Fundamental Entropy
#Physics
$\displaystyle \sigma(U,V,N)\Rightarrow \mathrm{d}\sigma=\frac{ \partial \sigma }{ \partial U }\mathrm{d}U+\frac{ \partial \sigma }{ \partial V }\mathrm{d}V+\frac{ \partial \sigma }{ \partial N }\mathrm{d}N=\frac{1}{\tau}\mathrm{d}U+\frac{P}{\tau}\mathrm{d}V-\frac{\mu}{\tau}\mathrm{d}N$
- The terms are thermal, mechanical, and diffusive
$\displaystyle \mathrm{d}\sigma=\frac{1}{\tau}\mathrm{d}U+\frac{P}{\tau}\mathrm{d}V-\sum \mu \mathrm{d}N$
- As a result of taking the above derivative and using definitions of [fundamental temperature] and [pressure]
Topics
$\displaystyle \sigma=\frac{S}{k_{B}}=\ln \Omega$
- Has no units
- $\displaystyle k_{B}$ is the [Boltzmann constant]
$\displaystyle \sigma=-\left( \frac{ \partial F }{ \partial \tau } \right)_{V}=\frac{ \partial }{ \partial \tau }(\tau \ln Z)$
- $\displaystyle F$ is the [Helmholtz Free Energy]
$\sigma=\frac{U}{\tau}+\ln Z$
- $\displaystyle Z$ is the [partition function]