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 for a particular energy sate $\epsilon _{k}$ can be written as

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

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

Recall that, the mean particle number in a certain energy state $\epsilon _{k}$ is minus the derivative of the thermodynamic potential $\Omega _{k}$ with respect to the chemical potential $\mu$, at V and T constant. Therefore

$f(\epsilon _{k})=-{\partial \Omega _{k} \over \partial \mu }=...$

$f(\epsilon _{k})={1 \over {e^{(\epsilon _{k}-\mu )/kT}+1}}$