= Boltzmann distribution
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{title2=$P_i=Z^{-1}e^{-\beta E_i}$}
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The <Boltzmann distribution> assigns <probabilities> to individual <microstates> in a <canonical ensemble>, a <statistical ensemble> for thermal contact. The system exchanges <energy> with a <thermal reservoir> at fixed positive absolute <temperature> $T$, while its particle number $N$ and external parameters, such as <volume>, stay fixed. The <thermal reservoir> is large enough that these exchanges barely change its <temperature>; the coupling permits <thermal equilibrium> but contributes negligible interaction <energy>. All the <microstates> considered have the same $N$. No <chemical potential> term is needed in their weights; fixed $N$ does not imply that the <chemical potential> is zero.
For a finite discrete set of <microstates> with <energies> $E_i$, write
$$
p_i=\frac{e^{-\beta E_i}}{Z},\qquad
\beta=\frac1{k_BT},\qquad
Z=\sum_i e^{-\beta E_i},\qquad
\sum_i p_i=1.
$$
Here $k_B$ is the <Boltzmann constant>, $\beta$ is the <inverse temperature>, $e^{-\beta E_i}$ is a <Boltzmann factor>, and $Z$ is the <partition function>. The <Boltzmann factor> is a weight; dividing by the <partition function> makes it a <probability>. https://ocw.mit.edu/courses/5-62-physical-chemistry-ii-spring-2008/ec4f40d5a0fa41df3692f076e850543d_02_562ln08.pdf[MIT 5.62 Physical Chemistry II, spring 2008, lecture 2, pages 4–7: fixed N, V, T and canonical probabilities]{ref}
The useful surprise is that a fixed <temperature> still permits <thermal fluctuations> in the system's <energy>. The <Boltzmann distribution> predicts their <probabilities> and <expected values> without tracking every microscopic exchange. Its <exponential function> also makes rare excited <microstates> sensitive to <temperature>: an <energy> gap much larger than $k_BT$ gives strong <Boltzmann suppression>, while warming increases their relative populations. A population ratio can therefore reveal a gap or serve as a <thermometer>, provided <thermal equilibrium> and the relevant <state degeneracies> are known.
Historically, <Josiah Willard Gibbs>'s \i[Elementary Principles in Statistical Mechanics] (1902) presents the <canonical ensemble> in Chapter IV. His equations 90–92 give an <exponential function> of <energy> and its normalization, in classical <phase space>. This classical formulation exhibits the same weighting principle. https://en.wikisource.org/wiki/Elementary_Principles_in_Statistical_Mechanics[Gibbs (1902), title page and preface]{ref} https://en.wikisource.org/wiki/Elementary_Principles_in_Statistical_Mechanics/Chapter_IV[Gibbs (1902), Chapter IV, equations 90–92]{ref}
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