Boltzmann distribution

The Boltzmann distribution is a fundamental concept in Classical Statistical Mechanics that describes the distribution of states (microstates) in a system at thermal equilibrium. It is the distribution that, given a fixed expected value, maximize entropy.

Given a system with states $X$ and a so called energy function $E : X \to R$ , Boltzmann distribution is given by the probability density function/probability mass function:

p (x) = \frac{e^{- \frac{E (x)}{k T}}}{Z}

where:

$k$ is Boltzmann's constant,
$T$ is a parameter, called the absolute temperature of the system, and
$Z$ is a normalization factor, called partition function.

For the discrete case, see this interactive simulation, and this other for the continuous case.