Wave damping as a critical phenomenon

The evolution of a monochromatic Langmuir wave in a collisionless plasma is studied using Vlasov simulations for a wide range of initial amplitudes. Three types of initial electron distributions are considered: Maxwellian, Lorentzian, and dilute warm Maxwellian plus dense cold component. It is shown that there exists a critical initial amplitude epsilon* that separates the damping and nondamping asymptotic regimes. Depending on the initial amplitude there are three main types of asymptotic evolution: (i) monotonic linear Landau damping below the threshold, epsilonmuch less thanepsilon*; (ii) the critical case epsilonapproximate toepsilon*, when the field damps algebraically as E(t)proportional tot(-3.26); (iii) at epsilon>epsilon* initial damping followed by a period of subsequent exponential growth and then irregular oscillations about a nonzero amplitude. This threshold is well described as a critical phenomenon, showing power-law dependencies on the distance from the threshold not only for field quantities, which are expected of second-order phase transitions in thermodynamics, but also for temporal ones. The critical exponent for both the initial damping and growth phases differ from those expected if the threshold is caused by O'Neil's particle trapping in the wave potential. However, trapping affects the critical exponents well above the threshold and explains the characteristic frequency of oscillations above the threshold. It is found that for a Maxwellian plasma the threshold amplitude epsilon* corresponds to the condition that the trapping (bounce) frequency omega(b) equals the modulus of the theoretical Landau damping rate \gamma(L)\; at the threshold, q(c)=omega(b)//gamma(L)\approximate to1. For Lorentzian and Maxwellian-plus-cold component plasmas this ratio is q(Lor)approximate to0.84 and q(CM)approximate to0.83, respectively. The temporal and the field scalings are thus interrelated, suggesting that the inclusion of the temporal dimension is vital for critical phenomena in collisionless plasmas, in contrast to thermodynamic systems where the very small characteristic time to achieve equilibrium removes time from the scaling. (C) 2004 American Institute of Physics.