Condensation of classical nonlinear waves

TL;DR

This paper investigates the formation of large-scale condensates in classical wave systems modeled by the defocusing nonlinear Schrödinger equation, revealing phase transition behaviors in different dimensions through theory and simulations.

Contribution

It introduces a thermodynamic wave turbulence framework to describe classical wave condensation, showing dimension-dependent phase transition phenomena and the subcritical nature of the transition.

Findings

Phase transition in 3D systems at low energy density.

No phase transition in 2D systems, similar to quantum Bose-Einstein condensation.

Numerical simulations confirm thermodynamic limit and theoretical predictions.

Abstract

We study the formation of a large-scale coherent structure (a condensate) in classical wave equations by considering the defocusing nonlinear Schr\"odinger equation as a representative model. We formulate a thermodynamic description of the condensation process by using a wave turbulence theory with ultraviolet cut-off. In 3 dimensions the equilibrium state undergoes a phase transition for sufficiently low energy density, while no transition occurs in 2 dimensions, in analogy with standard Bose-Einstein condensation in quantum systems. Numerical simulations show that the thermodynamic limit is reached for systems with $16^3$ computational modes and greater. On the basis of a modified wave turbulence theory, we show that the nonlinear interaction makes the transition to condensation subcritical. The theory is in quantitative agreement with the simulations.