Boltzmann Distribution

$P({\epsilon }_s)=\frac{e^{-\beta {\epsilon }_s}}{Z}$

• Microcanonical ensemble
• Probability of a system occupying a state $s$ with energy ${\epsilon }_s$. A state $s$ may refer to the spin of the spin magnet system for example
• $\beta$ is [coldness] of the system
• $Z$ is the [partition function]

$P(N,\epsilon )=\frac{e^{\frac{\mu N-\epsilon }{\tau }}}{\mathbb{Z}}$

• Boltzmann distribution for grand canonincal ensemble
• The probability of a system of $N$ particles at energy $\epsilon$
• $\mathbb{Z}$ is the [grand partition function]