--- trunk/tengDissertation/Introduction.tex 2006/04/10 05:35:55 2699 +++ trunk/tengDissertation/Introduction.tex 2006/04/11 03:38:09 2700 @@ -117,7 +117,7 @@ for a holonomic system of $f$ degrees of freedom, the \subsubsection{\label{introSection:equationOfMotionLagrangian}The Equations of Motion in Lagrangian Mechanics} -for a holonomic system of $f$ degrees of freedom, the equations of +For a holonomic system of $f$ degrees of freedom, the equations of motion in the Lagrangian form is \begin{equation} \frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - @@ -221,10 +221,217 @@ Statistical Mechanics concepts presented in this disse The thermodynamic behaviors and properties of Molecular Dynamics simulation are governed by the principle of Statistical Mechanics. The following section will give a brief introduction to some of the -Statistical Mechanics concepts presented in this dissertation. +Statistical Mechanics concepts and theorem presented in this +dissertation. -\subsection{\label{introSection:ensemble}Ensemble and Phase Space} +\subsection{\label{introSection:ensemble}Phase Space and Ensemble} + +Mathematically, phase space is the space which represents all +possible states. Each possible state of the system corresponds to +one unique point in the phase space. For mechanical systems, the +phase space usually consists of all possible values of position and +momentum variables. Consider a dynamic system in a cartesian space, +where each of the $6f$ coordinates and momenta is assigned to one of +$6f$ mutually orthogonal axes, the phase space of this system is a +$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , +\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and +momenta is a phase space vector. + +A microscopic state or microstate of a classical system is +specification of the complete phase space vector of a system at any +instant in time. An ensemble is defined as a collection of systems +sharing one or more macroscopic characteristics but each being in a +unique microstate. The complete ensemble is specified by giving all +systems or microstates consistent with the common macroscopic +characteristics of the ensemble. Although the state of each +individual system in the ensemble could be precisely described at +any instance in time by a suitable phase space vector, when using +ensembles for statistical purposes, there is no need to maintain +distinctions between individual systems, since the numbers of +systems at any time in the different states which correspond to +different regions of the phase space are more interesting. Moreover, +in the point of view of statistical mechanics, one would prefer to +use ensembles containing a large enough population of separate +members so that the numbers of systems in such different states can +be regarded as changing continuously as we traverse different +regions of the phase space. The condition of an ensemble at any time +can be regarded as appropriately specified by the density $\rho$ +with which representative points are distributed over the phase +space. The density of distribution for an ensemble with $f$ degrees +of freedom is defined as, +\begin{equation} +\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). +\label{introEquation:densityDistribution} +\end{equation} +Governed by the principles of mechanics, the phase points change +their value which would change the density at any time at phase +space. Hence, the density of distribution is also to be taken as a +function of the time. + +The number of systems $\delta N$ at time $t$ can be determined by, +\begin{equation} +\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. +\label{introEquation:deltaN} +\end{equation} +Assuming a large enough population of systems are exploited, we can +sufficiently approximate $\delta N$ without introducing +discontinuity when we go from one region in the phase space to +another. By integrating over the whole phase space, +\begin{equation} +N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f +\label{introEquation:totalNumberSystem} +\end{equation} +gives us an expression for the total number of the systems. Hence, +the probability per unit in the phase space can be obtained by, +\begin{equation} +\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int +{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. +\label{introEquation:unitProbability} +\end{equation} +With the help of Equation(\ref{introEquation:unitProbability}) and +the knowledge of the system, it is possible to calculate the average +value of any desired quantity which depends on the coordinates and +momenta of the system. Even when the dynamics of the real system is +complex, or stochastic, or even discontinuous, the average +properties of the ensemble of possibilities as a whole may still +remain well defined. For a classical system in thermal equilibrium +with its environment, the ensemble average of a mechanical quantity, +$\langle A(q , p) \rangle_t$, takes the form of an integral over the +phase space of the system, +\begin{equation} +\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho +(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho +(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} +\label{introEquation:ensembelAverage} +\end{equation} + +There are several different types of ensembles with different +statistical characteristics. As a function of macroscopic +parameters, such as temperature \textit{etc}, partition function can +be used to describe the statistical properties of a system in +thermodynamic equilibrium. + +As an ensemble of systems, each of which is known to be thermally +isolated and conserve energy, Microcanonical ensemble(NVE) has a +partition function like, +\begin{equation} +\Omega (N,V,E) = e^{\beta TS} +\label{introEqaution:NVEPartition}. +\end{equation} +A canonical ensemble(NVT)is an ensemble of systems, each of which +can share its energy with a large heat reservoir. The distribution +of the total energy amongst the possible dynamical states is given +by the partition function, +\begin{equation} +\Omega (N,V,T) = e^{ - \beta A} +\label{introEquation:NVTPartition} +\end{equation} +Here, $A$ is the Helmholtz free energy which is defined as $ A = U - +TS$. Since most experiment are carried out under constant pressure +condition, isothermal-isobaric ensemble(NPT) play a very important +role in molecular simulation. The isothermal-isobaric ensemble allow +the system to exchange energy with a heat bath of temperature $T$ +and to change the volume as well. Its partition function is given as +\begin{equation} +\Delta (N,P,T) = - e^{\beta G}. + \label{introEquation:NPTPartition} +\end{equation} +Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. + +\subsection{\label{introSection:liouville}Liouville's theorem} + +The Liouville's theorem is the foundation on which statistical +mechanics rests. It describes the time evolution of phase space +distribution function. In order to calculate the rate of change of +$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we +consider the two faces perpendicular to the $q_1$ axis, which are +located at $q_1$ and $q_1 + \delta q_1$, the number of phase points +leaving the opposite face is given by the expression, +\begin{equation} +\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } +\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 +}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 +\ldots \delta p_f . +\end{equation} +Summing all over the phase space, we obtain +\begin{equation} +\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho +\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + +\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( +{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial +\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 +\ldots \delta q_f \delta p_1 \ldots \delta p_f . +\end{equation} +Differentiating the equations of motion in Hamiltonian formalism +(\ref{introEquation:motionHamiltonianCoordinate}, +\ref{introEquation:motionHamiltonianMomentum}), we can show, +\begin{equation} +\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} ++ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , +\end{equation} +which cancels the first terms of the right hand side. Furthermore, +divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta +p_f $ in both sides, we can write out Liouville's theorem in a +simple form, +\begin{equation} +\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f +{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + +\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . +\label{introEquation:liouvilleTheorem} +\end{equation} +Liouville's theorem states that the distribution function is +constant along any trajectory in phase space. In classical +statistical mechanics, since the number of particles in the system +is huge, we may be able to believe the system is stationary, +\begin{equation} +\frac{{\partial \rho }}{{\partial t}} = 0. +\label{introEquation:stationary} +\end{equation} +In such stationary system, the density of distribution $\rho$ can be +connected to the Hamiltonian $H$ through Maxwell-Boltzmann +distribution, +\begin{equation} +\rho \propto e^{ - \beta H} +\label{introEquation:densityAndHamiltonian} +\end{equation} + +Liouville's theorem can be expresses in a variety of different forms +which are convenient within different contexts. For any two function +$F$ and $G$ of the coordinates and momenta of a system, the Poisson +bracket ${F, G}$ is defined as +\begin{equation} +\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial +F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - +\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial +q_i }}} \right)}. +\label{introEquation:poissonBracket} +\end{equation} +Substituting equations of motion in Hamiltonian formalism( +\ref{introEquation:motionHamiltonianCoordinate} , +\ref{introEquation:motionHamiltonianMomentum} ) into +(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's +theorem using Poisson bracket notion, +\begin{equation} +\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ +{\rho ,H} \right\}. +\label{introEquation:liouvilleTheromInPoissin} +\end{equation} +Moreover, the Liouville operator is defined as +\begin{equation} +iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial +p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial +H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} +\label{introEquation:liouvilleOperator} +\end{equation} +In terms of Liouville operator, Liouville's equation can also be +expressed as +\begin{equation} +\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho +\label{introEquation:liouvilleTheoremInOperator} +\end{equation} + + \subsection{\label{introSection:ergodic}The Ergodic Hypothesis} Various thermodynamic properties can be calculated from Molecular @@ -239,22 +446,23 @@ statistical ensemble are identical \cite{Frenkel1996, ensemble average. It states that time average and average over the statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. \begin{equation} -\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty } -\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma -{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq +\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } +\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma +{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp \end{equation} -where $\langle A \rangle_t$ is an equilibrium value of a physical -quantity and $\rho (p(t), q(t))$ is the equilibrium distribution -function. If an observation is averaged over a sufficiently long -time (longer than relaxation time), all accessible microstates in -phase space are assumed to be equally probed, giving a properly -weighted statistical average. This allows the researcher freedom of -choice when deciding how best to measure a given observable. In case -an ensemble averaged approach sounds most reasonable, the Monte -Carlo techniques\cite{metropolis:1949} can be utilized. Or if the -system lends itself to a time averaging approach, the Molecular -Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} -will be the best choice\cite{Frenkel1996}. +where $\langle A(q , p) \rangle_t$ is an equilibrium value of a +physical quantity and $\rho (p(t), q(t))$ is the equilibrium +distribution function. If an observation is averaged over a +sufficiently long time (longer than relaxation time), all accessible +microstates in phase space are assumed to be equally probed, giving +a properly weighted statistical average. This allows the researcher +freedom of choice when deciding how best to measure a given +observable. In case an ensemble averaged approach sounds most +reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be +utilized. Or if the system lends itself to a time averaging +approach, the Molecular Dynamics techniques in +Sec.~\ref{introSection:molecularDynamics} will be the best +choice\cite{Frenkel1996}. \section{\label{introSection:geometricIntegratos}Geometric Integrators} A variety of numerical integrators were proposed to simulate the @@ -442,8 +650,6 @@ Suppose that a Hamiltonian system has a form with $H = \label{introEquation:SymplecticFlowComposition} \end{equation} Suppose that a Hamiltonian system has a form with $H = T + V$ - - \section{\label{introSection:molecularDynamics}Molecular Dynamics}