Universal Nonthermal Power-law Distribution Functions from the Self-consistent Evolution of Collisionless Electrostatic Plasmas

Abstract Collisionless systems often exhibit nonthermal power-law tails in their distribution functions. Interestingly, collisionless plasmas in various physical scenarios (e.g., the ion population of the solar wind) feature a v −5 tail in their velocity ( v ) distribution, whose origin has been a long-standing puzzle. We show this power-law tail to be a natural outcome of the collisionless relaxation of driven electrostatic plasmas. Using a quasi-linear analysis of the perturbed Vlasov–Poisson equations, we show that the coarse-grained mean distribution function (DF), f 0 , follows a quasi-linear diffusion equation with a diffusion coefficient D ( v ) that depends on v through the plasma dielectric constant. If the plasma is isotropically forced on scales larger than the Debye length with a white-noise-like electric field, D ( v ) ∼ v 4 for σ < v < ω P / k , with σ the thermal velocity, ω P the plasma frequency, and k the characteristic wavenumber of the perturbation; the corresponding quasi-steady-state f 0 develops a v −( d + 2) tail in d dimensions ( v −5 tail in 3D), while the energy ( E ) distribution develops an E −2 tail independent of dimensionality. Any redness of the noise only alters the scaling in the high v end. Nonresonant particles moving slower than the phase velocity of the plasma waves ( ω P / k ) experience a Debye-screened electric field, and significantly less (power-law suppressed) acceleration than the near-resonant particles. Thus, a Maxwellian DF develops a power-law tail, while its core ( v < σ ) eventually also heats up but over a much longer timescale. We definitively show that self-consistency (ignored in test-particle treatments) is crucial for the emergence of the universal v −5 tail.