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 or the nonlinear Schrödinger equation, we observe states with correlation scales smaller than the system size but much larger than the excitation scale. We call a such state precondensate to distinguish it from condensate defined as a systemwide coherent state. Both condensate and precondensate are characterized by large-scale phase coherence and a narrow distribution of amplitudes. When one excites small scales, precondensate is achieved relatively quickly by an inverse cascade heating quasiequilibrium distribution of large-scale modes. The transition from the precondensate to the systemwide condensate requires a much longer time. The spectra of precondensate differ from quasiequilibrium 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.