Wave-particle interactions in quantum plasmas

Abstract

Wave-particle interaction (WPI) is one of the most fundamental processes in plasma physics in which one most prominent example is the Landau damping. Owing to its excellent energy-exchange mechanism, the WPI has gained increasing interest not only from theoretical points of view, but also its many important applications including plasma heating and plasma acceleration. In this review work, we present theoretical backgrounds of linear and nonlinear wave-particle interactions in quantum plasmas. Specifically, we focus on the wave-particle interactions for homogeneous plasma waves (i.e., waves with infinite extent rather than a localized pulse) as well as for propagating electrostatic waves in the weak and strong quantum regimes to demonstrate the modifications of several classical features including those associated with resonant and trapped particles. Finally, the future perspectives of WPI in quantum plasmas are presented.

Appendices

Appendix 1. Expressions for $A,~A_1,~B$ and C in $\gamma$

$$\begin{aligned} A=-\frac{16\pi e^3}{ A_0m^2k^3}{\int _C \frac{(v_p-v)\left[ (v_p-v)^2+ \frac{v_q^2}{2}\right] }{\left\{ (v_p-v)^2-v_q^2\right\} ^2 \left\{ \left( v_p-v+v_q \right) ^2-v_q^2\right\} \left\{ \left( v_p-v- v_q\right) ^2-v_q^2\right\} } F^{(0)}(v)\mathrm{d}v}, \end{aligned}$$

(167)

where

$$\begin{aligned} A_0={1-\frac{\pi e^2}{k^2m}\int _C \frac{F^{(0)}(v)}{(v_p-v)^2-\left( 2v_q \right) ^2 }}\mathrm{d}v. \end{aligned}$$

(168)

Also,

$$\begin{aligned}&A_1=-{12\pi e^3}\frac{\hbar }{k^2 m^2}\left[ \int _C \frac{1}{\left\{ \left( v_p-v+v_q\right) ^2-4v_q^2 \right\} \left\{ \left( v_p-v-v_q\right) ^2- 4v_q^2\right\} \mathrm{d}v }\right. \\&\quad \left. + \frac{3}{2}\int _C \frac{\left[ (v_p-v)^2+v_q^2\right] }{\left\{ (v_p-v)^2-4v_q^2\right\} \left\{ \left( v_p-v+2v_q\right) ^2-v_q^2\right\} \left\{ \left( v_p-v-2v_q\right) ^2- v_q^2\right\} }\right] F^{(0)}(v)dv,\\&B=\frac{4\pi e^4}{k^4 m}\int _C \left[ \frac{1}{\left\{ v_p-v+2v_q\right\} \left\{ v_p-v+v_q\right\} \left\{ \left( v_p-v+2v_q \right) ^2-v_q^2\right\} } \right. \\&\quad \left. +\frac{1}{\left\{ v_p-v-2v_q\right\} \left\{ v_p-v-v_q\right\} \left\{ \left( v_p-v-2v_q\right) ^2-v_q^2 \right\} }\right. \\&\quad \left. - \frac{2}{\left\{ (v_p-v)^2-v_q^2 \right\} ^2 }\right] F^{(0)}(v)\mathrm{d}v, \\&C(k,\omega ; v_g)=-\frac{4\pi e^4}{m \hbar ^2 k^2} \int _C \frac{1}{(v_p-v)^2-v_q^2} \frac{I(v)}{v-v_g}\mathrm{d}v \\&\quad =-\frac{4\pi e^4}{m \hbar ^2 k^4} \int _C \left[ \frac{v-v_g-v_q}{\left\{ \left( v_p-v+2v_q\right) ^2-v_q^2\right\} \left( v_p-v+v_q\right) ^2 \left( v-v_g-2v_q\right) } \right. \\&\quad \left. - \frac{v-v_g+ v_q}{\left\{ \left( v_p-v-2v_q\right) ^2-v_q^2\right\} \left( v_p-v-v_q \right) ^2 \left( v-v_g + 2v_q\right) } \right. \\&\quad \left. -2v_q \frac{ \left\{ \left( v_p-v\right) ^2+v_q^2 \right\} }{(v-v_g)\left\{ \left( v_p-v\right) ^2-v_q^2 \right\} ^3 }-4v_q\frac{v_p-v}{\left\{ \left( v_p-v\right) ^2-v_q^2 \right\} ^3}\right] F^{(0)}(v)\mathrm{d}v, \end{aligned}$$

where

$$\begin{aligned} I(v)= \frac{1}{k^2}\left[ \left( v-v_g+v_q \right) \frac{f^{(0)}\left( v+2v_q\right) -f^{(0)}(v)}{{\left\{ v_p-\left( v+v_q \right) \right\} }^2} +\left( v-v_g-v_q \right) \frac{f^{(0)}(v)-f^{(0)}\left( v-2v_q\right) }{{ \left\{ v_p-\left( v-v_q \right) \right\} }^2}\right] . \end{aligned}$$

(169)

Appendix 2. Reduced expressions for $\alpha ,~\beta ,~\gamma$ and D with the Fermi distribution at zero temperature

$$\begin{aligned}&\alpha =-\frac{8m \omega _{\rm p}^2}{3\hbar k^2 v_{\rm F}^3} \sum _{j=\pm 1} \left( v_p+jv_q \right) \log \left| \frac{v_p+jv_q-v_{\rm F}}{v_p+jv_q+v_{\rm F}}\right| , \end{aligned}$$

(170)

$$\begin{aligned}&\beta =1-\frac{3 m \omega _{\rm p}^2}{2 \hbar k^3 v_{\rm F}^2} \sum _{j=\pm 1}\left[ \left\{ 2\left( v_p+jv_q \right) +\left( v_p-v_g\right) \right\} \log \left| \frac{v_p+jv_q-v_{\rm F}}{v_p+jv_q+v_{\rm F}}\right| \right. \nonumber \\&\quad \left. -\left\{ v_{\rm F}^2-\left( v_p+jv_q \right) ^2-2\left( v_p+jv_q \right) \left( v_p-v_g\right) \right\} \frac{v_{\rm F}}{ v_{\rm F}^2- \left( v_P+jv_q \right) ^2} +\left( v_p-v_g \right) \frac{v_{\rm F} \left( v_p+jv_q \right) }{ v_{\rm F}^2-\left( v_p+jv_q \right) ^2}\right] , \end{aligned}$$

(171)

$$\begin{aligned}&\gamma =\left( \frac{1}{4}\frac{A A_1}{\hbar }-\frac{1}{2\hbar ^2}B+C\right) k^2, \end{aligned}$$

(172)

where

$$\begin{aligned} \begin{aligned}&A= - \frac{4em^2 \omega _{\rm p}^2}{A_0\hbar ^3 k^6 v_{\rm F}^3}\sum _{j=\pm 1}\left[ kv_{\rm F}v_q +\frac{\omega _{\rm p}}{6v_q} \left\{ \left( v_{\rm F}^2-\left( v_p+jv_q \right) ^2 \right) \left( 4-\frac{jk v_q}{2 \omega _{\rm p}} \right) \right. \right. \\&\quad \left. \left. -6v_q\left( v_p+jv_q \right) \left( 1-\frac{j k v_q}{ 2 \omega _{\rm p}} \right) \right\} \log \left| \frac{v_p+jv_q-v_{\rm F}}{ v_p+jv_q+v_{\rm F}}\right| \right. \\&\quad \left. +\frac{\omega _{\rm p}}{3 v_q} \left\{ v_{\rm F}^2-\left( v_p+j2v_q \right) ^2 \right\} \left( 1-\frac{jk v_q}{4\omega _{\rm p}} \right) \log \left| \frac{v_p+j2v_q-v_{\rm F}}{v_p+j2v_q+v_{\rm F}}\right| \right. \\&\quad \left. +i \pi v_q \omega _{\rm p} \left\{ v_{\rm F}^2-\left( v_p-2v_q \right) ^2 \right\} \left( 1+\frac{k v_q}{4\omega _{\rm p}} \right) \right] , \end{aligned} \end{aligned}$$

(173)

with

$$\begin{aligned} \begin{aligned}&A_0=1-\frac{3\omega _{\rm p}^2}{16 v_{\rm F}^2 k^2}\left[ 2-\sum _{j=\pm 1}\frac{jv_q}{ 4 v_{\rm F}}\left\{ v_{\rm F}^2-\left( v_p+jv_q\right) ^2 \right\} \log \left| \frac{v_p+j2v_q-v_{\rm F}}{v_p+j2v_q+v_{\rm F}}\right| \right] \\&\quad -i\frac{3\pi \omega _{\rm p}^2}{64 v_{\rm F}^3 v_q k^2}\left\{ v_{\rm F}^2-\left( v_p-2v_q \right) ^2 \right\} . \end{aligned} \end{aligned}$$

(174)

Also,

$$\begin{aligned} \begin{aligned}&A_1=-\frac{e}{m}\frac{9\omega _{\rm p}^2 m^3}{4\hbar ^2 v_{\rm F}^3k^5} \sum _{j=\pm 1} \left[ \frac{j}{4}\left\{ v_{\rm F}^2-\left( v_p+jv_q\right) ^2 \right\} \log \left| \frac{v_p+jv_q-v_{\rm F}}{v_p+jv_q+v_{\rm F}}\right| \right. \\&\quad \left. - \frac{2j}{3}\left\{ v_{\rm F}^2-\left( v_p+j3v_q \right) ^2 \right\} \log \left| \frac{v_p+j3v_q-v_{\rm F}}{v_p+j3v_q+v_{\rm F}}\right| +j\left\{ v_{\rm F}^2-\left( v_p+j2v_q \right) ^2 \right\} \log \left| \frac{v_p+j2v_q-v_{\rm F}}{v_p+j2v_q+v_{\rm F}}\right| \right] \\&\quad -i\frac{9\pi \omega _{\rm p}^2}{16v_{\rm F}^3k^3}\frac{e }{v_q^2}\left[ \frac{2}{3} \left\{ v_{\rm F}^2-\left( v_p-3v_q \right) ^2 \right\} -\left\{ v_{\rm F}^2-\left( v_p-2v_q \right) ^2 \right\} \right] , \end{aligned} \end{aligned}$$

(175)

$$\begin{aligned} \begin{aligned}&B=-\frac{e^2}{k^7}\frac{3\omega _{\rm p}^2m^3}{2v_{\rm F}^3\hbar ^3}\sum _{j=\pm 1}\left[ 16 v_{\rm F} v_q+4\left\{ v_{\rm F}^2-\left( v_p+j2v_q\right) ^2 \right\} \log \left| \frac{v_p+j2v_q-v_{\rm F}}{v_p+j2v_q+v_{\rm F}} \right| \right. \\&\quad \left. +\left\{ v_{\rm F}^2-\left( v_p+j3v_q\right) ^2 \right\} \log \left| \frac{v_p+j3v_q-v_{\rm F}}{v_p+j3v_q+v_{\rm F}}\right| -\left\{ v_{\rm F}^2-\left( v_p+jv_q \right) ^2 -8j v_q\left( v_p+jv_q\right) \right\} \log \left| \frac{v_p+jv_q-v_{\rm F}}{v_p+jv_q+v_{\rm F}}\right| \right] \\&\quad -i \frac{3 \pi e^2 \omega _{\rm p}^2}{4 k^4 v_{\rm F}^3}\left[ -\frac{1}{ v_q^3} \left\{ v_{\rm F}^2-\left( v_p-2v_q\right) ^2 \right\} + \frac{1}{ 4 v_q^3}\left\{ v_{\rm F}^2-\left( v_p-3v_q \right) ^2 \right\} \right] , \end{aligned} \end{aligned}$$

(176)

$$\begin{aligned} \begin{aligned}&C=-\frac{3}{4} \frac{e^2}{\hbar ^2 k^4}\frac{\omega _{\rm p}^2}{v_{\rm F}^3} \sum _{j=\pm 1} \left[ -\frac{1}{8 v_q^3} \frac{v_p+j2v_q-v_g}{ v_p+jv_q-v_g} \left\{ v_{\rm F}^2-\left( v_p+j3v_q \right) ^2 \right\} \log \left| \frac{v_p+j3v_q-v_{\rm F}}{v_p+j3v_q+v_{\rm F}}\right| +jM_j \log \left| \frac{v_p+jv_q-v_{\rm F}}{v_p+jv_q+v_{\rm F}}\right| \right. \\&\quad \left. +jN_j \frac{2v_{\rm F}}{v_{\rm F}^2-\left( v_p+jv_q \right) ^2 } -\left( \frac{1}{ 2 v_q} \frac{v_p-v_g}{v_p-jv_q-v_g} -\frac{1}{2} \frac{1}{ v_p+jv_q-v_g}-\frac{1}{2 v_q}\right) \frac{2v_{\rm F}\left( v_p+jv_q \right) }{v_{\rm F}^2-\left( v_p+jv_q\right) ^2 }\right. \\&\quad \left. -v_q \frac{v_{\rm F}^2-\left( v_g +jv_q\right) ^2 }{\left( v_p-jv_q- v_g \right) ^3 \left( v_p+jv_q-v_g\right) }\log \left| \frac{ v_g +jv_q-v_{\rm F}}{v_g+jv_q+v_{\rm F}}\right| +\frac{2v_q}{k^2} \frac{ \left\{ \left( \omega -k v_g\right) ^2 +\frac{\hbar ^2 k^4}{4m^2} \right\} \left( v_{\rm F}^2-v_g^2\right) }{\left( v_p+v_q-v_g \right) ^3 \left( v_p-v_q-v_g\right) ^3} log \frac{v_g-v_{\rm F}}{v_g+v_{\rm F}}\right] , \end{aligned} \end{aligned}$$

(177)

with

$$\begin{aligned} \begin{aligned}&M_{1,-1}= \frac{1}{4v_q^2 \left( v_p-v_g\mp 2v_q \right) ^2 }\left[ \left( v_p-v_g\mp 3v_q \right) \left\{ v_{\rm F}^2-3\left( v_p\pm v_q\right) ^2+2\left( v_g\pm v_q \right) \left( v_p\pm v_q\right) \right\} \right. \\&\quad \left. \mp 4v_q\left( v_p-v_g\mp v_q \right) \left( 3v_p-v_g\pm 2v_q \right) -2 \left( v_p\pm v_q \right) \left( v_p-v_g\right) \left( v_p-v_g\mp 3v_q \right) \right. \\&\quad \left. +\left\{ 2\left( v_p-v_g\mp v_q \right) +\left( v_p-v_g\mp 3v_q \right) \pm 2\left( v_q-v_g \right) \left( v_q-v_g\mp 3v_q \right) ^2\left( \frac{1}{2v_q}\mp \frac{1}{v_p- v_g\mp 2v_q} \right) \right\} \left\{ v_{\rm F}^2-\left( v_p\pm v_q \right) ^2 \right\} \right] \\&\quad +\frac{1}{4v_q^2 \left( v_p- v_g\pm v_q \right) ^2} \left[ 5\left( v_p-v_g\pm v_q \right) \left\{ v_{\rm F}^2-\left( v_p\pm v_q \right) ^2 \right\} \pm 3v_q \left\{ v_{\rm F}^2-\left( v_p\pm v_q\right) ^2 \right\} \right. \\&\quad \left. \mp 4v_q^2\left( v_p \pm v_q\right) -\left( 3v_p-3v_g\pm 5v_q \right) \left\{ v_{\rm F}^2\mp 2v_q\left( v_p\pm v_q \right) - \left( v_p\pm v_q \right) ^2 \right\} \right. \\&\quad \left. +2v_q^2\frac{v_{\rm F}^2-\left( v_p\pm v_q\right) ^2 }{v_p-v_g\pm v_q} \right] \pm \frac{v_p}{v_q^2}, \end{aligned} \end{aligned}$$

(178)

$$\begin{aligned} \begin{aligned}&N_{1,-1}=\pm \frac{1}{4v_q^2 \left( v_p-v_g\mp v_q \right) ^2}\left[ 2v_q\left( v_p-v_g\mp v_q \right) \left\{ v_{\rm F}^2-3\left( v_p\pm v_q\right) ^2 -2\left( v_p-v_g\right) \left( v_p\pm v_q \right) \right\} \right. \\&\quad \left. +\left\{ v_{\rm F}^2-\left( v_p\pm v_q \right) ^2 \right\} \left( v_p-v_g\right) \left( v_p-v_g\mp 3v_q \right) \right] \\&\quad \mp \frac{1}{4v_q{\left( v_p-v_g\mp v_q \right) ^2}}\left[ \left( v_p-v_g\mp 3v_q \right) \left\{ v_{\rm F}^2-\left( v_p\pm v_q \right) ^2 \right\} \pm 4v_q\left( v_p\pm v_q \right) \left( v_p-v_g\pm v_q \right) \right] \\&\quad \pm \frac{1}{4v_q^2}\left[ v_p\pm v_q- \left\{ v_{\rm F}^2-\left( v_p\pm v_q \right) ^2 \right\} \right] , \end{aligned} \end{aligned}$$

(179)

$$\begin{aligned} D=\frac{3 e^2 \pi \omega _{\rm p}^2}{4 k\hbar m v_{\rm F}^3}\left[ (v_{\rm F}^2-v_g^2) \frac{(v_p-v_g)^2+v_q^2}{{\left\{ (v_p- v_g)^2-v_q^2 \right\} ^3 }} +\frac{1}{8 v_q^4}\left\{ v_{\rm F}^2 -\left( v_p-3v_q \right) ^2 \right\} \frac{ v_p-v_g-2v_q}{ v_p-v_g-v_q}\right] . \end{aligned}$$

(180)

This expression (180) of D is obtained by using the following relations.

$$\begin{aligned}&\lim _{\nu _g\rightarrow 0} \frac{1}{\Omega -Kv+i\nu _g}=\frac{1}{\Omega -Kv} -i\pi \frac{1}{|K|}\delta \left( v-\frac{\Omega }{K} \right) , \nonumber \\&\quad \lim _{\nu _3\rightarrow 0} \frac{1}{\omega -kv-3kv_q+i\nu _3 }= \frac{1}{\omega -kv-3kv_q} \nonumber \\&\quad -i\pi \frac{1}{|K|} \delta \left( v-v_p+3v_q \right) , \end{aligned}$$

(181)

and we have made use of $\Omega /K\rightarrow v_g$. The infinitesimal quantities $|\nu _g|$ and $|\nu _3|$ are taken to anticipate the Landau damping terms associated with the group velocity and three-plasmon resonances.

Thus, the reduced expressions of P,  Q and R can be obtained from the relations $P=\beta /\alpha$, $Q=\gamma /\alpha$ and $R=D/\alpha$.