condensed matter - Bose-Einstein condensate in 1D

I've read that for a Bose-Einstein gas in 1D there's no condensation. Why this happenes? How can I prove that?

Answer

The claim is often that there is no condensation in $d<3$. The other answers are correct, but let's be clear, there are actually two assumptions present in the claim:

1. 
   

   Assume you have $N$ noninteracting bosons in $d$-dimensions in a hypervolume $L^d$

   
   


2. 
   

   Assume that these bosons have an energy-momentum relationship of $E(p) = Ap^s$.

   
   

Now, the way we calculate the critical temperature ($1/\beta_c$) for BEC requires satisfying the equation

$$\int_0^\infty \frac{\rho(E)dE}{e^{\beta_c E}-1}=N$$

where $\rho(E)$ is the density of states. Whether this integral is convergent or not depends on the values of both $s$ and $d$. The details of the proof are up to you though. :)