Important Formulas for Statistical Mechanics

Thermal Wavelemgth

\begin{equation} \lambda = \frac{h}{\sqrt{2\pi m k_B T}} \end{equation}

Fugacity

\begin{equation} z = e^{\beta\mu}=e^{\frac{\mu}{k_B T}} \end{equation}

Classical Limit

Classical Formulas

\begin{equation} U = \frac{3}{2} PV \end{equation}

Corrected EOS

\begin{equation} PV = Nk_B T [1 -\zeta 2^{-5/2} \frac{n\lambda^3}{g}+ \cdots] \end{equation}

Fermi and Bose Functions

\begin{equation} f_k (z)=\sum_{l=1}^{\infty}(-1)^{l+1}\frac{z^l}{l^k} \end{equation} \begin{equation} g_k (z)=\sum_{l=1}^{\infty}\frac{z^l}{l^k} \end{equation}

Parametric EOS (Important)

For Fermi case: \begin{equation} \frac{\lambda^3 P}{k_B T} =g f_{5/2}(z) \end{equation} \begin{equation} \lambda^3 n=g f_{3/2}(z) \end{equation} \begin{equation} U = \frac{3}{2}PV \end{equation} For Bose case changing \(f(z)\) to \(g(z)\).

Fermi and Bose Distribution

\begin{equation} \bar n_j = \frac{g_j}{e^{-\alpha +\beta\epsilon_j}-\zeta} \end{equation} where \(\alpha = \beta\mu\) and \(\beta = \frac{1}{k_B T}\).

Partition Functions

\begin{equation} Z_N(V,T) = \sum_s e^{-\beta E_s}= Tr(e^{-\beta \hat H}) \end{equation} \begin{equation} \Xi (z,V,T) = \sum_{N=0}^{\infty}z^N Z_N(V,T)=Tr(e^{-\beta(\hat H-\mu \hat N)}) \end{equation}

Density Operator

\begin{equation} \hat\rho = \frac{1}{Z}e^{-\beta\hat H} \end{equation} \begin{equation} \hat\rho = \frac{1}{\Xi}e^{-\beta(\hat H-\mu\hat N)} \end{equation}

Von Neumann Entropy

\begin{equation} S=-Tr(\rho \ln\rho) \end{equation}

Quantum Ideal Gas

Grand Partition Function

\begin{equation} \Xi = \prod_i (1-\zeta e^{-\beta(E_i -\mu)})^{-\zeta} \end{equation}

Average Particle Number

\begin{equation} \langle N\rangle = \frac{1}{\beta}\frac{\partial \ln \Xi}{\partial \mu}=\sum_{i}\frac{1}{e^{\beta(E_i -\mu)}-\zeta} \end{equation}

Fermi Gas

Fermi Energy

\begin{equation} E_F = \frac{\hbar^2}{2m}(\frac{6\pi^2}{g}\frac{N}{V})^{\frac{2}{3}} = \frac{\hbar^2 k_F^2}{2m} \end{equation}

Fermi Tmeperature

\begin{equation} T_F = \frac{E_F}{k_B} \end{equation}

Fermi Degeneracy Pressure

\begin{equation} P_0 = \frac{2}{5}\frac{N}{V}E_F \end{equation}

Black-Body Radiation

Partition Function

\begin{equation} Z=\prod_{\vec k,\alpha}\frac{e^{-\beta \hbar ck/2}}{1-e^{-\beta\hbar ck}} \end{equation}

Occupation Number

\begin{equation} \langle n_\alpha (k)\rangle = -\frac{1}{\beta}\frac{\partial}{\partial (\hbar\omega)}\ln Z = \frac{1}{e^{\beta\hbar\omega}-1} \end{equation}

Excitation Energy

\begin{equation} U=\frac{2V}{(2\pi)^3}\int d^3 \vec k \frac{\hbar ck}{e^{\beta\hbar ck}-1} \end{equation} \begin{equation} U=3PV \end{equation}

Energy Flux Density

\begin{equation} \phi = \frac{c E}{4V}=\frac{\pi^2 k_B^4 T^4}{60\hbar^3 c^2}=\sigma T^4 \end{equation}

Planck Distribution

\begin{equation} \epsilon (k,T)=\frac{\hbar c}{\pi^2}\frac{k^3}{e^{\beta\hbar\omega}-1} \end{equation} \begin{equation} \frac{E}{V}=\int \epsilon(k,T) dk \end{equation}