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Partition function for an interacting bosonic mode

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

  1. Wouter

    Wouter Guest

    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

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

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