Landau damping of longitudinal oscillation in ultra-relativistic plasmas by analytic function of nonextensive distribution

Abstract

The analytic function of relativistic nonextensive distribution is given, and employed to solve the Landau damping of longitudinal oscillation in ultra-relativistic plasmas. The unified expression of Landau damping which reduces to the result in relativistic Maxwellian distributed plasmas in the extensive limit is obtained, and find that Landau damping is relevant to both the number and energy of resonant particles, which described by temperature and nonextensive parameters in relativistic nonextensive distribution.

Appendix

Equation (39) is

$$ \Delta ( {\alpha,q} ) \equiv\frac{1}{\varLambda}\int _0^{u_{\max} } { \bigl[ {1 - ( {q - 1} )\alpha\sqrt{1 + u^2} } \bigr]^{\frac{2 - q}{q - 1}}u^2du} . $$

(55)

Let

$$ \Delta' \equiv\int_0^{u_{\max} } { \bigl[ {1 - ( {q - 1} )\alpha\sqrt{1 + u^2} } \bigr]^{\frac{2 - q}{q - 1}}u^2du} , $$

(56)

then

$$\begin{aligned} & \Delta' = l_1 \bigl\{ {l_2 + ( {q - 1} ) \bigl[ {l_3 ( {l_4 + l_5 + l_6 } ) - l_7 ( {l_8 + l_9 } )} \bigr]} \bigr\} ,\quad \frac{1}{2} < q \le1, \end{aligned}$$

(57)

$$\begin{aligned} & \Delta' = l_{10} \bigl\{ {l_{11} + \alpha \bigl[ {l_{12} ( {l_{13} + l_{14} } ) - l_{15} } \bigr]} \bigr\} ,\quad 1 \le q \le1 + 1 / \alpha, \end{aligned}$$

(58)

where

$$\begin{aligned} & l_1 = \frac{1}{16\alpha^2 ( {q - 1} )^2q ( {2q - 3} ) ( {2q - 1} )\Gamma ( {\frac{q - 2}{q - 1}} )}, \end{aligned}$$

(59)

$$\begin{aligned} & l_2 = 2^{2 + \frac{1}{1 - q}}q ( {\alpha- \alpha q} )^{\frac{q}{q - 1}} \bigl( {3 - 8q + 4q^2} \bigr)\Gamma \biggl( { \frac{2q - 1}{2 - 2q}} \biggr) \Gamma \biggl( { \frac{3 - 2q}{2 - 2q}} \biggr){}_2F_1 \biggl[ { \frac{2q - 1}{2 - 2q},\frac{3 - 2q}{2 - 2q};\frac{1}{2};\frac{1}{\alpha ^2 ( {q - 1} )^2}} \biggr], \end{aligned}$$

(60)

$$\begin{aligned} & l_3 = - i\alpha\pi \bigl( { - 6 + 13q - 9q^2 + 2q^3} \bigr)\Gamma \biggl( {\frac{2 - q}{1 - q}} \biggr), \end{aligned}$$

(61)

$$\begin{aligned} & l_4 = 8 \bigl[ {\alpha^2 ( {q - 1} )^2 - 1} \bigr]{}_2F_1 \biggl[ { \frac{3 - 2q}{2 - 2q},\frac{3}{2} + \frac{1}{2 - 2q};1;\alpha^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(62)

$$\begin{aligned} & l_5 = \bigl[ {8 - 4\alpha^2 \bigl( {q^2 - 1} \bigr)} \bigr]{}_2F_1 \biggl[ { \frac{3 - 2q}{2 - 2q},\frac{3}{2} + \frac{1}{2 - 2q},2,\alpha^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(63)

$$\begin{aligned} & l_6 = \alpha^2q ( {2q - 1} ){}_2F_1 \biggl[ {\frac{3 - 2q}{2 - 2q},\frac{3}{2} + \frac{1}{2 - 2q};3;\alpha^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(64)

$$\begin{aligned} & l_7 = 2^{3 + \frac{1}{1 - q}}q ( {\alpha- \alpha q} )^{\frac{1}{q - 1}}\Gamma \biggl( {\frac{3}{2} + \frac{1}{2 - 2q}} \biggr) \Gamma \biggl( {\frac{q}{2 - 2q}} \biggr), \end{aligned}$$

(65)

$$\begin{aligned} & l_8 = \bigl[ {\alpha^2 ( {q - 1} )^2 - 1} \bigr] ( {q - 1} ){}_2F_1 \biggl[ { \frac{3}{2} + \frac{1}{2 - 2q},\frac{q}{2 - 2q}; - \frac{1}{2}; \frac{1}{\alpha^2 ( {q - 1} )^2}} \biggr], \end{aligned}$$

(66)

$$\begin{aligned} & l_9 = \bigl[ {2 ( {q - 2} ) - \alpha^2 ( {q - 1} )^3} \bigr]{}_2F_1 \biggl[ { \frac{3}{2} + \frac{1}{2 - 2q},\frac{q}{2 - 2q};\frac{1}{2}; \frac{1}{\alpha^2 ( {q - 1} )^2}} \biggr], \end{aligned}$$

(67)

$$\begin{aligned} & l_{10} = \frac{ - i}{12\alpha^2\sqrt{\pi}q ( {2q - 1} )\Gamma ( {\frac{q - 2}{q - 1}} )}, \end{aligned}$$

(68)

$$\begin{aligned} & l_{11} = 12\sqrt{\pi}( {2q - 1} )\Gamma \biggl( { \frac{q - 2}{q - 1}} \biggr){}_3F_2 \biggl[ { - \frac{1}{2},1,\frac{3}{2};\frac{2q - 1}{2 ( {q - 1} )},\frac{3q - 2}{2 ( {q - 1} )}; \frac{1}{\alpha^2 ( {q - 1} )^2}} \biggr], \end{aligned}$$

(69)

$$\begin{aligned} & l_{12} = 3\pi^{3 / 2} ( {q - 2} )\Gamma \biggl( {\frac{2 - q}{1 - q}} \biggr), \end{aligned}$$

(70)

$$\begin{aligned} & l_{13} = 2 \bigl[ {\alpha^2 ( {q - 1} )^2 - 1} \bigr]{}_2F_1 \biggl[ { \frac{3 - 2q}{2 - 2q},\frac{3}{2} + \frac{1}{2 - 2q};1;\alpha ^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(71)

$$\begin{aligned} & l_{14} = \bigl[ {2 - \alpha^2 \bigl( {q^2 - 1} \bigr)} \bigr]{}_2F_1 \biggl[ { \frac{3 - 2q}{2 - 2q},\frac{3}{2} + \frac{1}{2 - 2q};2;\alpha ^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(72)

$$\begin{aligned} & l_{15} = 2^{2 + \frac{1}{1 - q}}\alpha q ( {2q - 1} )\Gamma \biggl( {\frac{3 - 2q}{2 - 2q}} \biggr)\Gamma \biggl( { \frac{q - 2}{2q - 2}} \biggr){}_3F_2 \biggl[ {1, \frac{q - 2}{2 ( {q - 1} )},\frac{2q - 3}{2q - 2};\frac{1}{2},\frac{5}{2}; \alpha^2 ( {q - 1} )^2} \biggr] . \end{aligned}$$

(73)

Equation (45) is

$$ S ( {\alpha,q} ) \equiv\frac{\alpha^2}{12\varLambda }\int _0^{u_{\max} } { \bigl[ {1 - ( {q - 1} )\alpha\sqrt{1 + u^2} } \bigr]} ^{\frac{1}{q - 1}}u^4du. $$

(74)

Let

$$ \varLambda' \equiv\int_0^{u_{\max} } { \bigl[ {1 - ( {q - 1} )\alpha\sqrt{1 + u^2} } \bigr]} ^{\frac{1}{q - 1}}u^4du, $$

(75)

then

$$\begin{aligned} & \varLambda'=J_1 \left \{ { \begin{array}{l} J_2 [ {2^{\frac{q}{q - 1}} ( {J_3 + J_4 } )J_5 - ( {J_6 - J_7 } )J_8 } ] \\ \quad{} + \alpha [ {J_9 - J_{10} ( {J_{11} ( {J_{12} - J_{13} } ) - J_{14} } )} ] \\ \end{array} } \right \},\quad\frac{4}{5} < q \le1; \end{aligned}$$

(76)

$$\begin{aligned} & \varLambda' = J_{15} \bigl\{ {J_{16} - 5\pi \bigl[ {J_{17} \bigl( { ( {q - 1} )J_{18} - J_{19} } \bigr) - J_{20} } \bigr]} \bigr\} ,\quad 1 \le q \le 1 + 1 / \alpha, \end{aligned}$$

(77)

where

$$\begin{aligned} & J_1 = - \frac{2^{\frac{1 + 6q}{1 - q}}}{\alpha^3 ( {q - 2} ) ( {q - 1} )^2 ( {3q - 2} ) ( {5q - 4} )\Gamma ( {\frac{1}{1 - q}} )}, \end{aligned}$$

(78)

$$\begin{aligned} & J_2 = - 3i2^{\frac{1 + q}{q - 1}}\sqrt{\pi}\Gamma \biggl[ { \frac {3 - 4q}{2 ( {q - 1} )}} \biggr]\Gamma \biggl[ {\frac{q - 2}{2 ( {q - 1} )}} \biggr], \end{aligned}$$

(79)

$$\begin{aligned} & J_3 = 2^{\frac{3q}{q - 1}} - 2^{\frac{1 + 2q}{q - 1}}q + 2^{\frac{2}{q - 1}}\alpha^2 \bigl( { - 9\times2^{\frac{q}{q - 1}} + 49 \times2^{\frac{1}{q - 1}}q - 21\times2^{\frac{q}{q - 1}}q^2 + 11 \times2^{\frac{1}{q - 1}}q^3} \bigr), \end{aligned}$$

(80)

$$\begin{aligned} & J_4 = \alpha^4\left ( { \begin{array}{l} 5\times2^{\frac{2 + q}{q - 1}} - 49\times2^{\frac{3}{q - 1}}q + 47\times 2^{\frac{2 + q}{q - 1}}q^2 \\ \quad{}- 11\times2^{\frac{3q}{q - 1}}q^3 + 5\times2^{\frac{3q}{q - 1}}q^4 - 7\times2^{\frac{3}{q - 1}}q^5 \end{array} } \right ), \end{aligned}$$

(81)

$$\begin{aligned} & J_5 = {}_2F_1 \biggl[ {1 + \frac{1}{2 - 2q},\frac{q - 2}{2 ( {q - 1} )};1;\alpha^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(82)

$$\begin{aligned} & J_6 = 2^{\frac{4q}{q - 1}} - 2^{\frac{1 + 3q}{q - 1}}q + \alpha^2 \bigl( { - 3\times2^{\frac{1 + 3q}{q - 1}} + 39\times2^{\frac{3 + q}{q - 1}}q - 73\times2^{\frac{4}{q - 1}}q^2 + 5\times2^{\frac{2 + 2q}{q - 1}}q^3} \bigr), \end{aligned}$$

(83)

$$\begin{aligned} & J_7 = 3\times2^{\frac{3}{q - 1}}\alpha^4 \left ( { \begin{array}{l} - 2^{\frac{q}{q - 1}} + 15\times2^{\frac{1}{q - 1}}q - 39\times 2^{\frac{1}{q - 1}}q^2 \\ \quad{}+23\times2^{\frac{q}{q - 1}}q^3 - 25\times2^{\frac{1}{q - 1}}q^4 + 5\times 2^{\frac{1}{q - 1}}q^5 \\ \end{array} } \right ), \end{aligned}$$

(84)

$$\begin{aligned} & J_8 = {}_2F_1 \biggl[ {1 + \frac{1}{2 - 2q},\frac{q - 2}{2 ( {q - 1} )};2;\alpha^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(85)

$$\begin{aligned} & \begin{aligned}[b] J_9 & = i2^{\frac{4}{q - 1}} \alpha^3\sqrt{\pi} \left ( \begin{array}{l} - 2^{4 + \frac{2}{q - 1}} + 21\times2^{\frac{2q}{q - 1}}q - 43\times 2^{\frac{2q}{q - 1}}q^2 \\ \quad{} +171\times2^{\frac{2}{q - 1}}q^3 - 41\times2^{\frac{1 + q}{q - 1}}q^4 + 15\times2^{\frac{2}{q - 1}}q^5 \\ \end{array} \right ) \\ &\quad \times\Gamma \biggl( {\frac{1}{2 - 2q}} \biggr)\Gamma \biggl[ { \frac{q - 2}{2 ( {q - 1} )}} \biggr]{}_2F_1 \biggl[ {1 + \frac{1}{2 - 2q},\frac{q - 2}{2 ( {q - 1} )};4;\alpha^2 ( {q - 1} )^2} \biggr], \end{aligned} \end{aligned}$$

(86)

$$\begin{aligned} & J_{10} = 3\times2^{\frac{2q}{q - 1}} ( {\alpha- \alpha q} )^{\frac{q}{q - 1}}, \end{aligned}$$

(87)

$$\begin{aligned} & J_{11} = 2^{\frac{q}{q - 1}}\Gamma \biggl( { \frac{3 - 2q}{2 - 2q}} \biggr)\Gamma \biggl( {\frac{q}{2 - 2q} - \frac{3}{2}} \biggr), \end{aligned}$$

(88)

$$\begin{aligned} & \begin{aligned}[b] J_{12} &= - 2^{\frac{2 + q}{q - 1}} + 7 \times2^{\frac{3}{q - 1}}q - 2^{\frac{3q}{q - 1}}q^2 + 3\times2^{\frac{3}{q - 1}}q^3 + \alpha^2\left ( { \begin{array}{l} 2^{\frac{2 + q}{q - 1}} - 11\times2^{\frac{3}{q - 1}}q + 3\times 2^{\frac{3q}{q - 1}}q^2 \\ \quad{} -13\times2^{\frac{2 + q}{q - 1}}q^3 + 7\times2^{\frac{2 + q}{q - 1}}q^4 - 3\times2^{\frac{3}{q - 1}}q^5 \\ \end{array} } \right ) \\ &\quad \times{}_2F_1 \biggl[ {1 + \frac{1}{2 - 2q}, - \frac{3}{2} + \frac {q}{2 - 2q}; - \frac{1}{2};\frac{1}{\alpha^2 ( {q - 1} )^2}} \biggr] , \end{aligned} \end{aligned}$$

(89)

$$\begin{aligned} & \begin{aligned}[b] J_{13} &= - 2^{\frac{1 + q}{q - 1}}q \bigl( {2^{\frac{q}{q - 1}} - 5\times2^{\frac{1}{q - 1}} + 3\times2^{\frac{1}{q - 1}}q^2} \bigr) + \alpha^2\left ( \begin{array}{l} 2^{\frac{2 + q}{q - 1}} - 11\times2^{\frac{3}{q - 1}}q + 3\times 2^{\frac{3q}{q - 1}}q^2 \\ \quad{} -13\times2^{\frac{2 + q}{q - 1}}q^3 + 7\times2^{\frac{2 + q}{q - 1}}q^4 - 3\times2^{\frac{3}{q - 1}}q^5 \\ \end{array} \right ) \\ &\quad \times{}_2F_1 \biggl[ {1 + \frac{1}{2 - 2q}, - \frac{3}{2} + \frac {q}{2 - 2q};\frac{1}{2};\frac{1}{\alpha^2 ( {q - 1} )^2}} \biggr] , \end{aligned} \end{aligned}$$

(90)

$$\begin{aligned} & \begin{aligned}[b] J_{14} &= \alpha \bigl( {2^{\frac{4q}{q - 1}} - 17\times2^{\frac{2 + 2q}{q - 1}}q + 13\times2^{\frac{1 + 3q}{q - 1}}q^2 - 67\times2^{\frac{4}{q - 1}}q^3 + 15\times2^{\frac{4}{q - 1}}q^4} \bigr) \\ &\quad \times\Gamma \biggl[ {\frac{q - 2}{2 ( {q - 1} )}} \biggr]\Gamma \biggl[ { \frac{q}{2 - 2q} - 2} \biggr]{}_2F_1 \biggl[ { \frac{q - 2}{2 ( {q - 1} )},\frac{q}{2 - 2q} - 2;\frac{1}{2};\frac{1}{\alpha^2 ( {q - 1} )^2}} \biggr], \end{aligned} \end{aligned}$$

(91)

$$\begin{aligned} & J_{15} = \frac{i2^{\frac{1 - 7q}{q - 1}}}{5\alpha^3\sqrt{\pi}( {q - 1} )^3 ( {3q - 2} ) ( {5q - 4} )\Gamma ( {\frac{1}{1 - q}} )}, \end{aligned}$$

(92)

$$\begin{aligned} & \begin{aligned}[b] J_{16} &= - 2^{6 + \frac{5}{q - 1}} \alpha^3 ( {q - 1} )^3 \bigl( {8 - 22q + 15q^2} \bigr)\Gamma \biggl[ {\frac{q - 2}{2 ( {q - 1} )}} \biggr]\Gamma \biggl( {\frac{1}{2 - 2q}} \biggr) \\ &\quad \times{}_3F_2 \biggl[ {1,\frac{1}{2 - 2q}, \frac{1}{1 - q} + \frac{q}{2 ( {q - 1} )};\frac{1}{2},\frac{7}{2}; \alpha^2 ( {q - 1} )^2} \biggr], \end{aligned} \end{aligned}$$

(93)

$$\begin{aligned} & J_{17} = 3\Gamma \biggl[ {\frac{3 - 4q}{2 ( {q - 1} )}} \biggr] \Gamma \biggl[ {\frac{q - 2}{2 ( {q - 1} )}} \biggr], \end{aligned}$$

(94)

$$\begin{aligned} & J_{18} =\left [ \begin{array}{l} 2^{\frac{5q}{q - 1}} + \alpha^2 ( { - 9\times2^{3 + \frac{5}{q - 1}} + 5\times2^{\frac{5q}{q - 1}}q - 11\times2^{3 + \frac{5}{q - 1}}q^2} ) \\ \quad{} +\alpha^4\left ( { \begin{array}{l} 5\times2^{3 + \frac{5}{q - 1}} - 11\times2^{4 + \frac{5}{q - 1}}q + 9\times2^{\frac{5q}{q - 1}}q^2 \\ \quad{} - 13\times2^{4 + \frac{5}{q - 1}}q^3 + 7\times2^{3 + \frac{5}{q - 1}}q^4 \end{array} } \right ) \end{array} \right ] {}_2F_1 \biggl[ {1 + \frac{1}{2 - 2q},\frac{q - 2}{2 ( {q - 1} )};1;\alpha^2 ( {q - 1} )^2} \biggr] , \end{aligned}$$

(95)

$$\begin{aligned} & J_{19} =\left [ \begin{array}{l} 2^{\frac{5q}{q - 1}} ( {q - 1} ) - 2^{2 + \frac{5}{q - 1}}\alpha ^2 ( { - 12 + 45q - 53q^2 + 20q^3} ) \\ \quad{} +3\alpha^4\left ( \begin{array}{l} - 2^{2 + \frac{5}{q - 1}} + 2^{\frac{5q}{q - 1}}q - 23\times2^{2 + \frac{5}{q - 1}}q^2 \\ \quad{} +31\times2^{2 + \frac{5}{q - 1}}q^3 - 5\times2^{4 + \frac{5}{q - 1}}q^4 + 5\times2^{2 + \frac{5}{q - 1}}q^5 \end{array} \right ) \end{array} \right ] \\ &\hphantom{J_{19} =} \times {}_2F_1 \biggl[ {1 + \frac{1}{2 - 2q},\frac{q - 2}{2 ( {q - 1} )};2;\alpha^2 ( {q - 1} )^2} \biggr], \end{aligned}$$

(96)

$$\begin{aligned} & J_{20} = \frac{2^{\frac{5q}{q - 1}}\alpha ( {8 - 22q + 15q^2} )\Gamma ( {\frac{1}{1 - q}} )\Gamma ( {\frac {1}{q - 1}} )}{\Gamma [ {2 + \frac{1}{2 ( {q - 1} )}} ]\Gamma [ {1 + \frac{q}{2 ( {q - 1} )}} ]} {}_3F_2 \biggl[ { - \frac{3}{2},1,\frac{3}{2};2 + \frac{1}{2 ( {q - 1} )},1 + \frac{q}{2 ( {q - 1} )};\frac{1}{\alpha^2 ( {q - 1} )^2}} \biggr] . \end{aligned}$$

(97)