# Cheat sheets/Statistical mechanics

Statistical mechanics is a branch of mathematical physics that studies, using probability theory, the average behaviour of a mechanical system where the state of the system is uncertain. A common use of statistical mechanics is in explaining the thermodynamic behaviour of large systems.[1]

## Thermal wavelength

${\displaystyle \lambda ={\frac {h}{\sqrt {2\pi mk_{B}T}}}=h{\sqrt {\frac {\beta }{2\pi m}}}=\hbar {\sqrt {\frac {2\pi \beta }{m}}}}$
${\displaystyle {1 \over \lambda }={\frac {\sqrt {2\pi mk_{B}T}}{h}}}$

## Partition function

Canonical: (in 3d)

${\displaystyle Z=\int \prod _{i=1}^{N}{\frac {d^{3}q_{i}d^{3}p_{i}}{h^{3}}}e^{-\beta {\frac {p_{i}^{2}}{2m}}}e^{-\beta U(q_{i})}=\int \prod _{i=1}^{N}{\frac {d^{3}q_{i}d^{3}p_{i}}{h^{3}}}e^{-\beta H}}$