Thermodynamics of weakly nonlinear langmuir waves, spontaneous symmetry breaking, and nambu-goldstone mode generation

A thermodynamic formulation of one-dimensional, weak Langmuir-wave (LW) turbulence is presented. A nonlinear Schrödinger equation, which has a non-local interaction term stemming from nonlinear Landau damping, is considered as a model equation. By considering the nonlinear Landau damping as a system-bath interaction, a non-equilibrium counterpart of canonical picture is applied to weakly nonlinear LWs. The background equilibrium electrons are regarded as a heat bath in contact with the wave-turbulence system (the perturbation part of the electron probability distribution), which is the system of interest. The nonequilibrium free energy of the LWs is defined using a plasmon-pair approximation, in which all LW quanta are treated as pairs created by modulational instabilities that retain phase information. The proposed thermodynamic theory predicts spontaneous symmetry breaking and resultant Nambu- Goldstone mode generation, depending upon the density of the LW quanta. Particle-in-cell simulations confirm the prediction as supercontinuum formation. Our formulation provides a novel approach to a wide class of wave turbulence.