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Say at a critical temperature $T_c$ there is a phase transition.

If we had water, $T_c$ for the liquid-solid transition is $0 ^\circ$C. If we left a block of ice at $1 ^\circ$C for a bit, it will melt completely so that the whole thing is in liquid form. The two states (solid and liquid) will coexist for some time, depending on how much heat is being pumped in I presume, but then only once of them will remain.

For a Bose-Einstein condensate (BEC), however, we find this kind of graph:

enter image description here

where $N$ is the total number of particles and $N_0$ is the number or particles that collapsed to a BEC. At any given temperature, their ratio is fixed and the two states therefore coexist.

A book that I was reading compared the BEC phase transition to the vapour-liquid one, where droplet of condensed fluid coexist with gas... really?

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    $\begingroup$ Your title seems unrelated to the body or your question, but to answer your title question: phases can coexist because they are in equilibrium with each other. Equilibrium is characterised by the absence of a driving force or influence for preference of any particular state. But I think you know this? $\endgroup$ – AngusTheMan Aug 1 '15 at 0:38
  • $\begingroup$ So how come that all ice melts into water as time progresses, therefore their coexistence comes to an end, whilst in a BEC the coexistence remains? $\endgroup$ – SuperCiocia Aug 1 '15 at 1:32
  • $\begingroup$ Ice melts into water only if there is net thermal input and the amount that melts is directly related to the latent heat of fusion that applies under the conditions concerned. $\endgroup$ – Russell McMahon Aug 1 '15 at 6:58
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The transition between the two phases is called (not unreasonably!) a phase transition, and phase transitions come in two flavours: first order and second order.

First order phase transitions (generally) have a sharp transition temperature. Steam condensing to water is a first order phase transition and as you've pointed out occurs at 100ºC (at one atmosphere).

Second order phase transitions (generally) occur over a range of temperatures. Formation of superfluid helium is an example of a second order phase transition, and this starts at about 2.2K and continues smoothly all the way to absolute zero.

I think formation of a BEC is basically a second order transition, though the neat division into purely first order and purely second order breaks down in the messy real world that physicists are forced to inhabit. Anyhow, thats why BEC formation doesn't have a precise transition temperature.

Strictly speaking your book is wrong, because condensation of steam and formation of a BEC are phase transitions with different orders.

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