# Partition function for an interacting bosonic mode

Discussion in 'Physics' started by Wouter, Oct 8, 2018.

1. ### WouterGuest

Let us assume a single bosonic mode, in equilibrium with a reservoir.

For a non-interacting Bose gas, the partition function becomes

$\mathcal{Z_\text{nonint}}=\sum_{N=0}^\infty e^{-\beta(\epsilon-\mu)N}=\frac{1}{1-e^{-\beta(\epsilon-\mu)}}$.

from which it is easy to obtain a free energy $F=-T\log(\mathcal{Z})$ and particle number statistics $N=-\frac{\partial F}{\partial \mu}=\frac{1}{e^{\beta(\epsilon-\mu)}-1}$ yielding the Bose-Einstein distribution.

How is this modified for finite two-particle interaction, i.e, how to evaluate

$\mathcal{Z_\text{int}}=\sum_{N=0}^\infty e^{-\beta\left[(\epsilon-\mu)N+\frac{U}{2}N^2\right]}$ ?