Background: When introducing exponential families in class today, my professor said that the cumulant generating function in the definition is essentially a normalizing constant, or a renormalization term after applying an exponential tilt.
This sounded to me a lot like the partition function from statistical mechanics, which is/was usually described as a normalizing constant in the papers I have read about mentioning it.
Question: Are the Boltzmann distributions from statistical mechanics an exponential family? Are the partition functions of these distributions related to the cumulant generating functions in the definition of exponential families? (Or is the normalizing constant similarity only superficial?)
Note: There are several relevant pages on Wikipedia which reference each other as external links, but none of them explain explicitly the relationship between the two concepts.
Wikipedia pages (1)(2)(3)(4)(5)
Definitions: An $s$-parameter exponential family is a family of probability densities $\{ p_{\eta}: \eta \in \Xi \}$ with respect to the measure $\mu$ on the measurable space $(X, \mathscr{F})$ of the following form: $$ p_{\eta}(x) = \exp \{ \eta^T T(x) - A(\eta) \} h(x) $$ where:
- $T: X \to \mathbb{R}^s$ is the sufficient statistic.
- $h: X \to \mathbb{R}$ is the carrier/base density.
- $\eta \in \Xi \subseteq \mathbb{R}^s$ is the natural parameter.
- $A: \Xi \to \mathbb{R}$ is the cumulant generating function.
(See chapter 2 of Keener: Theoretical Statistics: Topics for a Core Course.)
The (Maxwell-)Boltzmann probability distribution corresponding to parameter $a>0$ has the PDF: $$p_a(x) =\sqrt{\frac{2}{\pi}}\frac{x^2 e^{-x^2/(2a^2)}}{a^3} \,. $$ The partition function corresponding to this (I think) is: $$ \int e^{-\beta H_a(x)} $$ for an appropriate function $H_a(x)$. All of the relevant links on Wikipedia reference to each other on the bottom of the page, but no explicit connection is given.
