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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)

Related questions: (1)(2)

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.

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    $\begingroup$ What makes you unable to answer this question, whether the function $h(x) g(\theta) e^{\eta(\theta) T(x)}$, with $h(x)=x^2$, $g(\theta)=\sqrt{\tfrac{2}{\pi}} \theta^{-3}$, $T(x)=x^2$, and $\eta(\theta) = \frac{-1}{2 a^2}$ is an exponential family or not? $\endgroup$ – Martijn Weterings Sep 1 '17 at 9:26
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    $\begingroup$ @MartijnWeterings (1) I was and still am having difficulty finding a probabilistic definition for the partition function -- the definition on Wikipedia is in terms of energy, but I don't know how to interpret that in terms of the probability density given. (2) Also, apparently the cumulant generating function for the distribution is actually $A(\eta +u) - A(\eta)$, and calling $A(\eta)$ the "cumulant generating function" is just shorthand/sloppiness/abuse of terminology. (3) Also I am not sure what the natural parameter space would be $\mathbb{R}_+$? Is $a$ even the natural parameter? $\endgroup$ – Chill2Macht Sep 1 '17 at 15:30
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Could you elaborate on those three issues in your question? It looks like something completely different what you are asking there. Regarding the Boltzman distribution or Maxwell Boltzman statistics: The distribution $p(a) = \frac{g_a}{Z(T)} e^{\tfrac{-E(a)}{kT}}$, with $Z(T) = \int_a g_a e^{\tfrac{-E(a)}{kT}}$ (or a sum instead of integral) shows clearly that the Boltzman distributions are exponential families.

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