Gibbs information and information theory

Question

In the study of statistics, a given family of probability densities depending smoothly upon a parameter $theta$ can be expressed in the form
$p(x,theta)=expleft[c(x)+sum_{r}theta^{r}S_{r}(x)-psi(theta)right],$
where the variable $x$ ranges over the sample space.
In models of statistical mechanics, we generally deal with Gibbs measures in the form
$p(x,theta)=expleft[sum_{r}theta^{r}S_{r}(x)-ln(psi(theta))right],$
where the $S_{r}$ determines the form of the action and $psi(theta)=lnZ(theta)$ is gibbs free energy,
and $Z$ the partition function.
Based on the above, what is the meaning of $x$ in $Z$?

Andrew · Answer

Eq. 2 of Ref [1] (which is by the same first author as Ref [2], which the OP mentioned in the comments) defines the Gibbs measure for the canonical ensemble
begin{equation}
p(H,beta) = q(x) expleft[-beta H(x)-W(beta)right]
end{equation}
where $x$ ranges over configuration space, $H(x)$ is the energy, $beta=1/kT$ is the inverse temperature, and $W(beta)$ is a normalization constant.
Let's compare this with the normal way to write Boltzman distribution. The Boltzman distribution gives the probability of finding the system in a given microstate, which is determined by the configuration space variables $x$.
begin{equation}
p(x,beta) =frac{e^{-beta H(x)}}{int dx e^{-beta H(x)}} = frac{e^{-beta H(x)}}{Z}
end{equation}
where the partition function is $Zequiv int dx e^{-beta H(x)}$.
It should be understood that $x$ here is not necessarily a single variable, but in general could be a vector of parameters. For instance, $x$ could represent the positions and momenta of $10^{23}$ particles, and in this case $int dx$ really should be understood as a huge integral over each of these $10^{23}$ configuration space variables.
Now this isn't quite the same thing as Eq 2, which gives the probability of the system having a certain energy $H$, rather than being in a given microstate labelled by $x$.
To convert, we need to include a factor of the density of states $rho(H)$. This function tells us how many states $rho(H)d H$ have an energy in the interval from $H$ to $H+dH$.
begin{equation}
p(H,beta) = frac{rho(H) e^{-beta H}}{Z} = rho(H) expleft[-beta H - log Zright]
end{equation}
Comparing this expression with the first one, we can identify the normalization $W(beta)$  with the log of the partition function $Z$
begin{equation}
W(beta) = log Z
end{equation}
Now it appears that the function $q(x)$ in the high temperature $betarightarrow 0$ limit is related to the density of states
begin{equation}
q(x) equiv rho(H(x))
end{equation}
To be honest I don't understand the notation $q(x)$, because each microstate should be equally likely in the high temperature limit. So I am not sure if the author is intending a distribution of microstates that violates the assumption that each accessible microstate is equally likely, or if $q(x)$ is meant to include some coarse graining over some subset of microstates, or something else.
References:
[1] https://cds.cern.ch/record/332060/files/9708032.pdf
[2] "Geometrical aspects of statistical mechanics" Physical Review E, 1995, volume 51, number 2.

Gibbs information and information theory

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