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Under the conditions of weak Langmuir turbulence, a self-consistentwave-particle Hamiltonian models the effective nonlinear interaction of aspectrum of M waves with N resonant out-of-equilibrium tail electrons. In orderto address its intrinsically nonlinear time-asymptotic behavior, a Monte Carlocode was built to estimate its equilibrium statistical mechanics in both thecanonical and microcanonical ensembles. First the single wave model isconsidered in the cold beam/plasma instability and in the O'Neil setting fornonlinear Landau damping. O'Neil's threshold, that separates nonzerotime-asymptotic wave amplitude states from zero ones, is associated to a secondorder phase transition. These two studies provide both a testbed for the MonteCarlo canonical and microcanonical codes, with the comparison with exactcanonical results, and an opportunity to propose quantitative results tolongstanding issues in basic nonlinear plasma physics. Then the properlyspeaking weak turbulence framework is considered through the case of a largespectrum of waves. Focusing on the small coupling limit, as a benchmark for thestatistical mechanics of weak Langmuir turbulence, it is shown that Monte Carlomicrocanonical results fully agree with an exact microcanonical derivation. Thewave spectrum is predicted to collapse towards small wavelengths together withthe escape of initially resonant particles towards low bulk plasma thermalspeeds. This study reveals the fundamental discrepancy between the long-timedynamics of single waves, that can support finite amplitude steady states, andof wave spectra, that cannot.Comment: 15 pages, 7 figures, to appear in Physics of Plasma

Equilibrium statistical mechanics for single waves and wave spectra in Langmuir wave-particle interaction