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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. As an ensemble of systems, each of which |
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is known to be thermally isolated and conserve energy, the |
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Microcanonical ensemble (NVE) has 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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|
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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 |
