Editing Bose–Einstein condensate (section)

=== Gross–Pitaevskii equation ===
{{Main|Gross–Pitaevskii equation}}

In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger equation, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments.

This approach originates from the assumption that the state of the BEC can be described by the unique wavefunction of the condensate <math>\psi(\vec{r})</math>. For a [[Schrödinger field|system of this nature]], <math>|\psi(\vec{r})|^2</math> is interpreted as the particle density, so the total number of atoms is <math>N=\int d\vec{r}|\psi(\vec{r})|^2</math>

Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using [[mean-field theory]], the energy (E) associated with the state <math>\psi(\vec{r})</math> is:

:<math>E=\int
d\vec{r}\left[\frac{\hbar^2}{2m}|\nabla\psi(\vec{r})|^2+V(\vec{r})|\psi(\vec{r})|^2+\frac{1}{2}U_0|\psi(\vec{r})|^4\right]</math>

Minimizing this energy with respect to infinitesimal variations in <math>\psi(\vec{r})</math>, and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a non-linear [[Schrödinger equation]]):

:<math>i\hbar\frac{\partial \psi(\vec{r})}{\partial t} = \left(-\frac{\hbar^2\nabla^2}{2m}+V(\vec{r})+U_0|\psi(\vec{r})|^2\right)\psi(\vec{r})</math>

where:

:{|cellspacing="0" cellpadding="0"
|-
| <math>\,m</math>
| &nbsp;is the mass of the bosons,
|-
| <math>\,V(\vec{r})</math>
| &nbsp;is the external potential, and
|-
| <math>\,U_0</math>
| &nbsp;represents the inter-particle interactions.
|}

In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for <math>\ T= 0</math>):

:<math> {\omega _p} = \sqrt {\frac{{{p^2}}}{{2m}}\left( {\frac{{{p^2}}}{{2m}} + 2{U_0}{n_0}} \right)} </math>

The Gross-Pitaevskii equation (GPE) provides a relatively good description of the behavior of atomic BEC's. However, GPE does not take into account the temperature dependence of dynamical variables, and is therefore valid only for <math>\ T= 0</math>.
It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature.

==== Numerical solution ====
The Gross-Pitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and
different numerical methods, such as split-step
[[Crank–Nicolson method|Crank–Nicolson]]<ref>{{cite journal
 |author=P. Muruganandam and S. K. Adhikari
 |year=2009
 |title=Fortran Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal=Comput. Phys. Commun.
 |volume=180 |issue=3 |pages=1888–1912
 |doi=10.1016/j.cpc.2009.04.015
|bibcode=2009CoPhC.180.1888M|arxiv=0904.3131|s2cid=7403553
 }}</ref>
and [[Fourier spectral]]<ref>{{cite journal
 |author=P. Muruganandam and S. K. Adhikari
 |year=2003
 |title=Bose–Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods
 |journal=J. Phys. B
 |volume=36 |issue= 12|pages=2501–2514
 |doi=10.1088/0953-4075/36/12/310
 |bibcode=2003JPhB...36.2501M|arxiv=cond-mat/0210177|s2cid=13180020
 }}</ref> methods, are used for its solution. There are different Fortran and C programs for its solution for [[contact interaction]]<ref>{{cite journal
 |author=D. Vudragovic|display-authors=et al
 |year=2012
 |title=C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=183 |issue=9 |pages=2021–2025
 |doi=10.1016/j.cpc.2012.03.022
  |bibcode=2012CoPhC.183.2021V|arxiv=1206.1361|s2cid=12031850
 }}</ref><ref>
{{cite journal
 |author=L. E. Young-S.|display-authors=et al
 |year=2016
 |title=OpenMP Fortran and C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=204 |issue=9 |pages=209–213
 |doi=10.1016/j.cpc.2016.03.015
 |bibcode=2016CoPhC.204..209Y|arxiv=1605.03958|s2cid=206999817
 }}</ref>
and long-range [[dipolar interaction]]<ref>
{{cite journal
 |author=K. Kishor Kumar|display-authors=et al
 |year=2015
 |title=Fortran and C Programs for the time-dependent dipolar Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=195 |pages=117–128
 |doi=10.1016/j.cpc.2015.03.024
 |bibcode=2015CoPhC.195..117K|arxiv=1506.03283|s2cid=18949735
 }}</ref> which can be freely used.

==== 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>

==== Other ====
However, it is clear that in a general case the behaviour of Bose–Einstein condensate can be described by coupled evolution equations for condensate density, superfluid velocity and distribution function of elementary excitations. This problem was solved in 1977 by Peletminskii et al. in microscopical approach. The Peletminskii equations are valid for any finite temperatures below the critical point. Years after, in 1985, Kirkpatrick and Dorfman obtained similar equations using another microscopical approach. The Peletminskii equations also reproduce Khalatnikov hydrodynamical equations for superfluid as a limiting case.