Bose–Einstein condensate

where ${\displaystyle v=V/N}$ is the per-particle volume, ${\displaystyle \lambda }$ is the thermal wavelength, ${\displaystyle f}$ is the fugacity, and


It is noticeable that ${\displaystyle g_{3/2}}$ is a monotonically growing function of ${\displaystyle f}$ in ${\displaystyle f\in [0,1]}$, 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 ${\displaystyle \langle n_{0}\rangle }$, the equation of state can be rewritten as


Because the left term on the second equation must always be positive, ${\displaystyle {\frac {\lambda ^{3}}{v}}>g_{3/2}(f)}$, and because ${\displaystyle g_{3/2}(f)\leq g_{3/2}(1)}$, 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 ${\displaystyle T<T_{\text{c}}}$ or ${\displaystyle \lambda >\lambda _{\text{c}}}$, 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 ${\displaystyle f=1}$, obtaining


So, if ${\displaystyle T\ll T_{\text{c}}}$, the fraction ${\displaystyle {\frac {\langle n_{0}\rangle }{N}}\approx 1}$, and if ${\displaystyle T\gg T_{\text{c}}}$, the fraction ${\displaystyle {\frac {\langle n_{0}\rangle }{N}}\approx 0}$. At temperatures near to absolute 0, particles tend to condense in the fundamental state, which is the state with momentum ${\displaystyle {\vec {p}}=0}$.