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biological systems, providing structural, thermodynamic and |
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dynamical information. |
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|
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\subsection{\label{introSection:classicalMechanics}Classical Mechanics} |
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\section{\label{introSection:classicalMechanics}Classical |
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Mechanics} |
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|
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Closely related to Classical Mechanics, Molecular Dynamics |
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simulations are carried out by integrating the equations of motion |
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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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\subsubsection{\label{introSection:newtonian}Newtonian Mechanics} |
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\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
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|
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\subsubsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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|
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Newtonian Mechanics suffers from two important limitations: it |
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describes their motion in special cartesian coordinate systems. |
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which arise in attempts to apply Newton's equation to complex |
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system, alternative procedures may be developed. |
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|
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\subsubsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
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\subsection{\label{introSection:halmiltonPrinciple}Hamilton's |
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|
Principle} |
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|
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Hamilton introduced the dynamical principle upon which it is |
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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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< |
\lable{introEquation:halmitonianPrinciple1} |
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> |
\label{introEquation:halmitonianPrinciple1} |
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\end{equation} |
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|
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For simple mechanical systems, where the forces acting on the |
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\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 } {K dt = 0} , |
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< |
\lable{introEquation:halmitonianPrinciple2} |
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> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
67 |
> |
\label{introEquation:halmitonianPrinciple2} |
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\end{equation} |
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|
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< |
\subsubsubsection{\label{introSection:equationOfMotionLagrangian}The |
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> |
\subsection{\label{introSection:equationOfMotionLagrangian}The |
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Equations of Motion in Lagrangian Mechanics} |
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|
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for a holonomic system of $f$ degrees of freedom, the equations of |
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\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 |
78 |
< |
\lable{introEquation:eqMotionLagrangian} |
78 |
> |
\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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< |
\subsubsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} |
83 |
> |
\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics} |
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|
|
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Arising from Lagrangian Mechanics, Hamiltonian Mechanics was |
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|
introduced by William Rowan Hamilton in 1833 as a re-formulation of |
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p_i = \frac{\partial L}{\partial \dot q_i} |
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|
\label{introEquation:generalizedMomenta} |
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\end{equation} |
94 |
< |
With the help of these momenta, we may now define a new quantity $H$ |
94 |
< |
by the equation |
94 |
> |
The Lagrange equations of motion are then expressed by |
95 |
|
\begin{equation} |
96 |
< |
H = p_1 \dot q_1 + \ldots + p_f \dot q_f - L, |
96 |
> |
p_i = \frac{{\partial L}}{{\partial q_i }} |
97 |
> |
\label{introEquation:generalizedMomentaDot} |
98 |
> |
\end{equation} |
99 |
> |
|
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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 |
102 |
> |
\begin{equation} |
103 |
> |
H = \sum\limits_k {p_k \dot q_k } - L , |
104 |
|
\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. |
108 |
|
|
109 |
+ |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
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+ |
one can obtain |
111 |
+ |
\begin{equation} |
112 |
+ |
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 |
114 |
+ |
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
115 |
+ |
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
116 |
+ |
\end{equation} |
117 |
+ |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
118 |
+ |
second and fourth terms in the parentheses cancel. Therefore, |
119 |
+ |
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
120 |
+ |
\begin{equation} |
121 |
+ |
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
122 |
+ |
\right)} - \frac{{\partial L}}{{\partial t}}dt |
123 |
+ |
\label{introEquation:diffHamiltonian2} |
124 |
+ |
\end{equation} |
125 |
+ |
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
126 |
+ |
find |
127 |
+ |
\begin{equation} |
128 |
+ |
\frac{{\partial H}}{{\partial p_k }} = q_k |
129 |
+ |
\label{introEquation:motionHamiltonianCoordinate} |
130 |
+ |
\end{equation} |
131 |
+ |
\begin{equation} |
132 |
+ |
\frac{{\partial H}}{{\partial q_k }} = - p_k |
133 |
+ |
\label{introEquation:motionHamiltonianMomentum} |
134 |
+ |
\end{equation} |
135 |
+ |
and |
136 |
+ |
\begin{equation} |
137 |
+ |
\frac{{\partial H}}{{\partial t}} = - \frac{{\partial L}}{{\partial |
138 |
+ |
t}} |
139 |
+ |
\label{introEquation:motionHamiltonianTime} |
140 |
+ |
\end{equation} |
141 |
+ |
|
142 |
+ |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
143 |
+ |
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
144 |
+ |
equation of motion. Due to their symmetrical formula, they are also |
145 |
+ |
known as the canonical equations of motions. |
146 |
+ |
|
147 |
|
An important difference between Lagrangian approach and the |
148 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
149 |
|
function of the generalized velocities $\dot q_i$ and the |
155 |
|
independent variables and it only works with 1st-order differential |
156 |
|
equations. |
157 |
|
|
158 |
+ |
\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
159 |
|
|
160 |
< |
\subsubsection{\label{introSection:canonicalTransformation}Canonical Transformation} |
160 |
> |
\subsection{\label{introSection:canonicalTransformation}Canonical |
161 |
> |
Transformation} |
162 |
|
|
163 |
< |
\subsection{\label{introSection:statisticalMechanics}Statistical Mechanics} |
163 |
> |
\section{\label{introSection:statisticalMechanics}Statistical |
164 |
> |
Mechanics} |
165 |
|
|
166 |
|
The thermodynamic behaviors and properties of Molecular Dynamics |
167 |
|
simulation are governed by the principle of Statistical Mechanics. |
168 |
|
The following section will give a brief introduction to some of the |
169 |
|
Statistical Mechanics concepts presented in this dissertation. |
170 |
|
|
171 |
< |
\subsubsection{\label{introSection::ensemble}Ensemble} |
171 |
> |
\subsection{\label{introSection::ensemble}Ensemble} |
172 |
|
|
173 |
< |
\subsubsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
173 |
> |
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
174 |
|
|
175 |
< |
\subsection{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
175 |
> |
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
176 |
|
|
177 |
< |
\subsection{\label{introSection:correlationFunctions}Correlation Functions} |
177 |
> |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
178 |
|
|
179 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
180 |
|
|