PX284 - L3 - distribution functions

considering a many-particle system with a single state at energy, $E$
the grand partition function is a sum over all possible configurations of the system:

Z = \sum_{n} \exp (- n β (E - μ))

where, $n$ is the occupation number, ie. the number of particles in the state, and $μ$ is the chemical potential

the mean occupation:

⟨ n ⟩ = \sum_{n} \frac{n \exp (- n β (E - μ))}{Z} = - \frac{1}{β Z} \frac{\partial Z}{\partial E} = - \frac{1}{β} \frac{\partial \ln Z}{\partial E}

for fermions: $n = 0$ or $1$ because of pauli's exclusion principle

\begin{aligned} ∴ Z & = 1 + \exp (- β (E - μ)) \\ \ln Z & = \ln (1 + \exp (- β (E - μ))) \\ ⟨ n ⟩ & = \frac{\exp (- β (E - μ))}{1 + \exp (- β (E - μ))} \\ = \frac{1}{\exp (β (E - μ)) + 1} \end{aligned}

for bosons: $n = 0, 1, 2, \dots$

∴ Z = \sum_{n = 0}^{\infty} \exp (- n β (E - μ))

this is a geometric progression with sum to infinity:

\sum_{0}^{\infty} a r^{n} = \frac{1}{1 - r}

so here,

\begin{aligned} Z & = \frac{1}{1 - \exp (- β (E - μ))} \\ \ln Z & = - \ln (1 - \exp (- β (E - μ))) \\ ⟨ n ⟩ & = \frac{\exp (- β (E - μ))}{1 - \exp (- β (E - μ))} \\ = \frac{1}{\exp (β (E - μ)) - 1} \end{aligned}

these expression can be shown to hold for a particular single-particle state in a general multi-state system
defining the distribution function, $f (E)$ , as the mean occupation of a single-particle system $⟨ n ⟩$ in such a system
for fermions:

f_{F D} (E) = \frac{1}{\exp (β (E - μ)) + 1}

this is called the fermi-dirac distribution, which runs from $0 \to 1$
for bosons:

f_{B E} (E) = \frac{1}{\exp (β (E - μ)) - 1}

this is called the bose-einstein distribution, which runs from $0 \to \infty$
$f (E)$ can be used to calculate the properties
number of particles between $E$ and $E + d E$ $= g (E) f (E) d E$
so, the total number of particles:

N = \int_{0}^{\infty} g (E) f (E) d E

and, the internal energy:

U = \int_{0}^{\infty} E g (E) f (E) d E