Boltzmann Distribution

#Physics

$\displaystyle P(\varepsilon_{s})=\frac{e^{-\beta\varepsilon_{s}}}{Z}$

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

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

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