For an ideal Bose gas we have the equation of state
where  is the per-particle volume,  is the thermal wavelength,  is the fugacity, and
It is noticeable that  is a monotonically growing function of  in , which are the only values for which the series converge.
Recognizing that the second term on the right-hand side contains the expression for the average occupation number of the fundamental state , the equation of state can be rewritten as
Because the left term on the second equation must always be positive, , and because , a stronger condition is
which defines a transition between a gas phase and a condensed phase. On the critical region it is possible to define a critical temperature and thermal wavelength:
recovering the value indicated on the previous section. The critical values are such that if  or , we are in the presence of a Bose–Einstein condensate.
Understanding what happens with the fraction of particles on the fundamental level is crucial. As so, write the equation of state for , obtaining
:  and equivalently 
So, if , the fraction , and if , the fraction . At temperatures near to absolute 0, particles tend to condense in the fundamental state, which is the state with momentum .