Editing Bose–Einstein condensate (section)

==== Weaknesses of Gross–Pitaevskii model ====
The Gross–Pitaevskii model of BEC is a physical [[approximation]] valid for certain classes of BECs. By construction, the [[Gross–Pitaevskii equation|GPE]] uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also neglects anomalous contributions to [[self-energy]].<ref>Beliaev, S. T. Zh. Eksp. Teor. Fiz. 34, 417–432 (1958) [Soviet Phys. JETP 7, 289 (1958)]; ibid. 34, 433–446 [Soviet Phys. JETP 7, 299 (1958)].</ref> These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate [[wavefunction]] acquires the terms containing higher-order powers of the wavefunction. Moreover, for some physical systems the amount of such terms turns out to be infinite, therefore, the equation becomes essentially non-polynomial. The examples where this could happen are the Bose–Fermi composite condensates,<ref name=Schick:1971/><ref name=Kolomeisky:1992/><ref name=Kolomeisky:2000/><ref name=Chui:2004/> effectively lower-dimensional condensates,<ref>
{{cite journal
 |author1=L. Salasnich
 |author2=A. Parola
 |author3=L. Reatto |name-list-style=amp
 |year=2002
 |title=Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates
 |journal=Phys. Rev. A
 |volume=65 |issue=4 |page=043614
 |arxiv=cond-mat/0201395
 |bibcode = 2002PhRvA..65d3614S
 |doi=10.1103/PhysRevA.65.043614|s2cid=119376582
 }}
</ref> and dense condensates and [[superfluid]] clusters and droplets.<ref>
{{cite journal
 |author1=A. V. Avdeenkov
 |author2=K. G. Zloshchastiev
 |year=2011
 |title=Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent
 |journal=J. Phys. B: At. Mol. Opt. Phys.
 |volume=44 |issue=19 |pages=195303
 |arxiv=1108.0847
 |bibcode=2011JPhB...44s5303A
 |doi=10.1088/0953-4075/44/19/195303|s2cid=119248001
 }}
</ref> It is found that one has to go beyond the Gross-Pitaevskii equation. For example, the logarithmic term <math>\psi \ln |\psi|^2 </math> found in the [[Logarithmic Schrödinger equation]] must be added to the Gross-Pitaevskii equation along with a [[Vitaly Ginzburg|Ginzburg]]–Sobyanin contribution to correctly determine that the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures in close agreement with experiment.<ref>{{cite journal|author1=T.C Scott|author2=K. G. Zloshchastiev|title=Resolving the puzzle of sound propagation in liquid helium at low temperatures|journal=Low Temperature Physics|volume=45|issue=12|year=2019|pages=1231–1236|doi=10.1063/10.0000200|arxiv=2006.08981|bibcode=2019LTP....45.1231S|s2cid=213962795}}</ref>