# The statistical physics of a simple continuous system

Let $$\mathbb{R}_{\geq 0}$$ be the set of real numbers greater or equal to zero. Assume an average value $$\overline{R} \in \mathbb{R}$$, called the prior. Then, the probability distribution $$p(r), \forall r \in \mathbb{R}_{\geq 0}$$ which maximizes the entropy based on the prior is the Gibbs distribution. Since $$\mathbb{R}$$ is an uncountable set, the partition function is an integral:

$$Z=\int_0^\infty e^{-\beta r}dr=\frac{1}{\beta}$$

The average value $$\overline{R}$$ is

$$\overline{R}=-\frac{1}{Z}\frac{\partial Z}{\partial \beta}=-\beta (- \beta^{-2})=\frac{1}{\beta}$$

and the entropy is

$$S=k_B(\ln [Z]+\beta \overline{R})=k_B\left(\ln[\beta^{-1}]+\beta \frac{1}{\beta}\right)=k_B \left( 1-\ln [\beta] \right)$$

Graphing the entropy yields: Why is the entropy negative when $$1<\ln[\beta]$$? Should the entropy not be greater than zero for all values of $$\beta$$? Error somewhere?

EDIT:

As requested in the comments, the Log Plot of S is and $$\beta$$ is a Lagrange multiplier.

EDIT-2:

As clarified in the comments, here is the plot of $$S$$ with respect to $$\ln{\beta}$$. • Two things: 1) please define $\beta$, and 2) plot the entropy on a log scale. It's weird that you've defined the average to be $\bar{R}$ but then introduce $\beta$ as a separate parameter. – DanielSank Mar 2 at 19:51
• @DanielSank I'm assuming $\beta=1/kT$ , at least this is what I have always seen. – Aaron Stevens Mar 2 at 20:00
• Alexandre, are you looking at a system whose energy varies linearly with respect to some parameter $r$, and where there is no degeneracy for any energies? – Aaron Stevens Mar 2 at 20:02
• Please make your question one cohesive post. An edit history is available for those who are interested. – Aaron Stevens Mar 2 at 20:08
• @Aaron correct, it physically corresponds to a system where the energy varies linearly and where there is no degeneracy. One way to think of it is an entropic force $F$ over a distance $x$. In this case $E=Fx$. – Alexandre H. Tremblay Mar 2 at 20:13