Fundamental Entropy

$\sigma (U,V,N)\Rightarrow \mathrm{d}\sigma =\frac{\partial \sigma }{\partial U}\mathrm{d}U+\frac{\partial \sigma }{\partial V}\mathrm{d}V+\frac{\partial \sigma }{\partial N}\mathrm{d}N=\frac{1}{\tau }\mathrm{d}U+\frac{P}{\tau }\mathrm{d}V-\frac{\mu }{\tau }\mathrm{d}N$

• The terms are thermal, mechanical, and diffusive

$\mathrm{d}\sigma =\frac{1}{\tau }\mathrm{d}U+\frac{P}{\tau }\mathrm{d}V-\sum \mu \mathrm{d}N$

• As a result of taking the above derivative and using definitions of [fundamental temperature] and [pressure]

Topics

• Sackur–Tetrode Equation

$\sigma =\frac{S}{k_B}=\ln \Omega$

• Has no units
• $k_B$ is the [Boltzmann constant]

$\sigma =-{\left(\frac{\partial F}{\partial \tau }\right)}_V=\frac{\partial }{\partial \tau }(\tau \ln Z)$

• $F$ is the [Helmholtz Free Energy]

$\sigma =\frac{U}{\tau }+\ln Z$

• $Z$ is the [partition function]