# Interacting particles

We are familiar with the grand partition function for the grand canonical ensemble. This makes me wonder: what kinds of modifications would be required if the particles interacted?

Thanks.

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## 1 Answer

None. The grand canonical partition function is $$Z_\mathrm{GC} = \mathrm{tr}(e^{-\beta (H - \mu N)})$$ where $H$ is the hamiltonian and $N$ is the number operator.

Interacting particles simply means that the Hamiltonian needs to involve terms that take into account the interaction energy between particles. The same is true for the hamiltonian in, say, the canonical ensemble if there are interactions between particles. In neither case are any "extra terms" necessary.

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Does the chemical potential not takes care of interaction influence? –  Vladimir Kalitvianski Mar 26 '13 at 18:57
@VladimirKalitvianski No. Take, for example, the interactions between particles in a non-deal gas in a box. The chemical potential does not take this into account, the chemical potential is related to changes in energy that can occur because the number of particles in the system can change. –  joshphysics Mar 26 '13 at 19:02
Why in a box? Do you mean interaction with walls via $L$-dependence of energies? –  Vladimir Kalitvianski Mar 26 '13 at 20:37
I mean the interactions between gas molecules. Take, for example, van der waals forces en.wikipedia.org/wiki/Van_der_Waals_force. –  joshphysics Mar 27 '13 at 8:26
I do not say that the Hamiltonian contains the interaction potential. I ask whether the chemical potential is interaction-dependent? –  Vladimir Kalitvianski Mar 28 '13 at 10:11