The precise fraction depends on the behaviour of the low-energy density of states of the particles (this in turn depends on the dimensionality of the system) and the mass, or masses (in case of anisotropy), of the particles. Then there is also Tc, which is proportional to a certain power of the density, the power being determined by the above-mentioned behaviour of the density of single-particle states; for instance, for an isotropic system in three dimensions, at low energies the density of single-particle states behaves like ε1/2, resulting in Tc being proportional to n2/3, where n denotes the particle density. Your above statement neglects the fact that all the relevant quantities are thermodynamically interrelated, by the condition that Tc is the temperature at which for a given density of particles, ones with a given mass and low-energy density of states, is to coincide with the energy of the lowest-lying single-particle state. Chemical potential is partly determined by the density of the particles in the system under investigation, so that one cannot arbitrarily dilute the system and hope to achieve BEC at a feasibly low temperature.

I can understand the inter-relationship among different quantities and I can understand that a given system is expected to have a single value of p=0 condensate.  My question can be restated as follows.  If superfluidity of He-II arises due to 10% of He-4 atoms in p=0 state, I can logically understand why BEC in trapped dilute Bose gases with about 60% or more particles in p=0 condensate should not exhibit superfluidity.  The question pertains to some other system which,   for its interaction and other parameters,  has only 0.1% particles in p=0 condensate.  Would we expect it to have superfluidity or not ?. 

The statement that He-II with 10% atoms in p=0 condensate implies that even 1% or even much lower % of atoms in p=0 condensate in some other system could be expected to exhibit superfluidity.    And if there is no law which decides the lower bound value of p=0  condensate, it is obvious to think that even 1 atom in p=0 state could be expected to show superfluidity, at least a logic says so.  What I mean is that in the framework of scientific logic attributing superfluidity with p=0 condensate is incorrect.  

Jatendra Singh, if the number of particles we are dealing with is macroscopically large, then naturally 10%, 1%, 0.1%, etc., of it is also macroscopically large. What is relevant here is whether there is phase coherence and the attendant off-diagonal long-range order in the density matrix (a la Yang, and Penrose and Onsager). Amongst others, Anderson discusses this very issue in his review article Considerations on the Flow of Superfluid Helium (Rev. Mod. Phys. 38, 298 (1966)) -- link attached below.

http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.38.298

Hello, I got this from Prof. John Lipa at Stanford Univ.:

"as they get more dilute, the transition temperature goes down. It can be seen when you add 3He atoms that are fermions. also why it was hard to get Bose condensation in cold atom clouds with lasers- they are very dilute."

Once I read John Lipa's answer, I am thinking the limit is set by how low your apparatus can get to if you keep making the Helium more dilute since the transition temp. is going down to achieve superfluidity.

Not sure if this answers the question, but I found it interesting. I am asking another person at JPL. Not sure if he will respond.

Yatendra,

I believe there is no particular lower bound. The onset of superfluidity (like superconductivity) proceeds via a Shafroth precursor "state" (although not a state in the ordinary sense) as the temperature lowers. This appears as a competition between coherent-breaking correlations and synergetic rearrangements of the precursor.

Of course the precursor appears more distinct in superconductivity than in superfluidity, but the process, formulated in terms of Yang's Off-Diagonal Long-Range Order, ODLRO, in the approriately reduced density matrix, should be analogous.

@ Behnam . 

Thanks for sending the Ref of Anderson's review and your useful comments.  The idea of p=0 condensate has been popular with all physicists working on the subject of superfluidity of bosonic systems and this has been so in spite of the fact that all efforts made over the last seventy five years or so have not helped in concluding the theory  of superfluidity and even the nature of condensate responsible for superfluidity.   In fact these efforts have not been able to establish whether there exists p=0 condensate in He-II and BEC state of trapped dilute Bose gases beyond a point of doubt.   We have a remarkable phenomenological theory (the two fluid theory of Landau) which does not talk about p=0 condensate yet explains most of the properties of He-II and in addition the similar theory (Tisza's theory) which attributed superfluidity with p=0 condensate but could not succeed to this level.  However,  the popular belief in the relation of p=0 condensate with superfluidity has been growing.    The success of Landau's theory itself is a good proof that the p=0 condensate does not exist in a system of interacting bosons like He-4 atoms.   You are right that coherence, macroscopic wave function ODLRO in single particle density matrix etc are more essntial aspects of a superfluid but for these aspects we need not stick to p=0 condensate;  we can think of pair condensate as we do for superconductors.     I really want to understand why simple suggestion of London that p=0 condensate is so dear to physicists in spite of its failure to render a viable theory.  

I am really looking for a suitable discussion on this question.      

@ Allen Thanks for your discussion and related information. 

@ Erkki,  Thanks for bringing Shafroth precursor to my notice which I missed some how.   I would appreciate if you could state what it is or provide me its main reference. 

@ Yatendra Singh - You are welcome. Regarding the p=0 condensation, one should realise that interaction amongst bosons give rise to at least partial depletion of the p=0 condensate. This can be explicitly shown for the case of bosons interacting through a hard-core potential and in the limit of  n a3

Yatendra,

One key ref. is M. R. Shafroth, Phys. Rev. 100, 463 (1955). He is also referenced in some recent textbooks like Solid-State Physics: Introduction to the Theory

 by James Patterson,Bernard Bailey.

He seems to be one of the first to make a connection between BE condensation and superconductivity, although believed today to be incomplete.