I am currently doing a question with regards to bosons gases and come across a sticking point. The question is as follows.

For a gas of bosons at temperature $T$ below the Bose-Einstein condensation temperature $T_c$ the specific heat is


where $T_c$ is a constant for a fixed particle number. Derive an expression of the mean energy.

Now my issue is not with the maths it with the limit of the integral. When I set up the integral as follows.

$$\int dE=\int 1.93R\left(\frac{T}{T_c}\right)^{\frac{3}{2}} dt$$

now what are the limits? I can't seem to understand what the lower limits can be. I have been reading up on wiki but can't find anything that gives any indication to what they could be. My thought originally where the lower limits to $E=0$ at $T=0$, but this seems counter-intuitive because even at zero Kelvin the bosons will still have some energy, so energy can be zero. If I carry out the integral without the limit I obviously get a constant, but I can seem to understand what that constant would represent. Could the constant be the chemical potential?

Is there a detail I'm missing with in the physics of boson gases?


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