# Partition function of weakly interacting Bose gas

In the Bogoliuobov theory of superfluidity, the hamiltonian of a system of weakly interacting bosons in second quantized notation is diagonalized with the following result:
$$\hat H = H_0 \hat1 + \sum_k E_k \hat\xi_k^+\hat \xi_k\$$ Where $H_0$ is the ground state energy (a constant) and $E_k$ is the energy spectrum of the collective excitations:
$$E_k = \sqrt{\epsilon_k (\epsilon_k + 2n\tilde U(\vec k))}$$ Now suppose I want to calculate the grand canonical partition function of this system. My initial approach was to imagine an ideal gas of these collective excitations (or quasiparticles) as the hamiltonian suggests, and just use :
$$Z_{GC} = Tr(e^{-\frac{\hat H - \mu \hat N}{k_BT}})$$ With N being the number operator of these quasi particles ($\hat N = \sum_k \hat\xi_k^+\hat \xi_k\$). But then it occured to me that:

1) As the collective excitations are not really particles, their particle number doesn't need to be conserved and the lagrange multiplier $\mu$ for this constraint is not needed (like a gas of photons in blackbody radiation). So I thought about setting $\mu=0$ like the gas of photons. But this doesn't seem right either because in the limit of large momentum (k), the quasiparticles can essentially be replaced with our real bosons, and the number of these real bosons is undoubtedly conserved.

2) The derivation of the above hamiltonian includes an approximation of taking the occupation number of the ground state to be large (macroscopic) like a Bose-Einstein condensate, and thereby taking the corresponding creation and annihilation operators of this particular state to be commuting. This means that our hamiltonian is not applicable to all temperatures.

So my question is :
Is there any way to derive an approximate grand canonical partition function for the system using the above hamiltonian to derive its thermodynamic properties; and if yes, how do we take the non conserving nature of the quasiparticles into account? Can we just set $\mu = 0$ like an ideal photon gas? In general, how can we study the thermodynamics of such a system from first principles?

## 1 Answer

Is there any way to derive an approximate grand canonical partition function for the system using the above hamiltonian to derive its thermodynamic properties; and if yes, how do we take the non conserving nature of the quasiparticles into account?

Yes, you start with the grand canonical partition function for the original particles. Doing this, is easiest using coherent states and working with a path integral. This gives you

$Z= \int D\phi D\phi^*e^{-S(\phi,\phi^*)}$ With $S(\phi,\phi^*) = \sum_{k,w}\phi^*(\epsilon_k-iw)\phi+\sum_{k1,k2,k3,k4}\phi^*_{k1}\phi^*_{k2}\phi_{k3}\phi_{k4}V(k1,k2,k3,k4)$

Here, $\epsilon$ contains the kinetic energy, potential energy and chemical potential of the real particles. $V$ decribes the interaction.

By doing a steepest decent approximation, you will derive a Gross Pitiveski equation for the classical solution and a non-diagonal second order action. This is basically assuming the occupation of the classical solution is large. After diagonalizing this action by the Bogolubov transform you get an energy as: $E_k = \sqrt{\epsilon_k (\epsilon_k + 2n\tilde U(\vec k))}$

This approximate second-order action is that for a Bose gas with 0 chemical potential.

You can study thermodynamic properties by repeating this process with what ever coupling you need to generate your response functions.

The moral of the story is you should start with the partition function fo the bare particles and do the classical-mean-field approximation to get the partition function for the Bogolubov particles.

If you don't like path integrals or coherent states you can check out Bogolubov's original paper, which is some sense required reading for this subject:

https://ufn.ru/dates/pdf/j_phys_ussr/j_phys_ussr_1947_11_1/3_bogolubov_j_phys_ussr_1947_11_1_23.pdf

• Thank you so much for your detailed answer. What you suggest is a bit beyond what we study in our graduate statistical mechanics course. I will definitely check the original paper by Bogolubov. But for the time being, I had another idea, can I just use the energy spectrum for $E_k$ and compute the canonical partition function with $Z=Tr(e^{-\hat H/k_BT})$ ? From that I can get the Helmholtz free energy, and subsequently other thermodynamic quantities. Nov 6, 2017 at 19:31
• I think so. That will definitely get you Helmholtz free energy, heat capacity and entropy. I think it will also get compressibility, but figuring out linear response like currents won't be as simple as directly coupling Bogolubov particles to a voltage in the partition function, you'll have to start with the real particle theory and go through the approximation again. I haven't done that calculation though so it might end up matching. Nov 6, 2017 at 19:52