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\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND} |
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\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
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As a special discipline of molecular modeling, Molecular dynamics |
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has proven to be a powerful tool for studying the functions of |
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biological systems, providing structural, thermodynamic and |
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dynamical information. |
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\section{\label{introSection:classicalMechanics}Classical |
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Mechanics} |
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sufficient to predict the future behavior of the system. |
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\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
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The discovery of Newton's three laws of mechanics which govern the |
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motion of particles is the foundation of the classical mechanics. |
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Newton¡¯s first law defines a class of inertial frames. Inertial |
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frames are reference frames where a particle not interacting with |
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other bodies will move with constant speed in the same direction. |
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With respect to inertial frames Newton¡¯s second law has the form |
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\begin{equation} |
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F = \frac {dp}{dt} = \frac {mv}{dt} |
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\label{introEquation:newtonSecondLaw} |
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\end{equation} |
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A point mass interacting with other bodies moves with the |
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acceleration along the direction of the force acting on it. Let |
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$F_ij$ be the force that particle $i$ exerts on particle $j$, and |
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$F_ji$ be the force that particle $j$ exerts on particle $i$. |
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Newton¡¯s third law states that |
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\begin{equation} |
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F_ij = -F_ji |
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\label{introEquation:newtonThirdLaw} |
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\end{equation} |
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|
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Conservation laws of Newtonian Mechanics play very important roles |
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in solving mechanics problems. The linear momentum of a particle is |
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conserved if it is free or it experiences no force. The second |
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conservation theorem concerns the angular momentum of a particle. |
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The angular momentum $L$ of a particle with respect to an origin |
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from which $r$ is measured is defined to be |
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\begin{equation} |
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L \equiv r \times p \label{introEquation:angularMomentumDefinition} |
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\end{equation} |
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The torque $\tau$ with respect to the same origin is defined to be |
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\begin{equation} |
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N \equiv r \times F \label{introEquation:torqueDefinition} |
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\end{equation} |
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Differentiating Eq.~\ref{introEquation:angularMomentumDefinition}, |
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\[ |
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\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times |
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\dot p) |
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\] |
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since |
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\[ |
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\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0 |
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\] |
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thus, |
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\begin{equation} |
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\dot L = r \times \dot p = N |
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\end{equation} |
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If there are no external torques acting on a body, the angular |
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momentum of it is conserved. The last conservation theorem state |
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that if all forces are conservative, Energy |
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\begin{equation}E = T + V \label{introEquation:energyConservation} |
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\end{equation} |
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is conserved. All of these conserved quantities are |
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important factors to determine the quality of numerical integration |
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scheme for rigid body \cite{Dullweber1997}. |
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|
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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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Newtonian Mechanics suffers from two important limitations: it |
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which arise in attempts to apply Newton's equation to complex |
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system, alternative procedures may be developed. |
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\subsection{\label{introSection:halmiltonPrinciple}Hamilton's |
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\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
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Principle} |
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Hamilton introduced the dynamical principle upon which it is |
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The actual trajectory, along which a dynamical system may move from |
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one point to another within a specified time, is derived by finding |
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the path which minimizes the time integral of the difference between |
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the kinetic, $K$, and potential energies, $U$. |
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the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
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\begin{equation} |
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\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
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\label{introEquation:halmitonianPrinciple1} |
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\label{introEquation:halmitonianPrinciple2} |
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\end{equation} |
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\subsection{\label{introSection:equationOfMotionLagrangian}The |
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\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
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Equations of Motion in Lagrangian Mechanics} |
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for a holonomic system of $f$ degrees of freedom, the equations of |
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For a holonomic system of $f$ degrees of freedom, the equations of |
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motion in the Lagrangian form is |
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\begin{equation} |
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\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - |
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Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
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Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
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equation of motion. Due to their symmetrical formula, they are also |
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known as the canonical equations of motions. |
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known as the canonical equations of motions \cite{Goldstein01}. |
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An important difference between Lagrangian approach and the |
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Hamiltonian approach is that the Lagrangian is considered to be a |
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appropriate for application to statistical mechanics and quantum |
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mechanics, since it treats the coordinate and its time derivative as |
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independent variables and it only works with 1st-order differential |
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equations. |
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equations\cite{Marion90}. |
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|
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\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
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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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\subsection{\label{introSection:canonicalTransformation}Canonical |
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Transformation} |
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\section{\label{introSection:statisticalMechanics}Statistical |
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Mechanics} |
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The thermodynamic behaviors and properties of Molecular Dynamics |
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The thermodynamic behaviors and properties of Molecular Dynamics |
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simulation are governed by the principle of Statistical Mechanics. |
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The following section will give a brief introduction to some of the |
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Statistical Mechanics concepts presented in this dissertation. |
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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}Ensemble} |
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\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
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|
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Mathematically, phase space is the space which represents all |
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possible states. Each possible state of the system corresponds to |
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one unique point in the phase space. For mechanical systems, the |
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phase space usually consists of all possible values of position and |
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momentum variables. Consider a dynamic system in a cartesian space, |
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where each of the $6f$ coordinates and momenta is assigned to one of |
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$6f$ mutually orthogonal axes, the phase space of this system is a |
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$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , |
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\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and |
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momenta is a phase space vector. |
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|
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A microscopic state or microstate of a classical system is |
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specification of the complete phase space vector of a system at any |
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instant in time. An ensemble is defined as a collection of systems |
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sharing one or more macroscopic characteristics but each being in a |
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unique microstate. The complete ensemble is specified by giving all |
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systems or microstates consistent with the common macroscopic |
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characteristics of the ensemble. Although the state of each |
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individual system in the ensemble could be precisely described at |
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any instance in time by a suitable phase space vector, when using |
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ensembles for statistical purposes, there is no need to maintain |
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distinctions between individual systems, since the numbers of |
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systems at any time in the different states which correspond to |
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different regions of the phase space are more interesting. Moreover, |
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in the point of view of statistical mechanics, one would prefer to |
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use ensembles containing a large enough population of separate |
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members so that the numbers of systems in such different states can |
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be regarded as changing continuously as we traverse different |
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regions of the phase space. The condition of an ensemble at any time |
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can be regarded as appropriately specified by the density $\rho$ |
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with which representative points are distributed over the phase |
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space. The density of distribution for an ensemble with $f$ degrees |
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of freedom is defined as, |
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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 value which would change the density at any time at phase |
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space. Hence, the density of distribution is also to be taken as a |
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function of the time. |
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|
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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 are exploited, we can |
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sufficiently approximate $\delta N$ without introducing |
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discontinuity when we go from one region in the phase space to |
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another. By integrating over the whole phase space, |
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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 Equation(\ref{introEquation:unitProbability}) and |
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the knowledge of the system, it is possible to calculate the average |
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value of any desired quantity which depends on the coordinates and |
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momenta of the system. Even when the dynamics of the real system is |
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complex, or stochastic, or even discontinuous, the average |
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properties of the ensemble of possibilities as a whole may still |
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remain well defined. For a classical system in thermal equilibrium |
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with its environment, the ensemble average of a mechanical quantity, |
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$\langle A(q , p) \rangle_t$, takes the form of an integral over the |
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phase space of the system, |
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\begin{equation} |
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\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
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(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
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(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
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\label{introEquation:ensembelAverage} |
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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}, partition function can |
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be used to describe the statistical properties of a system in |
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thermodynamic equilibrium. |
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|
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As an ensemble of systems, each of which is known to be thermally |
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isolated and conserve energy, Microcanonical ensemble(NVE) has a |
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partition function like, |
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\begin{equation} |
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\Omega (N,V,E) = e^{\beta TS} |
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\label{introEqaution: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 experiment are carried out under constant pressure |
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condition, isothermal-isobaric ensemble(NPT) play a very important |
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role in molecular simulation. The isothermal-isobaric ensemble allow |
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the system to exchange energy with a heat bath of temperature $T$ |
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and to change the volume as well. Its partition function is given as |
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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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|
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\subsection{\label{introSection:liouville}Liouville's theorem} |
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|
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The Liouville's theorem is the foundation on which statistical |
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mechanics rests. It describes the time evolution of phase space |
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distribution function. In order to calculate the rate of change of |
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$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
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consider the two faces perpendicular to the $q_1$ axis, which are |
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located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
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leaving the opposite face is given by the expression, |
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\begin{equation} |
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\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
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\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
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}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 |
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\ldots \delta p_f . |
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\end{equation} |
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Summing all over the phase space, we obtain |
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\begin{equation} |
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\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho |
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\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + |
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\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( |
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{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial |
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\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 |
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\ldots \delta q_f \delta p_1 \ldots \delta p_f . |
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\end{equation} |
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Differentiating the equations of motion in Hamiltonian formalism |
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(\ref{introEquation:motionHamiltonianCoordinate}, |
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\ref{introEquation:motionHamiltonianMomentum}), we can show, |
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\begin{equation} |
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\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} |
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+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
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\end{equation} |
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which cancels the first terms of the right hand side. Furthermore, |
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divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
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p_f $ in both sides, we can write out Liouville's theorem in a |
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simple form, |
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\begin{equation} |
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\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f |
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{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + |
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\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
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\label{introEquation:liouvilleTheorem} |
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\end{equation} |
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|
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Liouville's theorem states that the distribution function is |
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constant along any trajectory in phase space. In classical |
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statistical mechanics, since the number of particles in the system |
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is huge, we may be able to believe the system is stationary, |
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\begin{equation} |
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\frac{{\partial \rho }}{{\partial t}} = 0. |
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\label{introEquation:stationary} |
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\end{equation} |
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In such stationary system, the density of distribution $\rho$ can be |
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connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
| 393 |
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distribution, |
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\begin{equation} |
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\rho \propto e^{ - \beta H} |
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\label{introEquation:densityAndHamiltonian} |
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\end{equation} |
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|
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Liouville's theorem can be expresses in a variety of different forms |
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which are convenient within different contexts. For any two function |
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$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
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bracket ${F, G}$ is defined as |
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\begin{equation} |
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\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
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F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
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\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial |
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q_i }}} \right)}. |
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\label{introEquation:poissonBracket} |
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\end{equation} |
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Substituting equations of motion in Hamiltonian formalism( |
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\ref{introEquation:motionHamiltonianCoordinate} , |
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\ref{introEquation:motionHamiltonianMomentum} ) into |
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(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
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theorem using Poisson bracket notion, |
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+ |
\begin{equation} |
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+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
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{\rho ,H} \right\}. |
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\label{introEquation:liouvilleTheromInPoissin} |
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\end{equation} |
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Moreover, the Liouville operator is defined as |
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\begin{equation} |
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iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial |
| 423 |
+ |
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial |
| 424 |
+ |
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} |
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\label{introEquation:liouvilleOperator} |
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+ |
\end{equation} |
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In terms of Liouville operator, Liouville's equation can also be |
| 428 |
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expressed as |
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+ |
\begin{equation} |
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+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
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+ |
\label{introEquation:liouvilleTheoremInOperator} |
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+ |
\end{equation} |
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+ |
|
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+ |
|
| 435 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
| 436 |
|
|
| 437 |
+ |
Various thermodynamic properties can be calculated from Molecular |
| 438 |
+ |
Dynamics simulation. By comparing experimental values with the |
| 439 |
+ |
calculated properties, one can determine the accuracy of the |
| 440 |
+ |
simulation and the quality of the underlying model. However, both of |
| 441 |
+ |
experiment and computer simulation are usually performed during a |
| 442 |
+ |
certain time interval and the measurements are averaged over a |
| 443 |
+ |
period of them which is different from the average behavior of |
| 444 |
+ |
many-body system in Statistical Mechanics. Fortunately, Ergodic |
| 445 |
+ |
Hypothesis is proposed to make a connection between time average and |
| 446 |
+ |
ensemble average. It states that time average and average over the |
| 447 |
+ |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
| 448 |
+ |
\begin{equation} |
| 449 |
+ |
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 450 |
+ |
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 451 |
+ |
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp |
| 452 |
+ |
\end{equation} |
| 453 |
+ |
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
| 454 |
+ |
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
| 455 |
+ |
distribution function. If an observation is averaged over a |
| 456 |
+ |
sufficiently long time (longer than relaxation time), all accessible |
| 457 |
+ |
microstates in phase space are assumed to be equally probed, giving |
| 458 |
+ |
a properly weighted statistical average. This allows the researcher |
| 459 |
+ |
freedom of choice when deciding how best to measure a given |
| 460 |
+ |
observable. In case an ensemble averaged approach sounds most |
| 461 |
+ |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
| 462 |
+ |
utilized. Or if the system lends itself to a time averaging |
| 463 |
+ |
approach, the Molecular Dynamics techniques in |
| 464 |
+ |
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 465 |
+ |
choice\cite{Frenkel1996}. |
| 466 |
+ |
|
| 467 |
+ |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 468 |
+ |
A variety of numerical integrators were proposed to simulate the |
| 469 |
+ |
motions. They usually begin with an initial conditionals and move |
| 470 |
+ |
the objects in the direction governed by the differential equations. |
| 471 |
+ |
However, most of them ignore the hidden physical law contained |
| 472 |
+ |
within the equations. Since 1990, geometric integrators, which |
| 473 |
+ |
preserve various phase-flow invariants such as symplectic structure, |
| 474 |
+ |
volume and time reversal symmetry, are developed to address this |
| 475 |
+ |
issue. The velocity verlet method, which happens to be a simple |
| 476 |
+ |
example of symplectic integrator, continues to gain its popularity |
| 477 |
+ |
in molecular dynamics community. This fact can be partly explained |
| 478 |
+ |
by its geometric nature. |
| 479 |
+ |
|
| 480 |
+ |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 481 |
+ |
A \emph{manifold} is an abstract mathematical space. It locally |
| 482 |
+ |
looks like Euclidean space, but when viewed globally, it may have |
| 483 |
+ |
more complicate structure. A good example of manifold is the surface |
| 484 |
+ |
of Earth. It seems to be flat locally, but it is round if viewed as |
| 485 |
+ |
a whole. A \emph{differentiable manifold} (also known as |
| 486 |
+ |
\emph{smooth manifold}) is a manifold with an open cover in which |
| 487 |
+ |
the covering neighborhoods are all smoothly isomorphic to one |
| 488 |
+ |
another. In other words,it is possible to apply calculus on |
| 489 |
+ |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
| 490 |
+ |
defined as a pair $(M, \omega)$ which consisting of a |
| 491 |
+ |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 492 |
+ |
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 493 |
+ |
space $V$ is a function $\omega(x, y)$ which satisfies |
| 494 |
+ |
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 495 |
+ |
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 496 |
+ |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
| 497 |
+ |
example of symplectic form. |
| 498 |
+ |
|
| 499 |
+ |
One of the motivations to study \emph{symplectic manifold} in |
| 500 |
+ |
Hamiltonian Mechanics is that a symplectic manifold can represent |
| 501 |
+ |
all possible configurations of the system and the phase space of the |
| 502 |
+ |
system can be described by it's cotangent bundle. Every symplectic |
| 503 |
+ |
manifold is even dimensional. For instance, in Hamilton equations, |
| 504 |
+ |
coordinate and momentum always appear in pairs. |
| 505 |
+ |
|
| 506 |
+ |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
| 507 |
+ |
\[ |
| 508 |
+ |
f : M \rightarrow N |
| 509 |
+ |
\] |
| 510 |
+ |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
| 511 |
+ |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
| 512 |
+ |
Canonical transformation is an example of symplectomorphism in |
| 513 |
+ |
classical mechanics. |
| 514 |
+ |
|
| 515 |
+ |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 516 |
+ |
|
| 517 |
+ |
For a ordinary differential system defined as |
| 518 |
+ |
\begin{equation} |
| 519 |
+ |
\dot x = f(x) |
| 520 |
+ |
\end{equation} |
| 521 |
+ |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
| 522 |
+ |
\begin{equation} |
| 523 |
+ |
f(r) = J\nabla _x H(r). |
| 524 |
+ |
\end{equation} |
| 525 |
+ |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
| 526 |
+ |
matrix |
| 527 |
+ |
\begin{equation} |
| 528 |
+ |
J = \left( {\begin{array}{*{20}c} |
| 529 |
+ |
0 & I \\ |
| 530 |
+ |
{ - I} & 0 \\ |
| 531 |
+ |
\end{array}} \right) |
| 532 |
+ |
\label{introEquation:canonicalMatrix} |
| 533 |
+ |
\end{equation} |
| 534 |
+ |
where $I$ is an identity matrix. Using this notation, Hamiltonian |
| 535 |
+ |
system can be rewritten as, |
| 536 |
+ |
\begin{equation} |
| 537 |
+ |
\frac{d}{{dt}}x = J\nabla _x H(x) |
| 538 |
+ |
\label{introEquation:compactHamiltonian} |
| 539 |
+ |
\end{equation}In this case, $f$ is |
| 540 |
+ |
called a \emph{Hamiltonian vector field}. |
| 541 |
+ |
|
| 542 |
+ |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
| 543 |
+ |
\begin{equation} |
| 544 |
+ |
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 545 |
+ |
\end{equation} |
| 546 |
+ |
The most obvious change being that matrix $J$ now depends on $x$. |
| 547 |
+ |
The free rigid body is an example of Poisson system (actually a |
| 548 |
+ |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
| 549 |
+ |
energy. |
| 550 |
+ |
\begin{equation} |
| 551 |
+ |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 552 |
+ |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 553 |
+ |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 554 |
+ |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 555 |
+ |
\end{array}} \right) |
| 556 |
+ |
\end{equation} |
| 557 |
+ |
|
| 558 |
+ |
\begin{equation} |
| 559 |
+ |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
| 560 |
+ |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
| 561 |
+ |
\end{equation} |
| 562 |
+ |
|
| 563 |
+ |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 564 |
+ |
Let $x(t)$ be the exact solution of the ODE system, |
| 565 |
+ |
\begin{equation} |
| 566 |
+ |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
| 567 |
+ |
\end{equation} |
| 568 |
+ |
The exact flow(solution) $\varphi_\tau$ is defined by |
| 569 |
+ |
\[ |
| 570 |
+ |
x(t+\tau) =\varphi_\tau(x(t)) |
| 571 |
+ |
\] |
| 572 |
+ |
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 573 |
+ |
space to itself. In most cases, it is not easy to find the exact |
| 574 |
+ |
flow $\varphi_\tau$. Instead, we use a approximate map, $\psi_\tau$, |
| 575 |
+ |
which is usually called integrator. The order of an integrator |
| 576 |
+ |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
| 577 |
+ |
order $p$, |
| 578 |
+ |
\begin{equation} |
| 579 |
+ |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 580 |
+ |
\end{equation} |
| 581 |
+ |
|
| 582 |
+ |
The hidden geometric properties of ODE and its flow play important |
| 583 |
+ |
roles in numerical studies. Let $\varphi$ be the flow of Hamiltonian |
| 584 |
+ |
vector field, $\varphi$ is a \emph{symplectic} flow if it satisfies, |
| 585 |
+ |
\begin{equation} |
| 586 |
+ |
'\varphi^T J '\varphi = J. |
| 587 |
+ |
\end{equation} |
| 588 |
+ |
According to Liouville's theorem, the symplectic volume is invariant |
| 589 |
+ |
under a Hamiltonian flow, which is the basis for classical |
| 590 |
+ |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
| 591 |
+ |
field on a symplectic manifold can be shown to be a |
| 592 |
+ |
symplectomorphism. As to the Poisson system, |
| 593 |
+ |
\begin{equation} |
| 594 |
+ |
'\varphi ^T J '\varphi = J \circ \varphi |
| 595 |
+ |
\end{equation} |
| 596 |
+ |
is the property must be preserved by the integrator. It is possible |
| 597 |
+ |
to construct a \emph{volume-preserving} flow for a source free($ |
| 598 |
+ |
\nabla \cdot f = 0 $) ODE, if the flow satisfies $ \det d\varphi = |
| 599 |
+ |
1$. Changing the variables $y = h(x)$ in a |
| 600 |
+ |
ODE\ref{introEquation:ODE} will result in a new system, |
| 601 |
+ |
\[ |
| 602 |
+ |
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 603 |
+ |
\] |
| 604 |
+ |
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 605 |
+ |
In other words, the flow of this vector field is reversible if and |
| 606 |
+ |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. When |
| 607 |
+ |
designing any numerical methods, one should always try to preserve |
| 608 |
+ |
the structural properties of the original ODE and its flow. |
| 609 |
+ |
|
| 610 |
+ |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 611 |
+ |
A lot of well established and very effective numerical methods have |
| 612 |
+ |
been successful precisely because of their symplecticities even |
| 613 |
+ |
though this fact was not recognized when they were first |
| 614 |
+ |
constructed. The most famous example is leapfrog methods in |
| 615 |
+ |
molecular dynamics. In general, symplectic integrators can be |
| 616 |
+ |
constructed using one of four different methods. |
| 617 |
+ |
\begin{enumerate} |
| 618 |
+ |
\item Generating functions |
| 619 |
+ |
\item Variational methods |
| 620 |
+ |
\item Runge-Kutta methods |
| 621 |
+ |
\item Splitting methods |
| 622 |
+ |
\end{enumerate} |
| 623 |
+ |
|
| 624 |
+ |
Generating function tends to lead to methods which are cumbersome |
| 625 |
+ |
and difficult to use\cite{}. In dissipative systems, variational |
| 626 |
+ |
methods can capture the decay of energy accurately\cite{}. Since |
| 627 |
+ |
their geometrically unstable nature against non-Hamiltonian |
| 628 |
+ |
perturbations, ordinary implicit Runge-Kutta methods are not |
| 629 |
+ |
suitable for Hamiltonian system. Recently, various high-order |
| 630 |
+ |
explicit Runge--Kutta methods have been developed to overcome this |
| 631 |
+ |
instability \cite{}. However, due to computational penalty involved |
| 632 |
+ |
in implementing the Runge-Kutta methods, they do not attract too |
| 633 |
+ |
much attention from Molecular Dynamics community. Instead, splitting |
| 634 |
+ |
have been widely accepted since they exploit natural decompositions |
| 635 |
+ |
of the system\cite{Tuckerman92}. The main idea behind splitting |
| 636 |
+ |
methods is to decompose the discrete $\varphi_h$ as a composition of |
| 637 |
+ |
simpler flows, |
| 638 |
+ |
\begin{equation} |
| 639 |
+ |
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 640 |
+ |
\varphi _{h_n } |
| 641 |
+ |
\label{introEquation:FlowDecomposition} |
| 642 |
+ |
\end{equation} |
| 643 |
+ |
where each of the sub-flow is chosen such that each represent a |
| 644 |
+ |
simpler integration of the system. Let $\phi$ and $\psi$ both be |
| 645 |
+ |
symplectic maps, it is easy to show that any composition of |
| 646 |
+ |
symplectic flows yields a symplectic map, |
| 647 |
+ |
\begin{equation} |
| 648 |
+ |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 649 |
+ |
'\phi ' = \phi '^T J\phi ' = J. |
| 650 |
+ |
\label{introEquation:SymplecticFlowComposition} |
| 651 |
+ |
\end{equation} |
| 652 |
+ |
Suppose that a Hamiltonian system has a form with $H = T + V$ |
| 653 |
+ |
|
| 654 |
+ |
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 655 |
+ |
|
| 656 |
+ |
As a special discipline of molecular modeling, Molecular dynamics |
| 657 |
+ |
has proven to be a powerful tool for studying the functions of |
| 658 |
+ |
biological systems, providing structural, thermodynamic and |
| 659 |
+ |
dynamical information. |
| 660 |
+ |
|
| 661 |
+ |
\subsection{\label{introSec:mdInit}Initialization} |
| 662 |
+ |
|
| 663 |
+ |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 664 |
+ |
|
| 665 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 666 |
|
|
| 667 |
+ |
A rigid body is a body in which the distance between any two given |
| 668 |
+ |
points of a rigid body remains constant regardless of external |
| 669 |
+ |
forces exerted on it. A rigid body therefore conserves its shape |
| 670 |
+ |
during its motion. |
| 671 |
+ |
|
| 672 |
+ |
Applications of dynamics of rigid bodies. |
| 673 |
+ |
|
| 674 |
+ |
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 675 |
+ |
|
| 676 |
+ |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 677 |
+ |
|
| 678 |
+ |
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
| 679 |
+ |
|
| 680 |
+ |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
| 681 |
+ |
|
| 682 |
|
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 683 |
|
|
| 684 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 685 |
|
|
| 686 |
+ |
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 687 |
+ |
|
| 688 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
| 689 |
|
|
| 690 |
< |
\subsection{\label{introSection:hydroynamics}Hydrodynamics} |
| 690 |
> |
\begin{equation} |
| 691 |
> |
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 692 |
> |
\label{introEquation:bathGLE} |
| 693 |
> |
\end{equation} |
| 694 |
> |
where $H_B$ is harmonic bath Hamiltonian, |
| 695 |
> |
\[ |
| 696 |
> |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 697 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
| 698 |
> |
\] |
| 699 |
> |
and $\Delta U$ is bilinear system-bath coupling, |
| 700 |
> |
\[ |
| 701 |
> |
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 702 |
> |
\] |
| 703 |
> |
Completing the square, |
| 704 |
> |
\[ |
| 705 |
> |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
| 706 |
> |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 707 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 708 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
| 709 |
> |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 710 |
> |
\] |
| 711 |
> |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
| 712 |
> |
\[ |
| 713 |
> |
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 714 |
> |
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 715 |
> |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 716 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 717 |
> |
\] |
| 718 |
> |
where |
| 719 |
> |
\[ |
| 720 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 721 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 722 |
> |
\] |
| 723 |
> |
Since the first two terms of the new Hamiltonian depend only on the |
| 724 |
> |
system coordinates, we can get the equations of motion for |
| 725 |
> |
Generalized Langevin Dynamics by Hamilton's equations |
| 726 |
> |
\ref{introEquation:motionHamiltonianCoordinate, |
| 727 |
> |
introEquation:motionHamiltonianMomentum}, |
| 728 |
> |
\begin{align} |
| 729 |
> |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 730 |
> |
&= m\ddot x |
| 731 |
> |
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
| 732 |
> |
\label{introEq:Lp5} |
| 733 |
> |
\end{align} |
| 734 |
> |
, and |
| 735 |
> |
\begin{align} |
| 736 |
> |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
| 737 |
> |
&= m\ddot x_\alpha |
| 738 |
> |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
| 739 |
> |
\end{align} |
| 740 |
> |
|
| 741 |
> |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
| 742 |
> |
|
| 743 |
> |
\[ |
| 744 |
> |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
| 745 |
> |
\] |
| 746 |
> |
|
| 747 |
> |
\[ |
| 748 |
> |
L(x + y) = L(x) + L(y) |
| 749 |
> |
\] |
| 750 |
> |
|
| 751 |
> |
\[ |
| 752 |
> |
L(ax) = aL(x) |
| 753 |
> |
\] |
| 754 |
> |
|
| 755 |
> |
\[ |
| 756 |
> |
L(\dot x) = pL(x) - px(0) |
| 757 |
> |
\] |
| 758 |
> |
|
| 759 |
> |
\[ |
| 760 |
> |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
| 761 |
> |
\] |
| 762 |
> |
|
| 763 |
> |
\[ |
| 764 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
| 765 |
> |
\] |
| 766 |
> |
|
| 767 |
> |
Some relatively important transformation, |
| 768 |
> |
\[ |
| 769 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
| 770 |
> |
\] |
| 771 |
> |
|
| 772 |
> |
\[ |
| 773 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
| 774 |
> |
\] |
| 775 |
> |
|
| 776 |
> |
\[ |
| 777 |
> |
L(1) = \frac{1}{p} |
| 778 |
> |
\] |
| 779 |
> |
|
| 780 |
> |
First, the bath coordinates, |
| 781 |
> |
\[ |
| 782 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
| 783 |
> |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
| 784 |
> |
}}L(x) |
| 785 |
> |
\] |
| 786 |
> |
\[ |
| 787 |
> |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
| 788 |
> |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
| 789 |
> |
\] |
| 790 |
> |
Then, the system coordinates, |
| 791 |
> |
\begin{align} |
| 792 |
> |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 793 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
| 794 |
> |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
| 795 |
> |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
| 796 |
> |
}}\omega _\alpha ^2 L(x)} \right\}} |
| 797 |
> |
% |
| 798 |
> |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
| 799 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
| 800 |
> |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
| 801 |
> |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
| 802 |
> |
\end{align} |
| 803 |
> |
Then, the inverse transform, |
| 804 |
> |
|
| 805 |
> |
\begin{align} |
| 806 |
> |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
| 807 |
> |
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 808 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
| 809 |
> |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
| 810 |
> |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
| 811 |
> |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 812 |
> |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
| 813 |
> |
% |
| 814 |
> |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 815 |
> |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 816 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 817 |
> |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
| 818 |
> |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
| 819 |
> |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
| 820 |
> |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
| 821 |
> |
(\omega _\alpha t)} \right\}} |
| 822 |
> |
\end{align} |
| 823 |
> |
|
| 824 |
> |
\begin{equation} |
| 825 |
> |
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
| 826 |
> |
(t)\dot x(t - \tau )d\tau } + R(t) |
| 827 |
> |
\label{introEuqation:GeneralizedLangevinDynamics} |
| 828 |
> |
\end{equation} |
| 829 |
> |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
| 830 |
> |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
| 831 |
> |
\[ |
| 832 |
> |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 833 |
> |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
| 834 |
> |
\] |
| 835 |
> |
For an infinite harmonic bath, we can use the spectral density and |
| 836 |
> |
an integral over frequencies. |
| 837 |
> |
|
| 838 |
> |
\[ |
| 839 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
| 840 |
> |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
| 841 |
> |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
| 842 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
| 843 |
> |
\] |
| 844 |
> |
The random forces depend only on initial conditions. |
| 845 |
> |
|
| 846 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 847 |
> |
So we can define a new set of coordinates, |
| 848 |
> |
\[ |
| 849 |
> |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 850 |
> |
^2 }}x(0) |
| 851 |
> |
\] |
| 852 |
> |
This makes |
| 853 |
> |
\[ |
| 854 |
> |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
| 855 |
> |
\] |
| 856 |
> |
And since the $q$ coordinates are harmonic oscillators, |
| 857 |
> |
\[ |
| 858 |
> |
\begin{array}{l} |
| 859 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 860 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 861 |
> |
\end{array} |
| 862 |
> |
\] |
| 863 |
> |
|
| 864 |
> |
\begin{align} |
| 865 |
> |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
| 866 |
> |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
| 867 |
> |
(t)q_\beta (0)} \right\rangle } } |
| 868 |
> |
% |
| 869 |
> |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
| 870 |
> |
\right\rangle \cos (\omega _\alpha t)} |
| 871 |
> |
% |
| 872 |
> |
&= kT\xi (t) |
| 873 |
> |
\end{align} |
| 874 |
> |
|
| 875 |
> |
\begin{equation} |
| 876 |
> |
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 877 |
> |
\label{introEquation:secondFluctuationDissipation} |
| 878 |
> |
\end{equation} |
| 879 |
> |
|
| 880 |
> |
\section{\label{introSection:hydroynamics}Hydrodynamics} |
| 881 |
> |
|
| 882 |
> |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 883 |
> |
\subsection{\label{introSection:analyticalApproach}Analytical |
| 884 |
> |
Approach} |
| 885 |
> |
|
| 886 |
> |
\subsection{\label{introSection:approximationApproach}Approximation |
| 887 |
> |
Approach} |
| 888 |
> |
|
| 889 |
> |
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 890 |
> |
Body} |
