Bose-Einstein condensation and Berezinskii-Kosterlitz-Thouless transition in 2D Nonlinear Schr\"{o}dinger model

We analyse the Bose-Einstein condensation process and the Berezinskii-Kosterlitz-Thouless phase transition within the Gross-Pitaevskii model and their interplay with wave turbulence theory. By using numerical experiments we study how the condensate fraction and the first order correlation function behave with respect to the mass, the energy and the size of the system. By relating the free-particle energy to the temperature we are able to estimate the Berezinskii-Kousterlitz-Thouless transition temperature. Below this transition we observe that for a fixed temperature the superfluid fraction appears to be size-independent leading to a power-law dependence of the condensate fraction with respect to the system size.