Understanding the behaviour of an interacting Bose gas From the Bose-Einstein distribution it follows that a non-interacting Bose gas condenses into the Bose-Einstein condensate below a certain critical temperature. What happens when interactions are introduced in a Bose gas is not dealt with in introductory Statistical mechanics courses. Hence my questions are pretty naive and basic.
How is(are) the interaction(s) quantitatively modelled in a Bose gas and how does it change the behaviour (compared to the noninteracting Bose gas) when the temperature is lowered? Is there a way to physical understand the change in the behaviour, if any?
As a minor comment, I have been informed that in case of Fermi gases, the role of interactions shifts the effective mass from $m\to m^*$ and the energy levels (effectively mapping the interacting system to a system of quasiparticles that still obey FD statistics). Does the similar thing happen here too?
 A: 1) First note that a non-interacting Bose gas is an idealization. If the gas was truly non-interacting, then it would be impossible to cool by evaporative cooling (or any other method that removes energy and requires the gas to re-equilibrate). 
2) In a Bose gas the short range part of the interaction has to be repulsive (otherwise the gas will collape at low temperature). A typical model is a repulsive delta function 
$$
V(x_1,x_2)=\frac{4\pi a}{m} \delta(x_1-x_2)
$$
controlled by the s-wave scattering length $a$. Indeed, for a dilute gas this is not a model but a systematic description of the low energy properties.
3) In a non-interacting gas Bose condensation takes place at the Einstein temperature
$$
 T_c = \frac{2\pi}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}.
$$
The leading shift due to (repulsive, $a>0$) interactions is 
$$
\Delta T_c \simeq 1.3 an^{1/3} T_c
$$
which, even in a strongly interacting gas like helium, is not a large shift.
4) The systematic study of perturbation theory in $a$ goes back to Bogoliubov. He found, for example, that the dispersion relation of quasi-particle in a Bose condensed fluid is 
$$
\epsilon_p = \frac{1}{2m}\sqrt{(p^2+8\pi an)^2-(8\pi an)^2}
$$
which smoothly interpolates between a Goldstone mode at low $p$, and non-interacting atoms at large $p$. 
5) And indeed, as remarked below, you can take the interaction in (2) and treat in a mean field approximation. This leads to a non-linear Schroedinger equation (the Gross-Pitaevski equation) for the condensate wave function. This equation can be used to study cloud profiles, collective modes, etc. 
