Gross-Pitaevsky equation

Because the first models of the physics of superconductivity were based on the phenomenon of Bose-Einstein condensation (BEC) it is not surprising that the existence of the Josephson effect has also been postulated for cold-atom systems in the BEC state [11]. In both superconductors and BEC cold-atom systems, the Josephson effect arises from the approximation of non-conservation of particle number, which gives rise to a phase type of order parameter F(x) and concomitant wave phenomena, described at T = 0 by the Gross-Pitaevski equation " ... 

Worth observing that Eq. (1.198) formally differs from the Landau version by the thermal density presence, p(r), yet noting that the original Gross-Pitaevsky equation is given in general by the terms of the superfluid wave-function... 

Putz, M. V. (2012c). From Kohn-Sham to Gross-Pitaevsky equation within Bose-Einstein condensation y-theory. Int. J. Chem. Model. 4(1), 1-11. 

Recovering Gross-Pitaevsky Equation Within Local Density Approximation of -DFT... 

The practical implementation of BEC and of its mean-field approximation usually makes use of thermodynamic limit constraint, namely for systems with many-to-infinite number of particles N oo). The so-called Thomas-Fermi approximation [46, 47] may be used such that the potential and the interaction energies are larger than the kinetic energy, which can be therefore neglected in the stationary Gross-Pitaevsky equation, reducing it to the algebraic form [41, 42] ... 

Within this framework, for bonding electrons (with their mass mo), one has the generic bosonic condensation Hamiltonian, rooted in the nonlinear Schrodinger or Gross-Pitaevsky equation (67),... 

It nevertheless produces a non-linear Schrodinger equation known as Gross-Pitaevsky (GP) equation (Gross, 1961 Pitaevsky, 1961)... 

The natural question arises whether the two type equations, Kohn-Sham and Gross-Pitaevsky, may be seen as equivalent, and in which conditions The present article likes to address such important issue in the nowadays pioneering developments of DFT of EEC (Vetter, 1997 Nunes, 1999 Albus et al., 2003 Kim Zubarev, 2003 Brand, 2004 Aigaman Band, 2011 Putz, 2011b-c, 2012c). As such, the main plan of the paper consists in ... 

The last term of Eq. (1.197) vanishes through the left multiplication of (1.196) with E (r) followed by spatial integration since providing, in fact, the average of the fluctuation field, this way leaving the Eq. (1.196) with the so called modified Gross-Pitaevsky (Hartree-Fock) equation... 

At this point, one would like to verify whether the P-DFT in general and the Kohn-Sham equation in particular provide a point of contact with the specific Bose-Einstein condensation equation, in the mean-field approximation, as the Gross-Pitaevsky does. To this end, one may start by noticing the difference between perturbation and interaction when speaking about the two DPT systems [43, 44] ... 

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