# Gibbs distribution and information theory

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Hi!

In the study of statistics, a given family of probability densities depending smoothly upon a parameter $\theta$ can be expressed in the form

$p(x,\theta)=exp\left[c(x)+\sum_{r}\theta^{r}S_{r}(x)-\psi(\theta)\right],$

where the variable $x$ ranges over the sample space.

In models of statistical mechanics, we generally deal with Gibbs measures in the form

$p(x,\theta)=exp\left[\sum_{r}\theta^{r}S_{r}(x)-ln(\psi(\theta))\right],$

where the $S_{r}$ determines the form of the action and $\psi(\theta)=lnZ(\theta)$ is the partition function.

Based on the above, what is the meaning of $x$ in $Z$?

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