  Under the conditions of weak Langmuir turbulence, a self-consistent
wave-particle Hamiltonian models the effective nonlinear interaction of a
spectrum of M waves with N resonant out-of-equilibrium tail electrons. In order
to address its intrinsically nonlinear time-asymptotic behavior, a Monte Carlo
code was built to estimate its equilibrium statistical mechanics in both the
canonical and microcanonical ensembles. First the single wave model is
considered in the cold beam/plasma instability and in the O'Neil setting for
nonlinear Landau damping. O'Neil's threshold, that separates nonzero
time-asymptotic wave amplitude states from zero ones, is associated to a second
order phase transition. These two studies provide both a testbed for the Monte
Carlo canonical and microcanonical codes, with the comparison with exact
canonical results, and an opportunity to propose quantitative results to
longstanding issues in basic nonlinear plasma physics. Then the properly
speaking weak turbulence framework is considered through the case of a large
spectrum of waves. Focusing on the small coupling limit, as a benchmark for the
statistical mechanics of weak Langmuir turbulence, it is shown that Monte Carlo
microcanonical results fully agree with an exact microcanonical derivation. The
wave spectrum is predicted to collapse towards small wavelengths together with
the escape of initially resonant particles towards low bulk plasma thermal
speeds. This study reveals the fundamental discrepancy between the long-time
dynamics of single waves, that can support finite amplitude steady states, and
of wave spectra, that cannot.
