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\title{Statistical Mechanics Quick Reference}
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\section*{Micro-Canonical Ensemble}
A completely Isolated System in V, N, T, and E.

\subsection*{Phase Space}

There is a Phase Space with $\sum_i^N 3p_i+3q_i = 6N$ dimensions.  Points
in this space represent all possible momenta and positions for all N
particles.  A given energy defines \emph{accessible states} in the space,
between E and $\delta E$.  Accessible states are a hypersphere in that
space. $\Omega (E)$ is the volume of accessible states.  It may also be
thought of as the number of accessible states.
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Entropy:

\[S = k_B \ln \Omega \]

This relation for S can be used to find many other things through
thermodynamic equations of state.

\section*{Canonical Ensemble}
A system kept at constant T. V and N are isolated.  Heat may flow in and
out to maintain T.  E may vary.  May be thought of as a box of isolated
particles inside of a microcanonical ensemble.  In that case, only the
portion of the phase space in which these N particles are in the
canonical box counts.

The Canonical Ensemble useful for weakly interacting systems all at 
some ``Temperature'' like equilibrium with their environment.

\subsection*{Partition Function}

\begin{eqnarray*} 
Z = \sum_i e^{-\beta E_i} = \sum_j g_j e^{-\beta E_j}
= \frac{1}{N! h^{3N}} \times \\
\int e^{-\beta H(p_1,\ldots,p_{3N},q_1,\ldots,q_{3N})}\,dp_1\cdots 
dp_{3N}\,dq_1 \cdots dq_{3N}
\end{eqnarray*}

Where $\beta = \frac{1}{k_B T}$ and $g_j$ is the degeneracy.
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The Boltzmann Distribution is given by:
\[P_i = \frac{e^{-\beta E_i}}{Z}\]
It gives the probability of any microstate $i$ (point in phase space).

\[\langle E \rangle = U = - \frac{\partial \ln Z}{\partial \beta}\]
\[ \sigma^2_E = \frac{\partial^2 \ln Z}{\partial \beta^2}  \]
\[  S = \frac{\partial (\beta \ln Z)}{\partial T} 
= \frac{\langle E \rangle}{T} + k_B \ln Z\]

Important is the relation between the Helmholtz Free Energy, $A$,
and $Z$. $A$ is the \emph{work available} 

\[A = -\frac{\ln Z}{\beta} \Rightarrow Z = e^{- \beta A}\]

Other useful quantities in terms of Z (Many derived through A):

\begin{eqnarray*}
P & = & \frac{1}{\beta} 
\left( \frac{\partial \ln Z}{\partial V} \right)_{N,T} \\
S & = & k (\ln Z + \beta U)\\
C_V & = & \frac{\sigma^2_E }{k T^2}\\
\mu & = & -\frac{1}{\beta} 
\left( \frac{\partial \ln Z}{\partial N} \right)_{N,T} \\
\end{eqnarray*}

\section*{Grand Canonical Ensemble}

A system kept at constant T and $\mu$, chemical potential.  V is
isolated.  Heat and particles may flow in and out to maintain T and 
$\mu$. E and N may vary.

\subsection*{Grand Partition Function}

\begin{eqnarray*}
\Xi \mbox{ or } \mathcal{Z} $ = $ 
\sum_{N=0}^\infty \sum_i e^{\beta \mu N}e^{-\beta E_i}  \\
$ = $ \sum_{N=0}^\infty z^N Z(N,V,T) \\
$ = $ \sum_{N=0}^\infty \frac{z^N}{N ! h^{3N}} 
\int d\Omega_N e^{-\beta H}
\end{eqnarray*}
$z$ is the fugacity (easiness of adding a particle to the system):
$z = e^{\beta \mu}$

It can be used similar to the normal Partition Function.

Useful Relations:

\begin{eqnarray*}
\langle E \rangle $ = $ (\frac{\mu}{\beta} \frac{\partial}{\partial \mu}
 - \frac{\partial}{\partial \beta}) \ln (\Xi) \\
\langle N \rangle $ = $ \frac{1}{\beta} \frac{\partial}{\partial \mu}
\ln (\Xi) \\
\sigma_N^2 $ = $ \frac{1}{\beta} 
\frac{\partial \langle N \rangle}{\partial \mu}
\end{eqnarray*}

Grand Free Energy is used just like the original free energy.
\[\Phi = -\frac{\ln \Xi}{\beta} \Rightarrow \Xi = e^{- \beta \Phi}\]
\[d\Phi = -SdT - PdV - Nd\mu\]

\section*{Entropy}

Entropy is the measure of disorder, ignorance, or irriversability of the system.
If a system is fully specified, then $S=0$.

\subsection*{Discrete}

For discrete states with distinct probabilities $p_i$,

\[S = k \sum_i p_i \ln p_i\]

$k$ is $k_b$ for thermal systems and can also be $k = \ln (2)$ for entropy
measured in bits (Information Entropy).

\section*{Acknowledgments}

Almost all of this is taken from Dr. Tanner's class lectures.  A few
parts are from Sethna's Book \emph{Statistical Mechanics: Entropy, Order
Parameters and Complexity}. Wikipedia helped too.

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