |
tim |
2872 |
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
822 |
|
|
to the Sprang splitting, we can obtain |
823 |
tim |
2779 |
\begin{eqnarray*} |
824 |
tim |
2778 |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
825 |
|
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
826 |
tim |
2779 |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
827 |
|
|
\end{eqnarray*} |
828 |
tim |
2872 |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0,\] the dominant local |
829 |
tim |
2702 |
error of Spring splitting is proportional to $h^3$. The same |
830 |
tim |
2872 |
procedure can be applied to a general splitting, of the form |
831 |
tim |
2702 |
\begin{equation} |
832 |
|
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
833 |
|
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
834 |
|
|
\end{equation} |
835 |
tim |
2872 |
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
836 |
|
|
order methods. Yoshida proposed an elegant way to compose higher |
837 |
tim |
2789 |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
838 |
|
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
839 |
|
|
fourth-order symmetric method can be constructed by composing, |
840 |
tim |
2702 |
\[ |
841 |
|
|
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
842 |
|
|
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
843 |
|
|
\] |
844 |
|
|
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
845 |
|
|
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
846 |
|
|
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
847 |
|
|
\begin{equation} |
848 |
|
|
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
849 |
tim |
2872 |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)}, |
850 |
tim |
2702 |
\end{equation} |
851 |
tim |
2872 |
if the weights are chosen as |
852 |
tim |
2702 |
\[ |
853 |
|
|
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
854 |
|
|
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
855 |
|
|
\] |
856 |
|
|
|
857 |
tim |
2694 |
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
858 |
|
|
|
859 |
tim |
2720 |
As one of the principal tools of molecular modeling, Molecular |
860 |
|
|
dynamics has proven to be a powerful tool for studying the functions |
861 |
|
|
of biological systems, providing structural, thermodynamic and |
862 |
|
|
dynamical information. The basic idea of molecular dynamics is that |
863 |
|
|
macroscopic properties are related to microscopic behavior and |
864 |
|
|
microscopic behavior can be calculated from the trajectories in |
865 |
|
|
simulations. For instance, instantaneous temperature of an |
866 |
|
|
Hamiltonian system of $N$ particle can be measured by |
867 |
|
|
\[ |
868 |
tim |
2725 |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
869 |
tim |
2720 |
\] |
870 |
|
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
871 |
|
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
872 |
|
|
the boltzman constant. |
873 |
tim |
2694 |
|
874 |
tim |
2720 |
A typical molecular dynamics run consists of three essential steps: |
875 |
|
|
\begin{enumerate} |
876 |
|
|
\item Initialization |
877 |
|
|
\begin{enumerate} |
878 |
|
|
\item Preliminary preparation |
879 |
|
|
\item Minimization |
880 |
|
|
\item Heating |
881 |
|
|
\item Equilibration |
882 |
|
|
\end{enumerate} |
883 |
|
|
\item Production |
884 |
|
|
\item Analysis |
885 |
|
|
\end{enumerate} |
886 |
|
|
These three individual steps will be covered in the following |
887 |
|
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
888 |
tim |
2801 |
initialization of a simulation. Sec.~\ref{introSection:production} |
889 |
tim |
2872 |
will discusse issues in production run. |
890 |
tim |
2801 |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
891 |
|
|
trajectory analysis. |
892 |
tim |
2719 |
|
893 |
tim |
2720 |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
894 |
tim |
2719 |
|
895 |
tim |
2819 |
\subsubsection{\textbf{Preliminary preparation}} |
896 |
tim |
2719 |
|
897 |
tim |
2720 |
When selecting the starting structure of a molecule for molecular |
898 |
|
|
simulation, one may retrieve its Cartesian coordinates from public |
899 |
|
|
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
900 |
|
|
thousands of crystal structures of molecules are discovered every |
901 |
|
|
year, many more remain unknown due to the difficulties of |
902 |
tim |
2872 |
purification and crystallization. Even for molecules with known |
903 |
|
|
structure, some important information is missing. For example, a |
904 |
tim |
2720 |
missing hydrogen atom which acts as donor in hydrogen bonding must |
905 |
|
|
be added. Moreover, in order to include electrostatic interaction, |
906 |
|
|
one may need to specify the partial charges for individual atoms. |
907 |
|
|
Under some circumstances, we may even need to prepare the system in |
908 |
tim |
2872 |
a special configuration. For instance, when studying transport |
909 |
|
|
phenomenon in membrane systems, we may prepare the lipids in a |
910 |
|
|
bilayer structure instead of placing lipids randomly in solvent, |
911 |
|
|
since we are not interested in the slow self-aggregation process. |
912 |
tim |
2694 |
|
913 |
tim |
2819 |
\subsubsection{\textbf{Minimization}} |
914 |
tim |
2705 |
|
915 |
tim |
2720 |
It is quite possible that some of molecules in the system from |
916 |
tim |
2872 |
preliminary preparation may be overlapping with each other. This |
917 |
|
|
close proximity leads to high initial potential energy which |
918 |
|
|
consequently jeopardizes any molecular dynamics simulations. To |
919 |
|
|
remove these steric overlaps, one typically performs energy |
920 |
|
|
minimization to find a more reasonable conformation. Several energy |
921 |
|
|
minimization methods have been developed to exploit the energy |
922 |
|
|
surface and to locate the local minimum. While converging slowly |
923 |
|
|
near the minimum, steepest descent method is extremely robust when |
924 |
|
|
systems are strongly anharmonic. Thus, it is often used to refine |
925 |
|
|
structure from crystallographic data. Relied on the gradient or |
926 |
|
|
hessian, advanced methods like Newton-Raphson converge rapidly to a |
927 |
|
|
local minimum, but become unstable if the energy surface is far from |
928 |
|
|
quadratic. Another factor that must be taken into account, when |
929 |
tim |
2720 |
choosing energy minimization method, is the size of the system. |
930 |
|
|
Steepest descent and conjugate gradient can deal with models of any |
931 |
tim |
2872 |
size. Because of the limits on computer memory to store the hessian |
932 |
|
|
matrix and the computing power needed to diagonalized these |
933 |
|
|
matrices, most Newton-Raphson methods can not be used with very |
934 |
|
|
large systems. |
935 |
tim |
2694 |
|
936 |
tim |
2819 |
\subsubsection{\textbf{Heating}} |
937 |
tim |
2720 |
|
938 |
|
|
Typically, Heating is performed by assigning random velocities |
939 |
tim |
2872 |
according to a Maxwell-Boltzman distribution for a desired |
940 |
|
|
temperature. Beginning at a lower temperature and gradually |
941 |
|
|
increasing the temperature by assigning larger random velocities, we |
942 |
|
|
end up with setting the temperature of the system to a final |
943 |
|
|
temperature at which the simulation will be conducted. In heating |
944 |
|
|
phase, we should also keep the system from drifting or rotating as a |
945 |
|
|
whole. To do this, the net linear momentum and angular momentum of |
946 |
|
|
the system is shifted to zero after each resampling from the Maxwell |
947 |
|
|
-Boltzman distribution. |
948 |
tim |
2720 |
|
949 |
tim |
2819 |
\subsubsection{\textbf{Equilibration}} |
950 |
tim |
2720 |
|
951 |
|
|
The purpose of equilibration is to allow the system to evolve |
952 |
|
|
spontaneously for a period of time and reach equilibrium. The |
953 |
|
|
procedure is continued until various statistical properties, such as |
954 |
|
|
temperature, pressure, energy, volume and other structural |
955 |
|
|
properties \textit{etc}, become independent of time. Strictly |
956 |
|
|
speaking, minimization and heating are not necessary, provided the |
957 |
|
|
equilibration process is long enough. However, these steps can serve |
958 |
|
|
as a means to arrive at an equilibrated structure in an effective |
959 |
|
|
way. |
960 |
|
|
|
961 |
|
|
\subsection{\label{introSection:production}Production} |
962 |
|
|
|
963 |
tim |
2872 |
The production run is the most important step of the simulation, in |
964 |
tim |
2725 |
which the equilibrated structure is used as a starting point and the |
965 |
|
|
motions of the molecules are collected for later analysis. In order |
966 |
|
|
to capture the macroscopic properties of the system, the molecular |
967 |
tim |
2872 |
dynamics simulation must be performed by sampling correctly and |
968 |
|
|
efficiently from the relevant thermodynamic ensemble. |
969 |
tim |
2720 |
|
970 |
tim |
2725 |
The most expensive part of a molecular dynamics simulation is the |
971 |
|
|
calculation of non-bonded forces, such as van der Waals force and |
972 |
|
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
973 |
|
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
974 |
|
|
which making large simulations prohibitive in the absence of any |
975 |
tim |
2872 |
algorithmic tricks. |
976 |
tim |
2720 |
|
977 |
tim |
2872 |
A natural approach to avoid system size issues is to represent the |
978 |
tim |
2725 |
bulk behavior by a finite number of the particles. However, this |
979 |
tim |
2872 |
approach will suffer from the surface effect at the edges of the |
980 |
|
|
simulation. To offset this, \textit{Periodic boundary conditions} |
981 |
|
|
(see Fig.~\ref{introFig:pbc}) is developed to simulate bulk |
982 |
|
|
properties with a relatively small number of particles. In this |
983 |
|
|
method, the simulation box is replicated throughout space to form an |
984 |
|
|
infinite lattice. During the simulation, when a particle moves in |
985 |
|
|
the primary cell, its image in other cells move in exactly the same |
986 |
|
|
direction with exactly the same orientation. Thus, as a particle |
987 |
|
|
leaves the primary cell, one of its images will enter through the |
988 |
|
|
opposite face. |
989 |
tim |
2789 |
\begin{figure} |
990 |
|
|
\centering |
991 |
|
|
\includegraphics[width=\linewidth]{pbc.eps} |
992 |
|
|
\caption[An illustration of periodic boundary conditions]{A 2-D |
993 |
|
|
illustration of periodic boundary conditions. As one particle leaves |
994 |
|
|
the left of the simulation box, an image of it enters the right.} |
995 |
|
|
\label{introFig:pbc} |
996 |
|
|
\end{figure} |
997 |
tim |
2725 |
|
998 |
|
|
%cutoff and minimum image convention |
999 |
|
|
Another important technique to improve the efficiency of force |
1000 |
tim |
2872 |
evaluation is to apply spherical cutoff where particles farther than |
1001 |
|
|
a predetermined distance are not included in the calculation |
1002 |
tim |
2725 |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1003 |
tim |
2730 |
discontinuity in the potential energy curve. Fortunately, one can |
1004 |
tim |
2872 |
shift simple radial potential to ensure the potential curve go |
1005 |
|
|
smoothly to zero at the cutoff radius. The cutoff strategy works |
1006 |
|
|
well for Lennard-Jones interaction because of its short range |
1007 |
|
|
nature. However, simply truncating the electrostatic interaction |
1008 |
|
|
with the use of cutoffs has been shown to lead to severe artifacts |
1009 |
|
|
in simulations. The Ewald summation, in which the slowly decaying |
1010 |
|
|
Coulomb potential is transformed into direct and reciprocal sums |
1011 |
|
|
with rapid and absolute convergence, has proved to minimize the |
1012 |
|
|
periodicity artifacts in liquid simulations. Taking the advantages |
1013 |
|
|
of the fast Fourier transform (FFT) for calculating discrete Fourier |
1014 |
|
|
transforms, the particle mesh-based |
1015 |
tim |
2789 |
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
1016 |
tim |
2872 |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
1017 |
|
|
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
1018 |
|
|
which treats Coulombic interactions exactly at short range, and |
1019 |
|
|
approximate the potential at long range through multipolar |
1020 |
|
|
expansion. In spite of their wide acceptance at the molecular |
1021 |
|
|
simulation community, these two methods are difficult to implement |
1022 |
|
|
correctly and efficiently. Instead, we use a damped and |
1023 |
|
|
charge-neutralized Coulomb potential method developed by Wolf and |
1024 |
|
|
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
1025 |
|
|
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
1026 |
tim |
2725 |
\begin{equation} |
1027 |
|
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1028 |
|
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1029 |
|
|
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
1030 |
|
|
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
1031 |
|
|
\end{equation} |
1032 |
|
|
where $\alpha$ is the convergence parameter. Due to the lack of |
1033 |
|
|
inherent periodicity and rapid convergence,this method is extremely |
1034 |
|
|
efficient and easy to implement. |
1035 |
tim |
2789 |
\begin{figure} |
1036 |
|
|
\centering |
1037 |
|
|
\includegraphics[width=\linewidth]{shifted_coulomb.eps} |
1038 |
|
|
\caption[An illustration of shifted Coulomb potential]{An |
1039 |
|
|
illustration of shifted Coulomb potential.} |
1040 |
|
|
\label{introFigure:shiftedCoulomb} |
1041 |
|
|
\end{figure} |
1042 |
tim |
2725 |
|
1043 |
|
|
%multiple time step |
1044 |
|
|
|
1045 |
tim |
2720 |
\subsection{\label{introSection:Analysis} Analysis} |
1046 |
|
|
|
1047 |
tim |
2872 |
Recently, advanced visualization technique have become applied to |
1048 |
tim |
2721 |
monitor the motions of molecules. Although the dynamics of the |
1049 |
|
|
system can be described qualitatively from animation, quantitative |
1050 |
tim |
2872 |
trajectory analysis are more useful. According to the principles of |
1051 |
|
|
Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics}, |
1052 |
|
|
one can compute thermodynamic properties, analyze fluctuations of |
1053 |
|
|
structural parameters, and investigate time-dependent processes of |
1054 |
|
|
the molecule from the trajectories. |
1055 |
tim |
2721 |
|
1056 |
tim |
2872 |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
1057 |
tim |
2721 |
|
1058 |
tim |
2872 |
Thermodynamic properties, which can be expressed in terms of some |
1059 |
tim |
2725 |
function of the coordinates and momenta of all particles in the |
1060 |
|
|
system, can be directly computed from molecular dynamics. The usual |
1061 |
|
|
way to measure the pressure is based on virial theorem of Clausius |
1062 |
|
|
which states that the virial is equal to $-3Nk_BT$. For a system |
1063 |
|
|
with forces between particles, the total virial, $W$, contains the |
1064 |
|
|
contribution from external pressure and interaction between the |
1065 |
|
|
particles: |
1066 |
|
|
\[ |
1067 |
|
|
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
1068 |
|
|
f_{ij} } } \right\rangle |
1069 |
|
|
\] |
1070 |
|
|
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
1071 |
|
|
distance $r_{ij}$. Thus, the expression for the pressure is given |
1072 |
|
|
by: |
1073 |
|
|
\begin{equation} |
1074 |
|
|
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
1075 |
|
|
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1076 |
|
|
\end{equation} |
1077 |
|
|
|
1078 |
tim |
2819 |
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
1079 |
tim |
2721 |
|
1080 |
|
|
Structural Properties of a simple fluid can be described by a set of |
1081 |
tim |
2872 |
distribution functions. Among these functions,the \emph{pair |
1082 |
tim |
2721 |
distribution function}, also known as \emph{radial distribution |
1083 |
tim |
2872 |
function}, is of most fundamental importance to liquid theory. |
1084 |
|
|
Experimentally, pair distribution function can be gathered by |
1085 |
|
|
Fourier transforming raw data from a series of neutron diffraction |
1086 |
|
|
experiments and integrating over the surface factor |
1087 |
|
|
\cite{Powles1973}. The experimental results can serve as a criterion |
1088 |
|
|
to justify the correctness of a liquid model. Moreover, various |
1089 |
|
|
equilibrium thermodynamic and structural properties can also be |
1090 |
|
|
expressed in terms of radial distribution function \cite{Allen1987}. |
1091 |
tim |
2721 |
|
1092 |
tim |
2872 |
The pair distribution functions $g(r)$ gives the probability that a |
1093 |
tim |
2721 |
particle $i$ will be located at a distance $r$ from a another |
1094 |
|
|
particle $j$ in the system |
1095 |
|
|
\[ |
1096 |
|
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1097 |
tim |
2874 |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
1098 |
tim |
2872 |
(r)}{\rho}. |
1099 |
tim |
2721 |
\] |
1100 |
|
|
Note that the delta function can be replaced by a histogram in |
1101 |
tim |
2881 |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
1102 |
|
|
the height of these peaks gradually decreases to 1 as the liquid of |
1103 |
|
|
large distance approaches the bulk density. |
1104 |
tim |
2721 |
|
1105 |
|
|
|
1106 |
tim |
2819 |
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
1107 |
|
|
Properties}} |
1108 |
tim |
2721 |
|
1109 |
|
|
Time-dependent properties are usually calculated using \emph{time |
1110 |
tim |
2872 |
correlation functions}, which correlate random variables $A$ and $B$ |
1111 |
|
|
at two different times, |
1112 |
tim |
2721 |
\begin{equation} |
1113 |
|
|
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1114 |
|
|
\label{introEquation:timeCorrelationFunction} |
1115 |
|
|
\end{equation} |
1116 |
|
|
If $A$ and $B$ refer to same variable, this kind of correlation |
1117 |
tim |
2872 |
function is called an \emph{autocorrelation function}. One example |
1118 |
|
|
of an auto correlation function is the velocity auto-correlation |
1119 |
|
|
function which is directly related to transport properties of |
1120 |
|
|
molecular liquids: |
1121 |
tim |
2725 |
\[ |
1122 |
|
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1123 |
|
|
\right\rangle } dt |
1124 |
|
|
\] |
1125 |
tim |
2872 |
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
1126 |
|
|
function, which is averaging over time origins and over all the |
1127 |
|
|
atoms, the dipole autocorrelation functions are calculated for the |
1128 |
|
|
entire system. The dipole autocorrelation function is given by: |
1129 |
tim |
2725 |
\[ |
1130 |
|
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1131 |
|
|
\right\rangle |
1132 |
|
|
\] |
1133 |
|
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
1134 |
|
|
by |
1135 |
|
|
\[ |
1136 |
|
|
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1137 |
|
|
\] |
1138 |
|
|
In principle, many time correlation functions can be related with |
1139 |
|
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
1140 |
|
|
scattering spectra of molecular liquids. In practice, one can |
1141 |
|
|
extract the IR spectrum from the intensity of dipole fluctuation at |
1142 |
|
|
each frequency using the following relationship: |
1143 |
|
|
\[ |
1144 |
|
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1145 |
|
|
i2\pi vt} dt} |
1146 |
|
|
\] |
1147 |
tim |
2721 |
|
1148 |
tim |
2693 |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1149 |
tim |
2692 |
|
1150 |
tim |
2705 |
Rigid bodies are frequently involved in the modeling of different |
1151 |
|
|
areas, from engineering, physics, to chemistry. For example, |
1152 |
|
|
missiles and vehicle are usually modeled by rigid bodies. The |
1153 |
|
|
movement of the objects in 3D gaming engine or other physics |
1154 |
tim |
2872 |
simulator is governed by rigid body dynamics. In molecular |
1155 |
|
|
simulations, rigid bodies are used to simplify protein-protein |
1156 |
|
|
docking studies\cite{Gray2003}. |
1157 |
tim |
2694 |
|
1158 |
tim |
2705 |
It is very important to develop stable and efficient methods to |
1159 |
tim |
2872 |
integrate the equations of motion for orientational degrees of |
1160 |
|
|
freedom. Euler angles are the natural choice to describe the |
1161 |
|
|
rotational degrees of freedom. However, due to $\frac {1}{sin |
1162 |
|
|
\theta}$ singularities, the numerical integration of corresponding |
1163 |
|
|
equations of motion is very inefficient and inaccurate. Although an |
1164 |
|
|
alternative integrator using multiple sets of Euler angles can |
1165 |
|
|
overcome this difficulty\cite{Barojas1973}, the computational |
1166 |
|
|
penalty and the loss of angular momentum conservation still remain. |
1167 |
|
|
A singularity-free representation utilizing quaternions was |
1168 |
|
|
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
1169 |
|
|
approach uses a nonseparable Hamiltonian resulting from the |
1170 |
|
|
quaternion representation, which prevents the symplectic algorithm |
1171 |
|
|
to be utilized. Another different approach is to apply holonomic |
1172 |
|
|
constraints to the atoms belonging to the rigid body. Each atom |
1173 |
|
|
moves independently under the normal forces deriving from potential |
1174 |
|
|
energy and constraint forces which are used to guarantee the |
1175 |
|
|
rigidness. However, due to their iterative nature, the SHAKE and |
1176 |
|
|
Rattle algorithms also converge very slowly when the number of |
1177 |
|
|
constraints increases\cite{Ryckaert1977, Andersen1983}. |
1178 |
tim |
2694 |
|
1179 |
tim |
2872 |
A break-through in geometric literature suggests that, in order to |
1180 |
tim |
2705 |
develop a long-term integration scheme, one should preserve the |
1181 |
tim |
2872 |
symplectic structure of the flow. By introducing a conjugate |
1182 |
|
|
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
1183 |
|
|
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
1184 |
|
|
proposed to evolve the Hamiltonian system in a constraint manifold |
1185 |
|
|
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
1186 |
|
|
An alternative method using the quaternion representation was |
1187 |
|
|
developed by Omelyan\cite{Omelyan1998}. However, both of these |
1188 |
|
|
methods are iterative and inefficient. In this section, we descibe a |
1189 |
tim |
2789 |
symplectic Lie-Poisson integrator for rigid body developed by |
1190 |
|
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1191 |
tim |
2705 |
|
1192 |
tim |
2872 |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
1193 |
|
|
The motion of a rigid body is Hamiltonian with the Hamiltonian |
1194 |
tim |
2713 |
function |
1195 |
tim |
2706 |
\begin{equation} |
1196 |
|
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
1197 |
|
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
1198 |
|
|
\label{introEquation:RBHamiltonian} |
1199 |
|
|
\end{equation} |
1200 |
|
|
Here, $q$ and $Q$ are the position and rotation matrix for the |
1201 |
|
|
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
1202 |
|
|
$J$, a diagonal matrix, is defined by |
1203 |
|
|
\[ |
1204 |
|
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
1205 |
|
|
\] |
1206 |
|
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1207 |
tim |
2872 |
constrained Hamiltonian equation is subjected to a holonomic |
1208 |
|
|
constraint, |
1209 |
tim |
2706 |
\begin{equation} |
1210 |
tim |
2726 |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1211 |
tim |
2706 |
\end{equation} |
1212 |
tim |
2872 |
which is used to ensure rotation matrix's unitarity. Differentiating |
1213 |
|
|
\ref{introEquation:orthogonalConstraint} and using Equation |
1214 |
|
|
\ref{introEquation:RBMotionMomentum}, one may obtain, |
1215 |
tim |
2706 |
\begin{equation} |
1216 |
tim |
2707 |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1217 |
tim |
2706 |
\label{introEquation:RBFirstOrderConstraint} |
1218 |
|
|
\end{equation} |
1219 |
|
|
|
1220 |
|
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1221 |
|
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1222 |
|
|
the equations of motion, |
1223 |
|
|
|
1224 |
tim |
2796 |
\begin{eqnarray} |
1225 |
|
|
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1226 |
|
|
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1227 |
|
|
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1228 |
|
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1229 |
|
|
\end{eqnarray} |
1230 |
|
|
|
1231 |
tim |
2707 |
In general, there are two ways to satisfy the holonomic constraints. |
1232 |
tim |
2872 |
We can use a constraint force provided by a Lagrange multiplier on |
1233 |
|
|
the normal manifold to keep the motion on constraint space. Or we |
1234 |
|
|
can simply evolve the system on the constraint manifold. These two |
1235 |
|
|
methods have been proved to be equivalent. The holonomic constraint |
1236 |
|
|
and equations of motions define a constraint manifold for rigid |
1237 |
|
|
bodies |
1238 |
tim |
2707 |
\[ |
1239 |
|
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1240 |
|
|
\right\}. |
1241 |
|
|
\] |
1242 |
tim |
2706 |
|
1243 |
tim |
2707 |
Unfortunately, this constraint manifold is not the cotangent bundle |
1244 |
tim |
2888 |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
1245 |
|
|
rotation group $SO(3)$. However, it turns out that under symplectic |
1246 |
tim |
2707 |
transformation, the cotangent space and the phase space are |
1247 |
tim |
2872 |
diffeomorphic. By introducing |
1248 |
tim |
2706 |
\[ |
1249 |
tim |
2707 |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1250 |
tim |
2706 |
\] |
1251 |
tim |
2707 |
the mechanical system subject to a holonomic constraint manifold $M$ |
1252 |
|
|
can be re-formulated as a Hamiltonian system on the cotangent space |
1253 |
|
|
\[ |
1254 |
|
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
1255 |
|
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
1256 |
|
|
\] |
1257 |
tim |
2706 |
|
1258 |
tim |
2707 |
For a body fixed vector $X_i$ with respect to the center of mass of |
1259 |
|
|
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
1260 |
|
|
given as |
1261 |
|
|
\begin{equation} |
1262 |
|
|
X_i^{lab} = Q X_i + q. |
1263 |
|
|
\end{equation} |
1264 |
|
|
Therefore, potential energy $V(q,Q)$ is defined by |
1265 |
|
|
\[ |
1266 |
|
|
V(q,Q) = V(Q X_0 + q). |
1267 |
|
|
\] |
1268 |
tim |
2713 |
Hence, the force and torque are given by |
1269 |
tim |
2707 |
\[ |
1270 |
tim |
2713 |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)}, |
1271 |
tim |
2707 |
\] |
1272 |
tim |
2713 |
and |
1273 |
tim |
2707 |
\[ |
1274 |
|
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
1275 |
|
|
\] |
1276 |
tim |
2713 |
respectively. |
1277 |
tim |
2695 |
|
1278 |
tim |
2707 |
As a common choice to describe the rotation dynamics of the rigid |
1279 |
tim |
2872 |
body, the angular momentum on the body fixed frame $\Pi = Q^t P$ is |
1280 |
|
|
introduced to rewrite the equations of motion, |
1281 |
tim |
2707 |
\begin{equation} |
1282 |
|
|
\begin{array}{l} |
1283 |
tim |
2888 |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1284 |
|
|
\dot Q = Q\Pi {\rm{ }}J^{ - 1} \\ |
1285 |
tim |
2707 |
\end{array} |
1286 |
|
|
\label{introEqaution:RBMotionPI} |
1287 |
|
|
\end{equation} |
1288 |
|
|
, as well as holonomic constraints, |
1289 |
|
|
\[ |
1290 |
|
|
\begin{array}{l} |
1291 |
|
|
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
1292 |
|
|
Q^T Q = 1 \\ |
1293 |
|
|
\end{array} |
1294 |
|
|
\] |
1295 |
tim |
2692 |
|
1296 |
tim |
2707 |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1297 |
|
|
so(3)^ \star$, the hat-map isomorphism, |
1298 |
|
|
\begin{equation} |
1299 |
|
|
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1300 |
|
|
{\begin{array}{*{20}c} |
1301 |
|
|
0 & { - v_3 } & {v_2 } \\ |
1302 |
|
|
{v_3 } & 0 & { - v_1 } \\ |
1303 |
|
|
{ - v_2 } & {v_1 } & 0 \\ |
1304 |
|
|
\end{array}} \right), |
1305 |
|
|
\label{introEquation:hatmapIsomorphism} |
1306 |
|
|
\end{equation} |
1307 |
|
|
will let us associate the matrix products with traditional vector |
1308 |
|
|
operations |
1309 |
|
|
\[ |
1310 |
|
|
\hat vu = v \times u |
1311 |
|
|
\] |
1312 |
|
|
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1313 |
|
|
matrix, |
1314 |
tim |
2888 |
|
1315 |
tim |
2890 |
\begin{eqnarray*} |
1316 |
tim |
2888 |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
1317 |
|
|
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
1318 |
|
|
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
1319 |
|
|
\label{introEquation:skewMatrixPI} |
1320 |
|
|
\end{eqnarray*} |
1321 |
|
|
|
1322 |
tim |
2707 |
Since $\Lambda$ is symmetric, the last term of Equation |
1323 |
tim |
2713 |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1324 |
|
|
multiplier $\Lambda$ is absent from the equations of motion. This |
1325 |
tim |
2872 |
unique property eliminates the requirement of iterations which can |
1326 |
tim |
2789 |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
1327 |
tim |
2707 |
|
1328 |
tim |
2872 |
Applying the hat-map isomorphism, we obtain the equation of motion |
1329 |
|
|
for angular momentum on body frame |
1330 |
tim |
2713 |
\begin{equation} |
1331 |
|
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
1332 |
|
|
F_i (r,Q)} \right) \times X_i }. |
1333 |
|
|
\label{introEquation:bodyAngularMotion} |
1334 |
|
|
\end{equation} |
1335 |
tim |
2707 |
In the same manner, the equation of motion for rotation matrix is |
1336 |
|
|
given by |
1337 |
|
|
\[ |
1338 |
tim |
2713 |
\dot Q = Qskew(I^{ - 1} \pi ) |
1339 |
tim |
2707 |
\] |
1340 |
|
|
|
1341 |
tim |
2713 |
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
1342 |
|
|
Lie-Poisson Integrator for Free Rigid Body} |
1343 |
tim |
2707 |
|
1344 |
tim |
2872 |
If there are no external forces exerted on the rigid body, the only |
1345 |
|
|
contribution to the rotational motion is from the kinetic energy |
1346 |
|
|
(the first term of \ref{introEquation:bodyAngularMotion}). The free |
1347 |
|
|
rigid body is an example of a Lie-Poisson system with Hamiltonian |
1348 |
|
|
function |
1349 |
tim |
2713 |
\begin{equation} |
1350 |
|
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1351 |
|
|
\label{introEquation:rotationalKineticRB} |
1352 |
|
|
\end{equation} |
1353 |
|
|
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and |
1354 |
|
|
Lie-Poisson structure matrix, |
1355 |
|
|
\begin{equation} |
1356 |
|
|
J(\pi ) = \left( {\begin{array}{*{20}c} |
1357 |
|
|
0 & {\pi _3 } & { - \pi _2 } \\ |
1358 |
|
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
1359 |
|
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
1360 |
|
|
\end{array}} \right) |
1361 |
|
|
\end{equation} |
1362 |
|
|
Thus, the dynamics of free rigid body is governed by |
1363 |
|
|
\begin{equation} |
1364 |
|
|
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
1365 |
|
|
\end{equation} |
1366 |
tim |
2707 |
|
1367 |
tim |
2713 |
One may notice that each $T_i^r$ in Equation |
1368 |
|
|
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1369 |
|
|
instance, the equations of motion due to $T_1^r$ are given by |
1370 |
|
|
\begin{equation} |
1371 |
|
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
1372 |
|
|
\label{introEqaution:RBMotionSingleTerm} |
1373 |
|
|
\end{equation} |
1374 |
|
|
where |
1375 |
|
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
1376 |
|
|
0 & 0 & 0 \\ |
1377 |
|
|
0 & 0 & {\pi _1 } \\ |
1378 |
|
|
0 & { - \pi _1 } & 0 \\ |
1379 |
|
|
\end{array}} \right). |
1380 |
|
|
\] |
1381 |
|
|
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1382 |
tim |
2707 |
\[ |
1383 |
tim |
2713 |
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
1384 |
|
|
Q(0)e^{\Delta tR_1 } |
1385 |
tim |
2707 |
\] |
1386 |
tim |
2713 |
with |
1387 |
tim |
2707 |
\[ |
1388 |
tim |
2713 |
e^{\Delta tR_1 } = \left( {\begin{array}{*{20}c} |
1389 |
|
|
0 & 0 & 0 \\ |
1390 |
|
|
0 & {\cos \theta _1 } & {\sin \theta _1 } \\ |
1391 |
|
|
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
1392 |
|
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1393 |
tim |
2707 |
\] |
1394 |
tim |
2719 |
To reduce the cost of computing expensive functions in $e^{\Delta |
1395 |
tim |
2872 |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
1396 |
|
|
propagator, |
1397 |
tim |
2713 |
\[ |
1398 |
|
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1399 |
|
|
) |
1400 |
|
|
\] |
1401 |
tim |
2720 |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1402 |
tim |
2872 |
manner. In order to construct a second-order symplectic method, we |
1403 |
|
|
split the angular kinetic Hamiltonian function can into five terms |
1404 |
tim |
2707 |
\[ |
1405 |
tim |
2713 |
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1406 |
|
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1407 |
tim |
2872 |
(\pi _1 ). |
1408 |
|
|
\] |
1409 |
|
|
By concatenating the propagators corresponding to these five terms, |
1410 |
|
|
we can obtain an symplectic integrator, |
1411 |
tim |
2713 |
\[ |
1412 |
|
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
1413 |
tim |
2707 |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1414 |
|
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1415 |
tim |
2713 |
_1 }. |
1416 |
tim |
2707 |
\] |
1417 |
|
|
|
1418 |
tim |
2713 |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
1419 |
|
|
$F(\pi )$ and $G(\pi )$ is defined by |
1420 |
tim |
2707 |
\[ |
1421 |
tim |
2713 |
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
1422 |
|
|
) |
1423 |
|
|
\] |
1424 |
|
|
If the Poisson bracket of a function $F$ with an arbitrary smooth |
1425 |
|
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
1426 |
|
|
conserved quantity in Poisson system. We can easily verify that the |
1427 |
|
|
norm of the angular momentum, $\parallel \pi |
1428 |
|
|
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
1429 |
|
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
1430 |
|
|
then by the chain rule |
1431 |
|
|
\[ |
1432 |
|
|
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
1433 |
|
|
}}{2})\pi |
1434 |
|
|
\] |
1435 |
|
|
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
1436 |
|
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
1437 |
tim |
2872 |
Lie-Poisson integrator is found to be both extremely efficient and |
1438 |
|
|
stable. These properties can be explained by the fact the small |
1439 |
|
|
angle approximation is used and the norm of the angular momentum is |
1440 |
|
|
conserved. |
1441 |
tim |
2713 |
|
1442 |
|
|
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
1443 |
|
|
Splitting for Rigid Body} |
1444 |
|
|
|
1445 |
|
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
1446 |
|
|
energy and potential energy, |
1447 |
|
|
\[ |
1448 |
|
|
H = T(p,\pi ) + V(q,Q) |
1449 |
|
|
\] |
1450 |
|
|
The equations of motion corresponding to potential energy and |
1451 |
|
|
kinetic energy are listed in the below table, |
1452 |
tim |
2776 |
\begin{table} |
1453 |
tim |
2889 |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
1454 |
tim |
2713 |
\begin{center} |
1455 |
|
|
\begin{tabular}{|l|l|} |
1456 |
|
|
\hline |
1457 |
|
|
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
1458 |
|
|
Potential & Kinetic \\ |
1459 |
|
|
$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\ |
1460 |
|
|
$\frac{d}{{dt}}p = - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\ |
1461 |
|
|
$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\ |
1462 |
|
|
$ \frac{d}{{dt}}\pi = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi = \pi \times I^{ - 1} \pi$\\ |
1463 |
|
|
\hline |
1464 |
|
|
\end{tabular} |
1465 |
|
|
\end{center} |
1466 |
tim |
2776 |
\end{table} |
1467 |
tim |
2872 |
A second-order symplectic method is now obtained by the composition |
1468 |
|
|
of the position and velocity propagators, |
1469 |
tim |
2713 |
\[ |
1470 |
|
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1471 |
|
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1472 |
|
|
\] |
1473 |
tim |
2719 |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
1474 |
tim |
2872 |
sub-propagators which corresponding to force and torque |
1475 |
|
|
respectively, |
1476 |
tim |
2713 |
\[ |
1477 |
tim |
2707 |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1478 |
tim |
2713 |
_{\Delta t/2,\tau }. |
1479 |
tim |
2707 |
\] |
1480 |
tim |
2713 |
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
1481 |
tim |
2872 |
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order |
1482 |
|
|
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the |
1483 |
|
|
kinetic energy can be separated to translational kinetic term, $T^t |
1484 |
|
|
(p)$, and rotational kinetic term, $T^r (\pi )$, |
1485 |
tim |
2713 |
\begin{equation} |
1486 |
|
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
1487 |
|
|
\end{equation} |
1488 |
|
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
1489 |
|
|
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
1490 |
tim |
2872 |
corresponding propagators are given by |
1491 |
tim |
2713 |
\[ |
1492 |
|
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
1493 |
|
|
_{\Delta t,T^r }. |
1494 |
|
|
\] |
1495 |
tim |
2872 |
Finally, we obtain the overall symplectic propagators for freely |
1496 |
|
|
moving rigid bodies |
1497 |
tim |
2713 |
\begin{equation} |
1498 |
|
|
\begin{array}{c} |
1499 |
|
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
1500 |
|
|
\circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \\ |
1501 |
|
|
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1502 |
|
|
\end{array} |
1503 |
|
|
\label{introEquation:overallRBFlowMaps} |
1504 |
|
|
\end{equation} |
1505 |
tim |
2707 |
|
1506 |
tim |
2685 |
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1507 |
tim |
2716 |
As an alternative to newtonian dynamics, Langevin dynamics, which |
1508 |
|
|
mimics a simple heat bath with stochastic and dissipative forces, |
1509 |
|
|
has been applied in a variety of studies. This section will review |
1510 |
tim |
2872 |
the theory of Langevin dynamics. A brief derivation of generalized |
1511 |
|
|
Langevin equation will be given first. Following that, we will |
1512 |
|
|
discuss the physical meaning of the terms appearing in the equation |
1513 |
|
|
as well as the calculation of friction tensor from hydrodynamics |
1514 |
|
|
theory. |
1515 |
tim |
2685 |
|
1516 |
tim |
2719 |
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
1517 |
tim |
2685 |
|
1518 |
tim |
2872 |
A harmonic bath model, in which an effective set of harmonic |
1519 |
tim |
2719 |
oscillators are used to mimic the effect of a linearly responding |
1520 |
|
|
environment, has been widely used in quantum chemistry and |
1521 |
|
|
statistical mechanics. One of the successful applications of |
1522 |
tim |
2872 |
Harmonic bath model is the derivation of the Generalized Langevin |
1523 |
|
|
Dynamics (GLE). Lets consider a system, in which the degree of |
1524 |
tim |
2719 |
freedom $x$ is assumed to couple to the bath linearly, giving a |
1525 |
|
|
Hamiltonian of the form |
1526 |
tim |
2696 |
\begin{equation} |
1527 |
|
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1528 |
tim |
2719 |
\label{introEquation:bathGLE}. |
1529 |
tim |
2696 |
\end{equation} |
1530 |
tim |
2872 |
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated |
1531 |
|
|
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
1532 |
tim |
2696 |
\[ |
1533 |
tim |
2719 |
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1534 |
|
|
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
1535 |
|
|
\right\}} |
1536 |
tim |
2696 |
\] |
1537 |
tim |
2719 |
where the index $\alpha$ runs over all the bath degrees of freedom, |
1538 |
|
|
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
1539 |
tim |
2872 |
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath |
1540 |
tim |
2719 |
coupling, |
1541 |
tim |
2696 |
\[ |
1542 |
|
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1543 |
|
|
\] |
1544 |
tim |
2872 |
where $g_\alpha$ are the coupling constants between the bath |
1545 |
tim |
2874 |
coordinates ($x_ \alpha$) and the system coordinate ($x$). |
1546 |
tim |
2872 |
Introducing |
1547 |
tim |
2696 |
\[ |
1548 |
tim |
2719 |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1549 |
|
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1550 |
|
|
\] and combining the last two terms in Equation |
1551 |
|
|
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
1552 |
|
|
Hamiltonian as |
1553 |
tim |
2696 |
\[ |
1554 |
|
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1555 |
|
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1556 |
|
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1557 |
|
|
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1558 |
|
|
\] |
1559 |
|
|
Since the first two terms of the new Hamiltonian depend only on the |
1560 |
|
|
system coordinates, we can get the equations of motion for |
1561 |
tim |
2872 |
Generalized Langevin Dynamics by Hamilton's equations, |
1562 |
tim |
2719 |
\begin{equation} |
1563 |
|
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
1564 |
|
|
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
1565 |
|
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)}, |
1566 |
|
|
\label{introEquation:coorMotionGLE} |
1567 |
|
|
\end{equation} |
1568 |
|
|
and |
1569 |
|
|
\begin{equation} |
1570 |
|
|
m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - |
1571 |
|
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
1572 |
|
|
\label{introEquation:bathMotionGLE} |
1573 |
|
|
\end{equation} |
1574 |
tim |
2696 |
|
1575 |
tim |
2719 |
In order to derive an equation for $x$, the dynamics of the bath |
1576 |
|
|
variables $x_\alpha$ must be solved exactly first. As an integral |
1577 |
|
|
transform which is particularly useful in solving linear ordinary |
1578 |
tim |
2872 |
differential equations,the Laplace transform is the appropriate tool |
1579 |
|
|
to solve this problem. The basic idea is to transform the difficult |
1580 |
tim |
2719 |
differential equations into simple algebra problems which can be |
1581 |
tim |
2872 |
solved easily. Then, by applying the inverse Laplace transform, also |
1582 |
|
|
known as the Bromwich integral, we can retrieve the solutions of the |
1583 |
tim |
2719 |
original problems. |
1584 |
tim |
2696 |
|
1585 |
tim |
2719 |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
1586 |
|
|
transform of f(t) is a new function defined as |
1587 |
tim |
2696 |
\[ |
1588 |
tim |
2719 |
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1589 |
tim |
2696 |
\] |
1590 |
tim |
2719 |
where $p$ is real and $L$ is called the Laplace Transform |
1591 |
|
|
Operator. Below are some important properties of Laplace transform |
1592 |
tim |
2696 |
|
1593 |
tim |
2789 |
\begin{eqnarray*} |
1594 |
|
|
L(x + y) & = & L(x) + L(y) \\ |
1595 |
|
|
L(ax) & = & aL(x) \\ |
1596 |
|
|
L(\dot x) & = & pL(x) - px(0) \\ |
1597 |
|
|
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
1598 |
|
|
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
1599 |
|
|
\end{eqnarray*} |
1600 |
|
|
|
1601 |
|
|
|
1602 |
tim |
2872 |
Applying the Laplace transform to the bath coordinates, we obtain |
1603 |
tim |
2789 |
\begin{eqnarray*} |
1604 |
|
|
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1605 |
|
|
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1606 |
|
|
\end{eqnarray*} |
1607 |
|
|
|
1608 |
tim |
2719 |
By the same way, the system coordinates become |
1609 |
tim |
2789 |
\begin{eqnarray*} |
1610 |
|
|
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1611 |
|
|
& & \mbox{} - \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1612 |
|
|
\end{eqnarray*} |
1613 |
tim |
2696 |
|
1614 |
tim |
2719 |
With the help of some relatively important inverse Laplace |
1615 |
|
|
transformations: |
1616 |
tim |
2696 |
\[ |
1617 |
tim |
2719 |
\begin{array}{c} |
1618 |
|
|
L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ |
1619 |
|
|
L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ |
1620 |
|
|
L(1) = \frac{1}{p} \\ |
1621 |
|
|
\end{array} |
1622 |
tim |
2696 |
\] |
1623 |
tim |
2719 |
, we obtain |
1624 |
tim |
2794 |
\begin{eqnarray*} |
1625 |
|
|
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
1626 |
tim |
2696 |
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1627 |
|
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
1628 |
tim |
2794 |
_\alpha t)\dot x(t - \tau )d\tau } } \right\}} \\ |
1629 |
|
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1630 |
|
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1631 |
|
|
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1632 |
|
|
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1633 |
|
|
\end{eqnarray*} |
1634 |
|
|
\begin{eqnarray*} |
1635 |
|
|
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1636 |
tim |
2696 |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1637 |
|
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1638 |
tim |
2794 |
t)\dot x(t - \tau )d} \tau } \\ |
1639 |
|
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1640 |
|
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1641 |
|
|
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1642 |
|
|
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1643 |
|
|
\end{eqnarray*} |
1644 |
tim |
2719 |
Introducing a \emph{dynamic friction kernel} |
1645 |
tim |
2696 |
\begin{equation} |
1646 |
tim |
2719 |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1647 |
|
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
1648 |
|
|
\label{introEquation:dynamicFrictionKernelDefinition} |
1649 |
|
|
\end{equation} |
1650 |
|
|
and \emph{a random force} |
1651 |
|
|
\begin{equation} |
1652 |
|
|
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
1653 |
|
|
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
1654 |
|
|
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
1655 |
|
|
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), |
1656 |
|
|
\label{introEquation:randomForceDefinition} |
1657 |
|
|
\end{equation} |
1658 |
|
|
the equation of motion can be rewritten as |
1659 |
|
|
\begin{equation} |
1660 |
tim |
2696 |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
1661 |
|
|
(t)\dot x(t - \tau )d\tau } + R(t) |
1662 |
|
|
\label{introEuqation:GeneralizedLangevinDynamics} |
1663 |
|
|
\end{equation} |
1664 |
tim |
2719 |
which is known as the \emph{generalized Langevin equation}. |
1665 |
|
|
|
1666 |
tim |
2819 |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
1667 |
tim |
2719 |
|
1668 |
|
|
One may notice that $R(t)$ depends only on initial conditions, which |
1669 |
|
|
implies it is completely deterministic within the context of a |
1670 |
|
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
1671 |
|
|
uncorrelated to $x$ and $\dot x$, |
1672 |
tim |
2696 |
\[ |
1673 |
tim |
2719 |
\begin{array}{l} |
1674 |
|
|
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
1675 |
|
|
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
1676 |
|
|
\end{array} |
1677 |
tim |
2696 |
\] |
1678 |
tim |
2719 |
This property is what we expect from a truly random process. As long |
1679 |
tim |
2872 |
as the model chosen for $R(t)$ was a gaussian distribution in |
1680 |
|
|
general, the stochastic nature of the GLE still remains. |
1681 |
tim |
2696 |
|
1682 |
tim |
2719 |
%dynamic friction kernel |
1683 |
|
|
The convolution integral |
1684 |
tim |
2696 |
\[ |
1685 |
tim |
2719 |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } |
1686 |
tim |
2696 |
\] |
1687 |
tim |
2719 |
depends on the entire history of the evolution of $x$, which implies |
1688 |
|
|
that the bath retains memory of previous motions. In other words, |
1689 |
|
|
the bath requires a finite time to respond to change in the motion |
1690 |
|
|
of the system. For a sluggish bath which responds slowly to changes |
1691 |
|
|
in the system coordinate, we may regard $\xi(t)$ as a constant |
1692 |
|
|
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes |
1693 |
|
|
\[ |
1694 |
|
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
1695 |
|
|
\] |
1696 |
|
|
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1697 |
|
|
\[ |
1698 |
|
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
1699 |
|
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
1700 |
|
|
\] |
1701 |
tim |
2872 |
which can be used to describe the effect of dynamic caging in |
1702 |
|
|
viscous solvents. The other extreme is the bath that responds |
1703 |
|
|
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
1704 |
|
|
taken as a $delta$ function in time: |
1705 |
tim |
2719 |
\[ |
1706 |
|
|
\xi (t) = 2\xi _0 \delta (t) |
1707 |
|
|
\] |
1708 |
|
|
Hence, the convolution integral becomes |
1709 |
|
|
\[ |
1710 |
|
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
1711 |
|
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
1712 |
|
|
\] |
1713 |
|
|
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1714 |
|
|
\begin{equation} |
1715 |
|
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
1716 |
|
|
x(t) + R(t) \label{introEquation:LangevinEquation} |
1717 |
|
|
\end{equation} |
1718 |
|
|
which is known as the Langevin equation. The static friction |
1719 |
|
|
coefficient $\xi _0$ can either be calculated from spectral density |
1720 |
tim |
2850 |
or be determined by Stokes' law for regular shaped particles. A |
1721 |
tim |
2719 |
briefly review on calculating friction tensor for arbitrary shaped |
1722 |
tim |
2720 |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
1723 |
tim |
2696 |
|
1724 |
tim |
2819 |
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
1725 |
tim |
2719 |
|
1726 |
|
|
Defining a new set of coordinates, |
1727 |
tim |
2696 |
\[ |
1728 |
|
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1729 |
|
|
^2 }}x(0) |
1730 |
tim |
2719 |
\], |
1731 |
|
|
we can rewrite $R(T)$ as |
1732 |
tim |
2696 |
\[ |
1733 |
tim |
2719 |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1734 |
tim |
2696 |
\] |
1735 |
|
|
And since the $q$ coordinates are harmonic oscillators, |
1736 |
tim |
2789 |
|
1737 |
|
|
\begin{eqnarray*} |
1738 |
|
|
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1739 |
|
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1740 |
|
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1741 |
|
|
\left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1742 |
|
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1743 |
|
|
& = &kT\xi (t) \\ |
1744 |
|
|
\end{eqnarray*} |
1745 |
|
|
|
1746 |
tim |
2719 |
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1747 |
tim |
2696 |
\begin{equation} |
1748 |
|
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1749 |
tim |
2719 |
\label{introEquation:secondFluctuationDissipation}. |
1750 |
tim |
2696 |
\end{equation} |
1751 |
tim |
2719 |
In effect, it acts as a constraint on the possible ways in which one |
1752 |
|
|
can model the random force and friction kernel. |