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In field theory, particles are waves or excitations that propagate on thefundamental state. In experiments or cosmological models one typically wants tocompute the out-of-equilibrium evolution of a given initial distribution ofsuch waves. Wave Turbulence deals with out-of-equilibrium ensembles of weaklynonlinear waves, and is therefore well-suited to address this problem. As anexample, we consider the complex Klein-Gordon equation with a Mexican-hatpotential. This simple equation displays two kinds of excitations around thefundamental state: massive particles and massless Goldstone bosons. The formerare waves with a nonzero frequency for vanishing wavenumber, whereas the latterobey an acoustic dispersion relation. Using wave turbulence theory, we derivewave kinetic equations that govern the coupled evolution of the spectra ofmassive and massless waves. We first consider the thermodynamic solutions tothese equations and study the wave condensation transition, which is theclassical equivalent of Bose-Einstein condensation. We then focus on nonlocalinteractions in wavenumber space: we study the decay of an ensemble massiveparticles into massless ones. Under rather general conditions, these masslessparticles accumulate at low wavenumber. We study the dynamics of wavescoexisting with such a strong condensate, and we compute rigorously a nonlocalKolmogorov-Zakharov solution, where particles are transferred non-locally tothe condensate, while energy cascades towards large wave numbers through localinteractions. This nonlocal cascading state constitute the intermediateasymptotics between the initial distribution of waves and the thermodynamicstate reached in the long-time limit.

Wave turbulence description of interacting particles: Klein-Gordon model with a Mexican-hat potential