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

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

Maxwell-Boltzmann

The probability distribution can be found by:

$f(p)=\left(e^{{E-\mu } \over {kt}}\right)^{-1}$

Fermi-Dirac

Suppose that our system has discrete energies and that $n_{k}$ is the number of particles occupying the energy level $\epsilon _{k}$. 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

$\Omega =-T\sum _{k}\log(\sum _{n_{k}=0}^{1}e^{n_{k}(\mu -\epsilon _{k})/kT})$

$\Omega =-T\sum _{k}\log(1+e^{(\mu -\epsilon _{k})/kT})$