Waveturbulence description of interacting particles: KleinGordon model with a Mexicanhat potential
Abstract
In field theory, particles are waves or excitations that propagate on the fundamental state. In experiments or cosmological models, one typically wants to compute the outofequilibrium evolution of a given initial distribution of such waves. Wave turbulence deals with outofequilibrium ensembles of weakly nonlinear waves, and is therefore well suited to address this problem. As an example, we consider the complex KleinGordon equation with a Mexicanhat potential. This simple equation displays two kinds of excitations around the fundamental state: massive particles and massless Goldstone bosons. The former are waves with a nonzero frequency for vanishing wave number, whereas the latter obey an acoustic dispersion relation. Using waveturbulence theory, we derive wave kinetic equations that govern the coupled evolution of the spectra of massive and massless waves. We first consider the thermodynamic solutions to these equations and study the wave condensation transition, which is the classical equivalent of BoseEinstein condensation. We then focus on nonlocal interactions in wavenumber space: we study the decay of an ensemble of massive particles into massless ones. Under rather general conditions, these massless particles accumulate at low wave number. We study the dynamics of waves coexisting with such a strong condensate, and we compute rigorously a nonlocal KolmogorovZakharov solution, where particles are transferred nonlocally to the condensate, while energy cascades towards large wave numbers through local interactions. This nonlocal cascading state constitutes the intermediate asymptotics between the initial distribution of waves and the thermodynamic state reached in the longtime limit.
 Publication:

Physical Review E
 Pub Date:
 July 2015
 DOI:
 10.1103/PhysRevE.92.012909
 arXiv:
 arXiv:1504.05394
 Bibcode:
 2015PhRvE..92a2909G
 Keywords:

 05.45.a;
 11.10.Lm;
 Nonlinear dynamics and chaos;
 Nonlinear or nonlocal theories and models;
 Nonlinear Sciences  Chaotic Dynamics;
 High Energy Physics  Theory