DFT adapted to Bose-Einstein Condensation

The specificity of DFT treated for BEC implies functionals of the couple [p(r),T (r)] in terms of the overall super-fluidic density p(r) and of the order parameter T (r), both defined through the 7V-particles ensemble statistical average (Vetter, 1997 Putz, 2011b-c, 2012c) 

one should learn DFT deals with two kinds of working systems one real- called Hohenberg-Kohn and one artificial - called Kohn-Sham the natural or Hohenberg-Kohn system ( ext. or HIC) includes the external V (X.r) and interacting W(X.r,r ) potentials, the last one being included in the HK-functional 

The specific auxiliary order parameter related potential is just considered formally since at the end it has to be set to zero due to the fact its coupling currents are vanishing for the macroscopic 

Note that stays for the Hohenberg-Kohn functional, V(r) is the external applied potential, p is the chemical potential, W die inter-particle pair interaction, the kinetic energy of the real system, and J(r) the external 

The non-interacting or Kohn-Sham (auxiliary) constmction employs the artificial KS potential that has the effective potential role, which includes also the external or HK potential already (Kohn Sham, 1965), see Eq. (1.160), but having no interaction potential any longer within the KS functional 

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