2nd Week: Properties of Astrophysical Plasmas B

Occupation probabilities

The 1st law of Thermodynamics in a system (or subsystem) with variable number of particles is

$dE=TdS-PdV+\mu dN$ ...

Fermi-Dirac

Suppose that our system has discrete energies and that $n_{k}$ is the number of particles occupying the energy level $E_{k}$. This two quantities must satisfy

$N=\sum _{k}n_{k}$ $E_{Total}=\sum _{k}n_{k}\epsilon _{k}$

. Since we are dealing with fermions, $n_{k}$ can be 0 or 1. The thermodynamic potential can be written as

$\Omega =-T\sum _{k}\log(\sum _{n_{k}}e^{(\mu -E_{k})/kT})$