Understanding the behaviour of an interacting Bose gas

Question

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?

Answer 1

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.