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

TL;DR

This paper investigates how coherence develops in two-dimensional wave turbulence governed by the Gross-Pitaevskii equation, focusing on the formation of precondensate and condensate states through inverse cascades and their spectral characteristics.

Contribution

It introduces the concept of precondensate in wave turbulence, analyzes its spectral properties, and models the transition dynamics to a system-wide condensate.

Findings

Precondensate forms rapidly via inverse cascade at small scales.

Spectra of precondensate show two characteristic bending points.

Temporal laws for length scales predict transition probabilities to condensate.

Abstract

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.