Evolution of inverse cascades and formation of precondensate in Gross-Pitaevskii turbulence in two dimensions

Here we study how coherence appears in a system driven by noise at small scales. In the wave turbulence modeled by the Gross-Pitaevskii / nonlinear Schr\"odinger equation, we observe states with correlation scales less than the system size but much larger than the excitation scale. We call such state precondensate to distinguish it from condensate defined as a system-wide coherent state. Both condensate and precondensate are characterized by large scale phase coherence and narrow distribution of amplitudes. When one excites small scales, precondensate is achieved relatively quickly by an inverse cascade heating quasi-equilibrium distribution of large-scale modes. The transition from the precondensate to the system-wide condensate requires much longer time. The spectra of precondensate differ from quasi-equilibrium and are characterized by two bending points, one on the scale of the average distance between vortex pairs, and the other on the scale of the distance between vortices in a pair. We suggest temporal evolution laws for both lengths and use them to predict the probability of the transition to condensate.