A comparison of weak-turbulence and PIC simulations of weak   electron-beam plasma interaction
  Quasilinear theory has long been used to treat the problem of a weak electron beam interacting with plasma and generating Langmuir waves. Its extension to weak-turbulence theory treats resonant interactions of these Langmuir waves with other plasma wave modes, in particular ion-sound waves. These are strongly damped in plasma of equal ion and electron temperatures, as sometimes seen in, for example, the solar corona and wind. Weak turbulence theory is derived in the weak damping limit, with a term describing ion-sound wave damping then added. In this paper we use the EPOCH particle-in-cell code to numerically test weak turbulence theory for a range of electron-ion temperature ratios. We find that in the cold ion limit the results agree well, but increasing ion temperature the three-wave resonance becomes broadened in proportion to the ion-sound wave damping rate. This may be important in, for example, the theory of solar radio bursts, where the spectrum of Langmuir waves is critical. Additionally we establish lower limits on the number of simulation particles needed to accurately reproduce the electron and wave distributions in their saturated states, and to reproduce their intermediate states and time evolution. 
Heather Ratcliffe, Christopher S Brady, Mohammad B Che Rozenan, Valery   Nakariakov
physics.plasm-ph