Dear Dr. Chauhan,

1) As usual in statistical physics, many of its notions have exact meaning in the so-called thermodynamic limit only.  This refers also to the phase transition of Bose-Einstein condensation (BEC). The well-known estimate for the temperature of this phase transition (in TeX notation and in units with Boltzmann constant equal to unity)

T_c \sim (\hbar^2/m)(N/V)^{2/3}

has exact meaning in the limit N\to\infty and V\to\infty but N/V=const. Only in this limit the thermodynamic quantities (as, for example, heat capacity) have singularities as functions of the temperature T. In practice, for finite N and V, we have a fast change of the behavior of these thermodynamic quantities in vicinity of T_c, and the greater are N and V, the smaller is the width of the transition region from one dependence to another one. For small N this width can become of the same order of magnitude as T_c and then it does not make sense to speak about "the phase transition of BEC".

2) If you take formally N=1 and V=a^3 ("a" is a linear size of the volume V) then you get T_c \sim \hbar^2/ma^2 \sim E_0

where E_0 is the ground state energy of a particle confined in the volume V. According to the laws of statistical physics, the probability to find the particle at this state is w_0 \sim \exp(-E_0/T) \sim \exp(-T_c/T).

This means that for T

@ Alberto

Dear Alberto A. Díaz ,     Thanks and definitely I will go through  those references......

Regarding the first part of the question, consider the chemical potential for a system of bosons as a function of volume, the number of particles and the temperature in the light of the fact that the zero of this function signals the BE condensation. Regarding the BE condensation of photons, recently Erik van der Wurff of the Institute for Theoretical Physics (ITF) of University of Utrecht has written a Master thesis on the subject matter, in which he also reviews the relevant literature. I attach the link to this very good thesis here below.

http://web.science.uu.nl/itf/Teaching/2014/2014vanderWurff.pdf

@Behnam Farid 

 Dear Farid, Thanks a lot for your reference....this is invaluable......

You are welcome Anil. I should perhaps add that the thesis has been awarded the 2014 Lorentz Master Thesis Award in Theoretical Physics.

@Behnam Farid

Oh that's great...Thanks

You are welcome Anil.

The minimum number of atoms to show BEC depends on the dimension and the shape(boundary conditions) of atoms are located in. For example, the minimum number is 35131 in a 2D box. There are 3D results, 7616 in a box and 10458 for a sphere.

For the details, see my paper '2D BEC under pressure'.

@Wonyoung Cho 

Thanks Dear, but still questions are there.....why so much atoms? why not few atoms in some box shows this effect?

Some special effects happen EMERGENTLY at the critical point.

Although a few is effective in some situation, it is not critical in this case.

Try to read an article linked below.

http://scienceblogs.com/principles/2013/10/29/one-two-many-lots-investigating-the-start-of-many-body-physics/

Dear Dr. Chauhan,

1) As usual in statistical physics, many of its notions have exact meaning in the so-called thermodynamic limit only.  This refers also to the phase transition of Bose-Einstein condensation (BEC). The well-known estimate for the temperature of this phase transition (in TeX notation and in units with Boltzmann constant equal to unity)

T_c \sim (\hbar^2/m)(N/V)^{2/3}

has exact meaning in the limit N\to\infty and V\to\infty but N/V=const. Only in this limit the thermodynamic quantities (as, for example, heat capacity) have singularities as functions of the temperature T. In practice, for finite N and V, we have a fast change of the behavior of these thermodynamic quantities in vicinity of T_c, and the greater are N and V, the smaller is the width of the transition region from one dependence to another one. For small N this width can become of the same order of magnitude as T_c and then it does not make sense to speak about "the phase transition of BEC".

2) If you take formally N=1 and V=a^3 ("a" is a linear size of the volume V) then you get T_c \sim \hbar^2/ma^2 \sim E_0

where E_0 is the ground state energy of a particle confined in the volume V. According to the laws of statistical physics, the probability to find the particle at this state is w_0 \sim \exp(-E_0/T) \sim \exp(-T_c/T).

This means that for T