Phase-space structures in quantum-plasma wave turbulence

The quasilinear theory of the Wigner-Poisson system in one spatial dimension is examined. Conservation laws and properties of the stationary solutions are determined. Quantum effects are shown to manifest themselves in transient periodic oscillations of the averaged Wigner function in velocity space. The quantum quasilinear theory is checked against numerical simulations of the bump-on-tail and two-stream instabilities. The predicted wavelength of the oscillations in velocity space agrees well with the numerical results.

Figure 1

Initial velocity distribution used in the simulations of the bump-on-tail instability, for both the Vlasov and Wigner cases.

Figure 2

(Color online) Simulations of the Wigner-Poisson and Vlasov-Poisson systems, at time ${\omega }_{p}t=200$, for $H=0$ (Vlasov, left frame), 1 (middle frame), and 2 (right frame). Initially monochromatic spectrum. Top panels: electron distribution function $f(x,v)$ in phase space. Bottom panels: spatially averaged electron distribution function $F\left(v\right)$ in velocity space.

Figure 3

Zoom on the spatially averaged electron distribution function $F\left(v\right)$ in velocity space for the same cases as in Fig. 2.

Figure 4

(Color online) Simulations of the Wigner-Poisson and Vlasov-Poisson systems, for ${\omega }_{p}t=500$, for $H=0$ (Vlasov, left panel) and 1 (Wigner, right panel). Initially broad wave number spectrum. Top panels: electron distribution function $f(x,v)$ in phase space. Bottom panels: spatially averaged electron distribution function $F\left(v\right)$ in velocity space.

Figure 5

Initial velocity distribution used in the simulations of the two-stream instability, for both the Vlasov and Wigner cases.

Figure 6

Zoom of the averaged Wigner distribution $F\left(v\right)$ for the two-stream instability at time ${\omega }_{p}t=110$. Top frame: $H=0$ (Vlasov); middle panel: $H=1$; bottom panel: $H=2$.