C.2 Bose-EINSTEIN Statistics

For Bosons the occupation number is not restricted, so one must sum $ n_{i}$ over all integers in (C.12)

\begin{displaymath}\begin{array}{ll} Z_\mathrm{B}\ &\displaystyle = \ \prod_{i=1...
...1-e^{-\beta(E_{i}- E_\mathrm{F} )}\right)}^{-1} \ . \end{array}\end{displaymath} (C.16)

The logarithm of (C.16) yields the thermodynamic potential

\begin{displaymath}\begin{array}{ll} \Omega_\mathrm{B}(T,V, E_\mathrm{F}) &\disp...
...left(1-e^{-\beta(E_{i}- E_\mathrm{F} )}\right)} \ . \end{array}\end{displaymath} (C.17)

The mean number of particles is obtained from $ \Omega_{0}$ by differentiating with respect to the FERMI energy, as in (C.5), by keeping $ T$ and $ V$ (equivalently the $ E_{i}$) fixed. As a result, the mean number of Bosons is given by the Bose-EINSTEIN distribution function

\begin{displaymath}\begin{array}{l}\displaystyle n_\mathrm{B} \ \equiv \ \sum_{i...
...{e^{\beta \left(E_{i}- E_\mathrm{F} \right)}-1} \ , \end{array}\end{displaymath} (C.18)

where $ n_{i}$ is the mean occupation number in the $ i$th state. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors