Relaxation of plasma waves in Fermi-degenerate quantum plasmas

Plasma waves in a Fermi-degenerate quantum plasma are studied in the framework of the Vlasov-Poisson self-consistent-field theory. A complete time-dependent analytical solution of the initial-value problem is obtained for a multistream model both by stationary-wave and Laplace-transform methods. In the continuum limit, the excitation spectrum can be expressed by the imaginary part of the response function to the initial perturbations. The relaxation of plasma waves is discussed for one-dimensional systems with both Fermi and Maxwellian statistics. Apart from the usual exponential Landau damping, regimes of sub- and superexponential damping can be identified due to the phase relaxation of single-particle excitations. In addition, beat waves and echoes are discussed.

Figure 1

Eigenfrequencies of electrostatic and ballistic modes. The mode frequencies are compared with the Lindhard dispersion relation (solid line) and the boundaries of the SPE continuum (dashed lines). Parameters: $N=30, p_F=1.5$.

Figure 2

Electrostatic potential $y=-k^2\widehat{\varphi }$ and weights $c_{\alpha }/\mathrm{\Delta }\omega$ per frequency interval for (a) $k_0=0.1$, (b) $k_0=0.9$, and (c) $k_0=2.5$. Vertical dashed lines mark the frequency given by the Lindhard dispersion relation (dark gray) and the cutoff frequencies of the SPE continuum (light gray). For $k_0=2.5$, the upper cutoff coincides with the Lindhard frequency. Parameters: $N=30, p_F=1.5, {\widehat{n}}_s\left(0\right)=\delta (k-k_0)$, and ${\widehat{u}}_s\left(0\right)=0$.

Figure 3

Weights $c_{\alpha }/\mathrm{\Delta }\omega$ per frequency interval for a velocity perturbation with $k_0=1.2$. The weights of the multistream model (dots) are compared with the excitation spectrum (60) of the continuum theory. Parameters: $N=300, p_F=1.5, {\widehat{n}}_s\left(0\right)=0$, and ${\widehat{u}}_s\left(0\right)=k_0^2\delta (k-k_0)$.

Figure 4

Magnitude of the electrostatic potential $y=-k^2\widehat{\varphi }$ for (a) $k_0=0.9$ and (b) $k_0=2.5$. For $k_0=2.5$ it is compared to the envelope ${\left(k{p}_{F}t\right)}^{-1}$ (dashed line). Parameters: $N=300, p_F=1.5, {\widehat{n}}_s\left(0\right)=\delta (k-k_0), {\widehat{u}}_s\left(0\right)=0$.

Figure 5

Eigenfrequencies of electrostatic and degenerate ballistic modes. The mode frequencies are compared with the Lindhard dispersion relation (solid line) and the artificial boundary of the SPE continuum (dashed line). The artificial boundary of the SPE continuum is introduced by the cutoff of the Maxwell distribution at momentum $p_{\text{max}}$. Parameters: $N=90, T=1.5$, and $p_{\text{max}}=4.5$.

Figure 6

Electrostatic potential $y=-k^2\widehat{\varphi }$ and weights $c_{\alpha }/\mathrm{\Delta }\omega$ per frequency interval for (a) $k_0=0.3$ and (b) $k_0=0.5$. Vertical dashed lines mark the frequency given by the Lindhard dispersion relation (dark gray). Parameters: $N=90, T=1.5, {\widehat{n}}_s\left(0\right)=\delta (k-k_0)$, and ${\widehat{u}}_s\left(0\right)=0$.

Figure 7

Exponential Landau damping of a potential perturbation $y=-k^2\widehat{\varphi }(k,t)$ with wave number $k_0=0.3$ for a Maxwellian velocity distribution. The numerical solution of the stationary-wave method (solid line) is compared with exponential decay (dashed line) with a damping constant $\gamma =0.01967$ according to the Landau-Lindhard theory. Parameters: $N=90, T=1.5^2/2, {\widehat{n}}_s\left(0\right)=\delta (k-k_0)$, and ${\widehat{u}}_s\left(0\right)=0$.

Figure 8

Superexponential damping of a potential perturbation $y=-k^2\widehat{\varphi }(k,t)$ with wave number $k_0=2.5$ for a Maxwellian velocity distribution. The stationary-wave solution (solid line) is compared with the asymptotic approximation (66) for large wave numbers (dashed line). Parameters: $N=90, T=1.5^2/2, {\widehat{n}}_s\left(0\right)=\delta (k-k_0)$, and ${\widehat{u}}_s\left(0\right)=0$.