2.1 Landau–Lifshitz equation

For a system of $N$ charged particles with the same mass $m$ and equal charge $e$ , for each particle the Lorentz–Dirac equation (Dirac Reference Dirac1938) is

(2.1) $$\begin{eqnarray}\displaystyle ma_{i}^{\unicode[STIX]{x1D707}}=\frac{e}{c}\unicode[STIX]{x1D641}_{(i)}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}v_{\unicode[STIX]{x1D708}i}+m\unicode[STIX]{x1D70F}_{o}\left[{\dot{a}}_{i}^{\unicode[STIX]{x1D707}}+\frac{a_{i}^{2}}{c^{2}}v_{i}^{\unicode[STIX]{x1D707}}\right],\quad i=1,\ldots ,N, & & \displaystyle\end{eqnarray}$$

where $c$ , $\unicode[STIX]{x1D641}_{(i)}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$ and $\unicode[STIX]{x1D70F}_{o}=2e^{2}/3mc^{3}$ are the speed of light, the field strength tensor due to the action of the other particles, the external field on the $i$ th particle and the characteristic time of the charge $e$ , respectively. The subscript ‘ $i$ ’ represents the $i$ th particle and $v_{i}^{\unicode[STIX]{x1D707}}$ , $a_{i}^{\unicode[STIX]{x1D707}}$ and ${\dot{a}}_{i}^{\unicode[STIX]{x1D707}}=\text{d}a_{i}^{\unicode[STIX]{x1D707}}/\text{d}\unicode[STIX]{x1D70F}_{i}$ are the $4$ -velocity, the $4$ -acceleration and the $4$ -hyper-acceleration of the $i$ th particle, respectively, with $\unicode[STIX]{x1D70F}_{i}$ the proper time of the $i$ th particle. The equation can be expressed as

(2.2) $$\begin{eqnarray}\displaystyle ma_{i}^{\unicode[STIX]{x1D707}}=\frac{e}{c}\unicode[STIX]{x1D641}_{(i)}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}v_{\unicode[STIX]{x1D708}i}+m\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{i}){\dot{a}}_{\unicode[STIX]{x1D708}i},\quad i=1,\ldots ,N, & & \displaystyle\end{eqnarray}$$

where

(2.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{i})=g^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}-\frac{v_{i}^{\unicode[STIX]{x1D707}}v_{i}^{\unicode[STIX]{x1D708}}}{c^{2}}, & & \displaystyle\end{eqnarray}$$

with $g^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$ representing the Minkowski metric and the kinematic relation $v^{\unicode[STIX]{x1D707}}{\dot{a}}_{\unicode[STIX]{x1D707}}+a^{\unicode[STIX]{x1D707}}a_{\unicode[STIX]{x1D707}}=0$ has been used. In order to incorporate the Landau–Lifshitz proposal in (2.2), the hyper-acceleration must be substituted by

(2.4) $$\begin{eqnarray}\displaystyle {\dot{a}}_{\unicode[STIX]{x1D708}i}\rightarrow \frac{e}{mc}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D70F}_{i}}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{(i)}v_{i}^{\unicode[STIX]{x1D6FC}}=\frac{e}{mc}\left[\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{(i)}a_{i}^{\unicode[STIX]{x1D6FC}}+v_{i}^{\unicode[STIX]{x1D70C}}v_{i}^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{(i)}}{\unicode[STIX]{x2202}x_{i}^{\unicode[STIX]{x1D70C}}}\right], & & \displaystyle\end{eqnarray}$$

and in a second step, in the first term of the right-hand side of (2.4), the acceleration $a_{i}^{\unicode[STIX]{x1D6FC}}$ , must be substituted by the Lorentz force; that is:

(2.5) $$\begin{eqnarray}\displaystyle {\dot{a}}_{\unicode[STIX]{x1D708}i}\rightarrow \frac{e}{mc}\left[\frac{e}{mc}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{(i)}\unicode[STIX]{x1D641}_{(i)}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}v_{\unicode[STIX]{x1D6FD}i}+v_{i}^{\unicode[STIX]{x1D70C}}v_{i}^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{(i)}}{\unicode[STIX]{x2202}x_{i}^{\unicode[STIX]{x1D70C}}}\right]. & & \displaystyle\end{eqnarray}$$

Therefore, the Landau–Lifshitz equation of motion for each particle is (Ares de Parga Reference Ares de Parga2006) $i=1,\ldots ,N,$

(2.6) $$\begin{eqnarray}\displaystyle ma_{i}^{\unicode[STIX]{x1D707}}=\frac{e}{c}\unicode[STIX]{x1D641}_{(i)}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}v_{\unicode[STIX]{x1D708}i}+m\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{i})\frac{e}{mc}\left[\frac{e}{mc}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{(i)}\unicode[STIX]{x1D641}_{(i)}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}v_{\unicode[STIX]{x1D6FD}i}+v_{i}^{\unicode[STIX]{x1D70C}}v_{i}^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{(i)}}{\unicode[STIX]{x2202}x_{i}^{\unicode[STIX]{x1D70C}}}\right]. & & \displaystyle\end{eqnarray}$$

On the other hand, the force applied on each particle must be decomposed into two parts: the external field and the field produced by the other particles on the $i$ th particle; that is

(2.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D641}_{(i)}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}+\mathop{\sum }_{j\neq i}\unicode[STIX]{x1D641}_{ji}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(x_{\unicode[STIX]{x1D70C}}), & & \displaystyle\end{eqnarray}$$

where

(2.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D641}_{ji}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(x_{\unicode[STIX]{x1D70C}}) & = & \displaystyle \frac{4\unicode[STIX]{x03C0}e}{c}\iiint \text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }\,\text{d}\unicode[STIX]{x1D70F}^{\prime }[v^{\prime \unicode[STIX]{x1D708}}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D707}}-v^{\prime \unicode[STIX]{x1D707}}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D708}}]D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \times \,\unicode[STIX]{x1D6FF}(x_{\unicode[STIX]{x1D70C}}^{\prime }-x_{\unicode[STIX]{x1D70C}j}(\unicode[STIX]{x1D70F}^{\prime }))\unicode[STIX]{x1D6FF}(v_{\unicode[STIX]{x1D70C}}^{\prime }-v_{\unicode[STIX]{x1D70C}j}(\unicode[STIX]{x1D70F}^{\prime })),\end{eqnarray}$$

where $D_{\text{ret}}$ represents the retarded Green function (Jackson Reference Jackson1998).

Finally, it is important to note that the equation of motion described by (2.6) does not depend on the hyper-acceleration and consequently the phase space is described by a $8N$ -dimensional space with the regular $N$ constraint, $v^{\unicode[STIX]{x1D707}}v_{\unicode[STIX]{x1D707}}=c^{2}$ . This point represents one of the different approaches to obtaining the relativistic Vlasov equation compared with the work done by Hakim & Mangeney (Reference Hakim and Mangeney1968).

2.2 Relativistic Vlasov equation including radiation reaction force

Different derivations of a hierarchy of statistical equations were developed during the last century, independently, by Bogolyubov, Born and Green, Kirkwood and Yvon (the so-called BBGKY hierarchy). Particularly, we can cite the work done by Bogolyubov (Reference Bogolyubov1946) in the early sixties. An important approach was done by Hakim I (Reference Hakim1967), Hakim II (Reference Hakim1967) which lead to a hierarchy of manifestly covariant equations for a system of charge particles including the radiation reaction force (Hakim & Mangeney Reference Hakim and Mangeney1968, Reference Hakim and Mangeney1971) with the constraint of the absence of any correlation between the particles and equations to first order in $ne^{2}\unicode[STIX]{x1D70F}_{o}$ . We will follow Hakim and Mangeney in order to obtain a relativistic covariant modified Vlasov equation but with the difference that we will start from a phase space without the hyper-acceleration as an independent variable and including a weak constant electric field. The fact of not considering the hyper-acceleration as an independent variable is because we consider the Landau–Lifshitz equation of motion, which does not contain such dependence.

Let us make a summary of the derivation of the modified Vlasov equation just to place emphasis on the differences between our method and the Hakim and Mangeney one. The phase space in special relativity must be constructed by following the dynamics of the particles in the system. By looking at the equation of motion, equation (2.6), it is clear that the phase space is composed of

(2.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}={\{x_{\unicode[STIX]{x1D707}}\}}^{N}\times {\{v_{\unicode[STIX]{x1D707}}\}}^{N}, & & \displaystyle\end{eqnarray}$$

accompanied by the constraint

(2.10) $$\begin{eqnarray}\displaystyle v_{\unicode[STIX]{x1D707}}v^{\unicode[STIX]{x1D707}}=c^{2}, & & \displaystyle\end{eqnarray}$$

which can be incorporated directly in the densities by using a delta function. Notice that unlike Hakim & Mangeney (Reference Hakim and Mangeney1968) the hyper-acceleration is not taken as part of the phase space since the equation of motion to be considered is the Landau–Lifshitz equation which is a second-order differential equation. Using the regular assumptions of the Gibbs ensemble (Hakim I Reference Hakim1967; Hakim II Reference Hakim1967), it is possible to define a random one-particle density as

(2.11) $$\begin{eqnarray}\displaystyle R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})=\mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D6FF}(x_{\unicode[STIX]{x1D707}}-x_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v_{\unicode[STIX]{x1D707}}-v_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F})), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}$ represents a parameter which will describe the trajectory of each particle by means of $v_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F})=\text{d}x_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F})/\text{d}\unicode[STIX]{x1D70F}$ . Also, following Hakim & Mangeney (Reference Hakim and Mangeney1968), we define the proper time-dependent density,

(2.12) $$\begin{eqnarray}\displaystyle D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})=N\langle R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})\rangle , & & \displaystyle\end{eqnarray}$$

where the brackets $\langle R_{1}\rangle$ represent the regular average weighted by random initial conditions (Hakim I Reference Hakim1967; Hakim II Reference Hakim1967). Also, we will need

(2.13) $$\begin{eqnarray}\displaystyle R_{2}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime }) & = & \displaystyle \mathop{\sum }_{i\neq j}\unicode[STIX]{x1D6FF}(x_{\unicode[STIX]{x1D707}}-x_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F}))\nonumber\\ \displaystyle & & \displaystyle \times \,\unicode[STIX]{x1D6FF}(v_{\unicode[STIX]{x1D707}}-v_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(x_{\unicode[STIX]{x1D707}}^{\prime }-x_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F}^{\prime }))\unicode[STIX]{x1D6FF}(v_{\unicode[STIX]{x1D707}}^{\prime }-v_{\unicode[STIX]{x1D707}i}(\unicode[STIX]{x1D70F}^{\prime })),\quad\end{eqnarray}$$

and

(2.14) $$\begin{eqnarray}\displaystyle D_{2}=(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })=N(N-1)\langle R_{2}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })\rangle . & & \displaystyle\end{eqnarray}$$

Finally, the relativistic one-particle density is

(2.15) $$\begin{eqnarray}\displaystyle {\mathcal{N}}_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}})=\int _{-\infty }^{\infty }D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}. & & \displaystyle\end{eqnarray}$$

In order to obtain the continuity equation, let us follow the Klimontovich method (Klimontovich Reference Klimontovich1960), but with a difference since the force on each particle will not be the Lorentz force but the Lorentz force plus the radiation reaction force due to Landau and Lifshitz represented by (2.6). Starting from equation (2.11) in the equivalent following form

(2.16) $$\begin{eqnarray}\displaystyle R_{1}(x^{\unicode[STIX]{x1D707}},v^{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})=\mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})), & & \displaystyle\end{eqnarray}$$

and partially differentiating it with respect to $\unicode[STIX]{x1D70F}$ , we have,

(2.17) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x^{\unicode[STIX]{x1D707}},v^{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}} & = & \displaystyle \mathop{\sum }_{i=1}^{N}\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=1}^{N}\left\{-\frac{\text{d}x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{-\frac{\text{d}v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}.\end{eqnarray}$$

Let us analyse the term in the second line of (2.17); first; it is necessary to take into account the property of the $\unicode[STIX]{x1D6FF}$ -function,

(2.18) $$\begin{eqnarray}\displaystyle f(a)\unicode[STIX]{x1D6FF}(a-b)=f(b)\unicode[STIX]{x1D6FF}(a-b). & & \displaystyle\end{eqnarray}$$

Then, since $\text{d}x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})/\text{d}\unicode[STIX]{x1D70F}=v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})$ , we obtain

(2.19) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \left\{-\frac{\text{d}x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad =-v^{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad =-v_{\unicode[STIX]{x1D707}}\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}[\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))]\right\}.\end{eqnarray}$$

Consequently,

(2.20) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{i=1}^{N}\left\{-\frac{\text{d}x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{i=1}^{N}-v^{\unicode[STIX]{x1D707}}\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}[\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))]\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =-v^{\unicode[STIX]{x1D707}}\left\{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\mathop{\sum }_{i=1}^{N}[\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))]\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =-v^{\unicode[STIX]{x1D707}}{\displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x^{\unicode[STIX]{x1D707}},v^{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}}.\end{eqnarray}$$

We must now analyse the term on the third line of (2.17),

(2.21) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{-\frac{\text{d}v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{-a_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\{-a_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\},\end{eqnarray}$$

where $a_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})$ enters inside the partial derivative since it depends only on $\unicode[STIX]{x1D70F}$ . Therefore,

(2.22) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{-\frac{\text{d}v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{i=1}^{N}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\{-a_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\}.\end{eqnarray}$$

Noticing that $a_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})=a_{i}^{\unicode[STIX]{x1D707}}(x_{i}^{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F}),v_{i}^{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F}))$ , we obtain

(2.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{-\frac{\text{d}v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{i=1}^{N}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left\{\begin{array}{@{}c@{}}-a_{i}^{\unicode[STIX]{x1D707}}(x_{i}^{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F}),v_{i}^{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F}))\\ \times \,\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\end{array}\right\}.\end{eqnarray}$$

By using (2.18), we arrive at

(2.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\left\{-\frac{\text{d}v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})}{\text{d}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left\{-a^{\unicode[STIX]{x1D707}}(x^{\unicode[STIX]{x1D70C}},v^{\unicode[STIX]{x1D70C}})\left[\begin{array}{@{}c@{}}\displaystyle \mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D6FF}(x^{\unicode[STIX]{x1D707}}-x_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\\[12.0pt] \times \,\unicode[STIX]{x1D6FF}(v^{\unicode[STIX]{x1D707}}-v_{i}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F}))\end{array}\right]\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}[a^{\unicode[STIX]{x1D707}}(x^{\unicode[STIX]{x1D70C}},v^{\unicode[STIX]{x1D70C}})R].\end{eqnarray}$$

Here, it is important to analyse the following cases. First, let us consider that the force is the Lorentz force; that is,

(2.25) $$\begin{eqnarray}\displaystyle a^{\unicode[STIX]{x1D707}}(x^{\unicode[STIX]{x1D70C}},v^{\unicode[STIX]{x1D70C}})=\frac{e}{mc}\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(x^{\unicode[STIX]{x1D70C}})v_{\unicode[STIX]{x1D708}}. & & \displaystyle\end{eqnarray}$$

We can calculate the partial derivative of $a^{\unicode[STIX]{x1D707}}$ with respect to $v^{\unicode[STIX]{x1D707}}$ ,

(2.26) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}a^{\unicode[STIX]{x1D707}}(x^{\unicode[STIX]{x1D70C}},v^{\unicode[STIX]{x1D70C}})=\frac{e}{mc}\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(x^{\unicode[STIX]{x1D70C}})\frac{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D708}}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}=\frac{e}{mc}\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(x_{\unicode[STIX]{x1D70C}})\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\frac{e}{mc}\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D707}}(x_{\unicode[STIX]{x1D70C}})=0. & & \displaystyle\end{eqnarray}$$

Therefore, in this case

(2.27) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}[a^{\unicode[STIX]{x1D707}}(x^{\unicode[STIX]{x1D70C}},v^{\unicode[STIX]{x1D70C}})R]=a^{\unicode[STIX]{x1D707}}(x^{\unicode[STIX]{x1D70C}},v^{\unicode[STIX]{x1D70C}})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}R. & & \displaystyle\end{eqnarray}$$

Finally for the Lorentz force case and by using (2.17), (2.20), (2.24) and (2.27), we obtain the classical Klimontovich result (Klimontovich Reference Klimontovich1960) for the continuity equation written as

(2.28) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}+a^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}=0. & & \displaystyle\end{eqnarray}$$

However, when the equation of motion is governed by the Landau–Lifshitz equation, equation (2.6), (2.27) cannot be used due to the reaction radiation term. Therefore, by using (2.17), (2.20) and (2.24), in the case where the motion of the particles is governed by the Landau–Lifshitz equation, the classical Klimontovich result must be modified and the continuity equation can be expressed as

(2.29) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}+\frac{\unicode[STIX]{x2202}a^{\unicode[STIX]{x1D707}}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}=0. & & \displaystyle\end{eqnarray}$$

Notice that a difference appears with the Hakim & Mangeney (Reference Hakim and Mangeney1968) dependence of $R_{1}$ since they obtained $R_{1}(x_{i}^{\unicode[STIX]{x1D707}},v_{i}^{\unicode[STIX]{x1D707}},{\dot{a}}_{\unicode[STIX]{x1D708}i})$ and our result is $R_{1}=R_{1}(x^{\unicode[STIX]{x1D707}},v^{\unicode[STIX]{x1D707}})$ . This is because the equation of motion is taken from the beginning as equation (2.6) which does not depend on ${\dot{a}}_{\unicode[STIX]{x1D708}i}$ . Therefore, by using (2.6) and (2.7), we arrive at,

(2.30) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{e}{mc}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}v_{\unicode[STIX]{x1D708}}\\ \displaystyle +\,\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})\left[\frac{e}{mc}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}v_{\unicode[STIX]{x1D6FD}}+v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}\right]R_{1}\end{array}\right]=0.\end{eqnarray}$$

Let us analyse the second line of (2.30). First, the strength tensor $\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$ must be decomposed into two terms corresponding to the external force and the internal force, see (2.7) and (2.8); second, the non-radiation terms are represented in the second line of (2.31). Finally, the radiation terms are represented in the third line of (2.31).

(2.31) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{e}{mc}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}[\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}v_{\unicode[STIX]{x1D708}}R_{1}]+\frac{e}{mc}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}[\unicode[STIX]{x1D641}_{\text{int}}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}v_{\unicode[STIX]{x1D708}}R_{1}]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{e}{mc}\unicode[STIX]{x1D70F}_{o}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left[\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})\left[\begin{array}{@{}c@{}}\displaystyle \frac{e}{mc}[\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{\text{ext}}+\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{\text{int}}]\\ \times \,[\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}+\unicode[STIX]{x1D641}_{\text{int}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}]v_{\unicode[STIX]{x1D6FD}}\\ \displaystyle +\,v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}[\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{\text{ext}}+\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{\text{int}}]}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}\end{array}\right]R_{1}\right]=0.\end{eqnarray}$$

By using (2.8), the first two lines of (2.31) are expressed in the first two lines of (2.32). The radiation term must be analysed as follows. Terms depending on a higher order of $e^{2}$ will be neglected. In this order of ideas, as we can notice in (2.8), the internal $\unicode[STIX]{x1D641}_{\text{int}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ depends on $e$ and consequently the product of the external $\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ with the internal $\unicode[STIX]{x1D641}_{\text{int}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ as well as the product of the internal $\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ with the internal $\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ will be neglected since there is a factor $e\unicode[STIX]{x1D70F}_{o}/mc$ in the radiation reaction term. Finally, the conserved terms are described in the third and fourth lines of (2.32).

(2.32) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}+\frac{e}{mc}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}[\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}v_{\unicode[STIX]{x1D708}}R_{1}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}v_{\unicode[STIX]{x1D708}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D707}}}\int \text{d}\unicode[STIX]{x1D70F}^{\prime }\,\text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }[\{v^{\unicode[STIX]{x1D708}\prime }\unicode[STIX]{x2202}^{\unicode[STIX]{x1D707}}-v^{\unicode[STIX]{x1D707}\prime }\unicode[STIX]{x2202}^{\unicode[STIX]{x1D708}}\}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,R_{2}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{e}{mc}\unicode[STIX]{x1D70F}_{o}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left[\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})\left[\frac{e}{mc}[\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{\text{ext}}][\unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}]v_{\unicode[STIX]{x1D6FD}}+v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}}^{\text{ext}}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}\right]R_{1}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}_{o}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left\{\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\int \text{d}\unicode[STIX]{x1D70F}^{\prime }\,\text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }\\ \displaystyle \left[\{v_{\unicode[STIX]{x1D6FC}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}-v_{\unicode[STIX]{x1D708}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })\right]R_{2}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })\end{array}\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =0.\end{eqnarray}$$

In the following, we abandon the inclusion of an external weak constant electric field since the dispersion relations are not changed as long as the Debye length is sufficiently small compared with the external weak constant field (see appendix A). Therefore, in the following,

(2.33) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D641}_{\text{ext}}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=0. & & \displaystyle\end{eqnarray}$$

We arrive at

(2.34) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}R_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}v_{\unicode[STIX]{x1D708}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D707}}}\int \text{d}\unicode[STIX]{x1D70F}^{\prime }\,\text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }[\{v^{\unicode[STIX]{x1D708}\prime }\unicode[STIX]{x2202}^{\unicode[STIX]{x1D707}}-v^{\unicode[STIX]{x1D707}\prime }\unicode[STIX]{x2202}^{\unicode[STIX]{x1D708}}\}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,R_{2}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}_{o}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left\{\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\int \text{d}\unicode[STIX]{x1D70F}^{\prime }\,\text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }\\ \displaystyle \times \left[\{v_{\unicode[STIX]{x1D6FC}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}-v_{\unicode[STIX]{x1D708}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })\right]R_{2}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })\end{array}\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =0.\end{eqnarray}$$

Considering that the correlation function vanishes or is of order $(ne^{2}\unicode[STIX]{x1D70F}_{o})^{2}$ ; that is,

(2.35) $$\begin{eqnarray}\displaystyle D_{2}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}},\unicode[STIX]{x1D70F};x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime },\unicode[STIX]{x1D70F}^{\prime })=D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})D_{1}(x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime };\unicode[STIX]{x1D70F}^{\prime }). & & \displaystyle\end{eqnarray}$$

By calculating the average of (2.12) and taking $N-1\simeq N$ , we arrive at

(2.36) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}Nv_{\unicode[STIX]{x1D708}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D707}}}\int \text{d}\unicode[STIX]{x1D70F}^{\prime }\,\text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }[\{v^{\unicode[STIX]{x1D708}\prime }\unicode[STIX]{x2202}^{\unicode[STIX]{x1D707}}-v^{\unicode[STIX]{x1D707}\prime }\unicode[STIX]{x2202}^{\unicode[STIX]{x1D708}}\}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})D_{1}(x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime };\unicode[STIX]{x1D70F}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}_{o}\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}N\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left\{\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\int \text{d}\unicode[STIX]{x1D70F}^{\prime }\,\text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }\\[9.0pt] \displaystyle \left[\{v_{\unicode[STIX]{x1D6FC}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}-v_{\unicode[STIX]{x1D708}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })\right]\\[10.0pt] \times \,D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})D_{1}(x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime };\unicode[STIX]{x1D70F}^{\prime })\end{array}\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =0.\end{eqnarray}$$

Considering that

(2.37) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D70F}\rightarrow \mp \infty }D_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}};\unicode[STIX]{x1D70F})=0, & & \displaystyle\end{eqnarray}$$

and integrating over the proper time $\unicode[STIX]{x1D70F}$ , we obtain

(2.38) $$\begin{eqnarray}\displaystyle & & \displaystyle v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}{\mathcal{N}}_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}+\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}Nv_{\unicode[STIX]{x1D708}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D707}}}{\mathcal{N}}_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}})\nonumber\\ \displaystyle & & \displaystyle \qquad \times \int \text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }[\{v^{\prime \unicode[STIX]{x1D708}}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D707}}-v^{\prime \unicode[STIX]{x1D707}}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D708}}\}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })]{\mathcal{N}}_{1}(x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D70F}_{o}\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}N\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left\{\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\int \text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }\\[8.0pt] \displaystyle \left[\{v_{\unicode[STIX]{x1D6FC}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}-v_{\unicode[STIX]{x1D708}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })\right]\\[9.0pt] \times \,{\mathcal{N}}_{1}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}}){\mathcal{N}}_{1}(x_{\unicode[STIX]{x1D707}}^{\prime },v_{\unicode[STIX]{x1D707}}^{\prime })\end{array}\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =0,\end{eqnarray}$$

which represents a modified relativistic Vlasov equation up to the $ne^{2}\unicode[STIX]{x1D70F}_{o}$ order. It has to be highlighted that no collision term is obtained because all of the interactions are electromagnetic and there are no collisions or friction forces. Nevertheless, the radiation reaction term can be taken as a collision term because without it we recover the usual Vlasov equation with no collisions.

3.1 First-order perturbation

We assume now that the plasma is in an equilibrium state described by the Jüttner distribution function (Jüttner Reference Jüttner1911; Synge Reference Synge1956; Cubero et al. Reference Cubero, Casado-Pascual, Dunkel, Talkner and Hänggi2007; López-Carrera & Ares de Parga Reference López-Carrera and Ares de Parga2008). Accordingly, a small perturbation can be described

(3.1) $$\begin{eqnarray}\displaystyle {\mathcal{N}}_{eq}+{\mathcal{N}}_{p}\simeq {\mathcal{N}}_{J}+{\mathcal{N}}_{p}, & & \displaystyle\end{eqnarray}$$

where ${\mathcal{N}}_{eq}$ , ${\mathcal{N}}_{p}$ and ${\mathcal{N}}_{J}$ represent the distribution function in equilibrium, the perturbed distribution function and the Jüttner distribution function (see appendix B). Therefore, applying this to (2.38), considering that ${\mathcal{N}}_{eq}\simeq {\mathcal{N}}_{J}$ is the solution of the unperturbed case which does not depend on $x_{\unicode[STIX]{x1D707}}$ , that is ${\mathcal{N}}_{eq}\neq {\mathcal{N}}_{eq}(x_{\unicode[STIX]{x1D707}})$ and taking into account that the derivatives of the perturbation and the product of the perturbations must both be neglected, we arrive at

(3.2) $$\begin{eqnarray}\displaystyle & & \displaystyle v^{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}{\mathcal{N}}_{p}(x_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}})}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D707}}}+\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}Nv_{\unicode[STIX]{x1D708}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D707}}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D707}})\int \text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,[\{v^{\prime \unicode[STIX]{x1D708}}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D707}}-v^{\prime \unicode[STIX]{x1D707}}\unicode[STIX]{x2202}^{\unicode[STIX]{x1D708}}\}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })]{\mathcal{N}}_{p}(x_{\unicode[STIX]{x1D706}}^{\prime },v_{\unicode[STIX]{x1D706}}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad =-\unicode[STIX]{x1D70F}_{o}\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\left\{\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}\int \text{d}^{4}x^{\prime }\,\text{d}^{4}v^{\prime }\\[9.0pt] \displaystyle \left[\{v_{\unicode[STIX]{x1D6FC}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}-v_{\unicode[STIX]{x1D708}}^{\prime }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\unicode[STIX]{x1D70C}}}D_{\text{ret}}(x_{\unicode[STIX]{x1D70C}}-x_{\unicode[STIX]{x1D70C}}^{\prime })\right]\\[9.0pt] \times \,{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D706}}){\mathcal{N}}_{p}(x_{\unicode[STIX]{x1D70E}}^{\prime },v_{\unicode[STIX]{x1D70E}}^{\prime })\end{array}\right\}.\end{eqnarray}$$

Let us use the Fourier transformation in order to obtain the $4$ -current. We have

(3.3) $$\begin{eqnarray}\displaystyle \tilde{{\mathcal{N}}}_{p}(k_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}})=\frac{1}{(2\unicode[STIX]{x03C0})^{2}}\int \text{d}^{4}x\exp [-\text{i}k^{\unicode[STIX]{x1D707}}x_{\unicode[STIX]{x1D707}}]{\mathcal{N}}_{p}(x_{\unicode[STIX]{x1D70C}},v_{\unicode[STIX]{x1D707}}^{\prime }). & & \displaystyle\end{eqnarray}$$

By using (3.2) and (3.3), we obtain

(3.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{i}k^{\unicode[STIX]{x1D707}}v_{\unicode[STIX]{x1D707}}\tilde{{\mathcal{N}}}_{p}(k_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}})+\text{i}\left(\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}\right)v_{\unicode[STIX]{x1D708}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D707}}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D707}})\int \text{d}^{4}v^{\prime }\frac{k^{\unicode[STIX]{x1D707}}v^{\prime \unicode[STIX]{x1D708}}-k^{\unicode[STIX]{x1D708}}v^{\prime \unicode[STIX]{x1D707}}}{-k^{\unicode[STIX]{x1D706}}k_{\unicode[STIX]{x1D706}}}{\mathcal{N}}_{p}(k_{\unicode[STIX]{x1D70C}},v_{\unicode[STIX]{x1D70C}}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad =\left(\frac{4\unicode[STIX]{x03C0}e^{2}}{mc^{2}}\unicode[STIX]{x1D70F}_{o}\right)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\{\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D706}})\}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \int \text{d}^{4}v^{\prime }\frac{v_{\unicode[STIX]{x1D6FC}}^{\prime }k_{\unicode[STIX]{x1D708}}-v_{v}^{\prime }k_{\unicode[STIX]{x1D6FC}}}{-k^{\unicode[STIX]{x1D706}}k_{\unicode[STIX]{x1D706}}}k_{\unicode[STIX]{x1D70C}}{\mathcal{N}}_{p}(k_{\unicode[STIX]{x1D70E}},v_{\unicode[STIX]{x1D70E}}^{\prime }).\end{eqnarray}$$

Finally, we arrive at

(3.5) $$\begin{eqnarray}\displaystyle \tilde{{\mathcal{N}}}_{p}(k_{\unicode[STIX]{x1D707}},v_{\unicode[STIX]{x1D707}}) & = & \displaystyle \frac{\unicode[STIX]{x1D714}_{p}^{2}}{n_{o}}v_{\unicode[STIX]{x1D708}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v_{\unicode[STIX]{x1D707}}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D707}})\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{1}{k^{\unicode[STIX]{x1D707}}v_{\unicode[STIX]{x1D707}}}\int \text{d}^{4}v^{\prime }\frac{k^{\unicode[STIX]{x1D707}}v^{\prime \unicode[STIX]{x1D708}}-k^{\unicode[STIX]{x1D708}}v^{\prime \unicode[STIX]{x1D707}}}{k^{\unicode[STIX]{x1D706}}k_{\unicode[STIX]{x1D706}}}{\mathcal{N}}_{p}(x_{\unicode[STIX]{x1D70C}},v_{\unicode[STIX]{x1D70C}}^{\prime })\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}\left(\frac{\unicode[STIX]{x1D714}_{p}^{2}}{n_{o}}\unicode[STIX]{x1D70F}_{o}\right)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v^{\unicode[STIX]{x1D707}}}\{\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(v_{\unicode[STIX]{x1D70C}})v^{\unicode[STIX]{x1D70C}}v^{\unicode[STIX]{x1D6FC}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D706}})\}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{1}{k^{\unicode[STIX]{x1D707}}v_{\unicode[STIX]{x1D707}}}\int \text{d}^{4}v^{\prime }\frac{v_{\unicode[STIX]{x1D6FC}}^{\prime }k_{\unicode[STIX]{x1D708}}-v_{v}^{\prime }k_{\unicode[STIX]{x1D6FC}}}{k^{\unicode[STIX]{x1D706}}k_{\unicode[STIX]{x1D706}}}k_{\unicode[STIX]{x1D70C}}{\mathcal{N}}_{p}(k_{\unicode[STIX]{x1D70E}},v_{\unicode[STIX]{x1D70E}}^{\prime }),\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{p}$ represents the regular plasma frequency. The Fourier component of the $4$ -current vector $J^{\unicode[STIX]{x1D6FD}}(k_{v})$ is

(3.6) $$\begin{eqnarray}\displaystyle J^{\unicode[STIX]{x1D6FD}}(k_{v})=\int \text{d}^{4}vv^{\unicode[STIX]{x1D6FD}}{\mathcal{N}}_{p}(k_{\unicode[STIX]{x1D708}},v_{\unicode[STIX]{x1D70E}}). & & \displaystyle\end{eqnarray}$$

By using (3.5), we arrive at

(3.7) $$\begin{eqnarray}\displaystyle J^{\unicode[STIX]{x1D6FD}}(k_{v})=\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\left\{\begin{array}{@{}c@{}}-I_{\unicode[STIX]{x1D706}}k^{\unicode[STIX]{x1D6FD}}-k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}+k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}g_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}\\ +\,\text{i}\unicode[STIX]{x1D70F}_{o}(k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}J_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}+\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D709}^{-1}g_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}-k^{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D709}^{-1})\end{array}\right\}J^{\unicode[STIX]{x1D706}}(k_{v}), & & \displaystyle\end{eqnarray}$$

where

(3.8) $$\begin{eqnarray}\displaystyle I_{\unicode[STIX]{x1D708}}(k_{\unicode[STIX]{x1D70C}},\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D70C}})=-\frac{1}{n_{o}}\int \text{d}v^{4}\frac{v_{v}}{k^{\unicode[STIX]{x1D706}}v_{\unicode[STIX]{x1D706}}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D707}}), & & \displaystyle\end{eqnarray}$$

and

(3.9) $$\begin{eqnarray}\displaystyle I_{\unicode[STIX]{x1D707}}^{v}(k_{\unicode[STIX]{x1D70C}},\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D70C}})=-\frac{1}{n_{o}}\int \text{d}v^{4}\frac{v_{\unicode[STIX]{x1D707}}v^{\unicode[STIX]{x1D708}}}{(k^{\unicode[STIX]{x1D706}}v_{\unicode[STIX]{x1D706}})^{2}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D70C}}), & & \displaystyle\end{eqnarray}$$

and

(3.10) $$\begin{eqnarray}\displaystyle J_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D708}}(k_{\unicode[STIX]{x1D70C}},\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D70C}})=-\frac{1}{n_{o}}\int \text{d}v^{4}\frac{v_{\unicode[STIX]{x1D707}}v^{\unicode[STIX]{x1D708}}}{k^{\unicode[STIX]{x1D706}}v_{\unicode[STIX]{x1D706}}}{\mathcal{N}}_{eq}(v_{\unicode[STIX]{x1D70C}}), & & \displaystyle\end{eqnarray}$$

where (see appendix B)

(3.11) $$\begin{eqnarray}\displaystyle n_{o}=\frac{K_{2}(m\unicode[STIX]{x1D709})}{K_{1}(m\unicode[STIX]{x1D709})}, & & \displaystyle\end{eqnarray}$$

where $K_{n}$ are is Kelvin function of order $n$ (Synge Reference Synge1956) and

(3.12a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}=\frac{1}{\unicode[STIX]{x1D705}T_{o}}\quad \text{and}\quad \unicode[STIX]{x1D709}^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D709}\bar{v}^{\unicode[STIX]{x1D707}}. & & \displaystyle\end{eqnarray}$$

It has to be noted that other works constrain the problem to one dimension and, consequently, they use the one-dimensional Jüttner distribution function (Zhang et al. Reference Zhang, Wu, Zhou, Zhu and He2013).

3.2 General dispersion relations

Once we obtain the expression for the $4$ -vector current, equation (3.7), we need just to calculate the determinant and equal it to zero; that is

(3.13) $$\begin{eqnarray}\displaystyle \det G_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}=\det \left\{\begin{array}{@{}c@{}}\displaystyle g_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}\left(1-w_{p}^{2}\frac{k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\text{i}\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D709}^{-1}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\right)+w_{p}^{2}(I_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}})\\[10.0pt] \displaystyle +\,w_{p}^{2}k^{\unicode[STIX]{x1D6FD}}\frac{I_{\unicode[STIX]{x1D706}}+\text{i}\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D709}^{-1}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\end{array}\right\}=0. & & \displaystyle\end{eqnarray}$$

In a reference frame (the laboratory frame) where

(3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}^{\unicode[STIX]{x1D707}}=(\unicode[STIX]{x1D709},\mathbf{0}), & & \displaystyle\end{eqnarray}$$

and choosing the $z$ -axis as the direction of propagation of the perturbation mode, we have

(3.15) $$\begin{eqnarray}\displaystyle k^{\unicode[STIX]{x1D707}}=(k^{0},0,0,k^{3})=(w,0,0,k). & & \displaystyle\end{eqnarray}$$

For the longitudinal mode, we have

(3.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \left\{\left[1-\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}G\right]+w_{p}^{2}\{I_{0}^{0}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{0}^{0}\}+\frac{w_{p}^{2}k^{0}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}(I_{0}+\text{i}\unicode[STIX]{x1D70F}_{o})\right\}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \left\{\left[1-\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}G\right]+w_{p}^{2}\{I_{3}^{3}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{3}^{3}\}+\frac{w_{p}^{2}k^{3}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}(I_{3})\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =w_{p}^{4}\left\{\frac{k^{3}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}(I_{0}+\text{i}\unicode[STIX]{x1D70F}_{o})+I_{0}^{3}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{0}^{3}\right\}\times \left\{I_{3}^{0}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{3}^{0}+\frac{k^{0}I_{3}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\right\},\end{eqnarray}$$

where

(3.17) $$\begin{eqnarray}\displaystyle G=k^{\unicode[STIX]{x1D708}}I_{\unicode[STIX]{x1D708}}+\text{i}\unicode[STIX]{x1D70F}_{o}k^{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D709}^{-1}. & & \displaystyle\end{eqnarray}$$

For the transverse mode, we have

(3.18) $$\begin{eqnarray}\displaystyle \left\{1-\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}G\right\}+w_{p}^{2}\{I_{1}^{1}+\text{i}\unicode[STIX]{x1D70F}_{o}J_{1}^{1}\}=0. & & \displaystyle\end{eqnarray}$$

We will now incorporate the continuity equation into the calculation of the dispersion relations. However, when the density of point particles is transformed to a continuous density, some properties can be lost and we must impose the continuity equation (see appendix C). Then, in (3.6) the continuity equation must be considered as a constraint embedded in the theory; that is,

(3.19) $$\begin{eqnarray}\displaystyle k^{\unicode[STIX]{x1D707}}J_{\unicode[STIX]{x1D707}}=wJ^{0}(k_{v})-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{J}(k_{v})=0. & & \displaystyle\end{eqnarray}$$

This is equivalent to the procedure used in (2.9) and (2.10). Indeed, if we consider (3.7), (3.13) and (3.19), we have a system of equations

(3.20) $$\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}c@{}}G_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}J^{\unicode[STIX]{x1D706}}=0\\ \det G_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}=0\\[3.0pt] k^{\unicode[STIX]{x1D707}}J_{\unicode[STIX]{x1D707}}=wJ^{0}-kJ^{3}=0\end{array}\right\}, & & \displaystyle\end{eqnarray}$$

and using (3.13), (3.14) and (3.15)), and noticing the parity of the integrals $I_{\unicode[STIX]{x1D706}}$ , $I_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}$ , $J_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FD}}$ , we arrive at

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle G_{0}^{1}=G_{0}^{2}=G_{1}^{0}=G_{2}^{0}=G_{0}^{1}=G_{2}^{1}=G_{3}^{1}=G_{0}^{2}=G_{1}^{2}=G_{3}^{2}=G_{1}^{3}=G_{2}^{3}=0\quad \text{and}\quad G_{1}^{1}=G_{2}^{2}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Then,

(3.22) $$\begin{eqnarray}\displaystyle G_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}=\left(\begin{array}{@{}cccc@{}}G_{0}^{0} & 0 & 0 & G_{3}^{0}\\ 0 & G_{1}^{1} & 0 & 0\\ 0 & 0 & G_{1}^{1} & 0\\ G_{0}^{3} & 0 & 0 & G_{3}^{3}\end{array}\right). & & \displaystyle\end{eqnarray}$$

Finally, we obtain,

(3.23) $$\begin{eqnarray}\displaystyle [G_{1}^{1}]^{2}[G_{0}^{0}G_{3}^{3}-G_{3}^{0}G_{0}^{3}]=0. & & \displaystyle\end{eqnarray}$$

For the transverse mode, we have

(3.24) $$\begin{eqnarray}\displaystyle G_{1}^{1}G_{1}^{1}=0, & & \displaystyle\end{eqnarray}$$

and for the longitudinal mode, by using (3.20), we arrive at

(3.25a-c ) $$\begin{eqnarray}\displaystyle G_{0}^{0}G_{3}^{3}-G_{3}^{0}G_{0}^{3}=0,\quad wJ^{0}-kJ^{3}=0,\quad \text{and}\quad G_{0}^{0}w+G_{3}^{0}J=0. & & \displaystyle\end{eqnarray}$$

Therefore, by combining the last equations, we obtain

(3.26a,b ) $$\begin{eqnarray}\displaystyle wG_{3}^{0}+kG_{0}^{0}=0\quad \text{and}\quad wG_{3}^{3}+kG_{0}^{3}=0. & & \displaystyle\end{eqnarray}$$

However, these two equations are equivalent as we can see easily. Therefore, we will just use the first of the two equations. In this case, and after a long calculation and using the relations obtained in the appendix of Hakim & Mangeney (Reference Hakim and Mangeney1971), we arrive at

(3.27) $$\begin{eqnarray}\displaystyle G_{0}^{0}=\left(1-w_{p}^{2}\frac{k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\text{i}\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D709}^{-1}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\right)+w_{p}^{2}(I_{0}^{0}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{0}^{0})+w_{p}^{2}k^{0}\frac{I_{0}+\text{i}\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D709}_{0}\unicode[STIX]{x1D709}^{-1}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}. & & \displaystyle\end{eqnarray}$$

Since we consider the system at rest,

(3.28) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}_{0}=\unicode[STIX]{x1D709}\bar{v}_{0}=\unicode[STIX]{x1D709}, & & \displaystyle\end{eqnarray}$$

we have

(3.29) $$\begin{eqnarray}\displaystyle M=k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\text{i}\unicode[STIX]{x1D70F}_{o}\unicode[STIX]{x1D709}^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D709}^{-1}=k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\text{i}\unicode[STIX]{x1D70F}_{o}w. & & \displaystyle\end{eqnarray}$$

Therefore,

(3.30) $$\begin{eqnarray}\displaystyle G_{0}^{0}=\left(1-\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}M\right)+w_{p}^{2}(I_{0}^{0}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{0}^{0})+w_{p}^{2}k^{0}\frac{I_{0}+\text{i}\unicode[STIX]{x1D70F}_{o}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}. & & \displaystyle\end{eqnarray}$$

Noticing that for $\unicode[STIX]{x1D707}=1$ , $2$ , $3$ , we have

(3.31a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D709}\bar{v}^{\unicode[STIX]{x1D707}}=0\quad \text{and}\quad I_{3}^{0}=J_{3}^{0}=0. & & \displaystyle\end{eqnarray}$$

Then,

(3.32) $$\begin{eqnarray}\displaystyle G_{3}^{0}=w_{p}^{2}(I_{3}^{0}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{3}^{0})+w_{p}^{2}k^{0}\frac{I_{3}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}. & & \displaystyle\end{eqnarray}$$

Therefore equation (3.26) can be written as,

(3.33) $$\begin{eqnarray}\displaystyle k\left(1-\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}M+w_{p}^{2}(I_{0}^{0}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{0}^{0})+w_{p}^{2}k^{0}\frac{I_{0}+\text{i}\unicode[STIX]{x1D70F}_{o}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\right)+ww_{p}^{2}\left(I_{3}^{0}-\text{i}\unicode[STIX]{x1D70F}_{o}J_{3}^{0}+k^{0}\frac{I_{3}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\right)=0, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where

(3.34) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle I_{3}^{0}=-\frac{1}{k}\left\{w\frac{\unicode[STIX]{x2202}I_{0}}{\unicode[STIX]{x2202}w}+I_{0}\right\},\\[15.0pt] \displaystyle I_{3}=\frac{1}{k}\left\{\frac{K_{1}(\unicode[STIX]{x1D6FC})}{K_{2}(\unicode[STIX]{x1D6FC})}-wI_{0}\right\}=\frac{1}{k}\{k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}-wI_{0}\},\\[15.0pt] \displaystyle I_{0}^{0}=\frac{\unicode[STIX]{x2202}I_{0}}{\unicode[STIX]{x2202}w},\\[3.0pt] \displaystyle J_{3}^{0}=-K\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}(K^{-1}I_{3})=-K\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\left(\frac{4\unicode[STIX]{x03C0}K_{2}(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D6FC}k}\left\{\frac{K_{1}(\unicode[STIX]{x1D6FC})}{K_{2}(\unicode[STIX]{x1D6FC})}-wI_{0}\right\}\right)\\[15.0pt] \displaystyle \quad =-K\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\left(\frac{1}{Kk}\{k^{\unicode[STIX]{x1D707}}I_{\unicode[STIX]{x1D707}}-wI_{0}\}\right),\\[15.0pt] \displaystyle J_{0}^{0}=-K\frac{\unicode[STIX]{x2202}K^{-1}I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with

(3.35) $$\begin{eqnarray}\displaystyle K=\frac{m\unicode[STIX]{x1D709}}{4\unicode[STIX]{x03C0}K_{2}(m\unicode[STIX]{x1D709})}=\frac{\unicode[STIX]{x1D6FC}}{4\unicode[STIX]{x03C0}K_{2}(\unicode[STIX]{x1D6FC})}. & & \displaystyle\end{eqnarray}$$

Finally, the general dispersion relation for the longitudinal mode can be expressed as

(3.36) $$\begin{eqnarray}\displaystyle \frac{k^{2}+w_{p}^{2}k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}} & = & \displaystyle \frac{2ww_{p}^{2}I_{o}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}+w_{p}^{2}\frac{\unicode[STIX]{x2202}I_{o}}{\unicode[STIX]{x2202}w}\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}\unicode[STIX]{x1D70F}_{o}\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\left(w+k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}I_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}-k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)I_{o}\right),\end{eqnarray}$$

where

(3.37) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}=m\unicode[STIX]{x1D709}=\frac{m}{\unicode[STIX]{x1D705}T_{o}}, & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D6FC}$ the so-called inverse temperature parameter. This result differs from Hakim & Mangeney (Reference Hakim and Mangeney1971) by an extra factor $w/k$ in the imaginary part which appears in their expression. However, Hakim and Mangeney always calculated the dispersion relations for $w\simeq k$ and the results are equivalent in this neighbourhood. Nevertheless, this implies a difference when the damping due to the radiation reaction force is calculated in the general case. This result permits us to understand some important aspects of the longitudinal modes.

4.1 Real part of the dispersion relation

Let us restrict our attention to the longitudinal case. First it has to be noted that in general $I_{0}$ is a complex function which can be written as:

(4.1) $$\begin{eqnarray}\displaystyle I_{0}=\text{Re}\,I_{0}+\text{i}\,\text{Im}\,I_{0}, & & \displaystyle\end{eqnarray}$$

which can be verified by making an inspection of (3.8) and noticing that the term

(4.2) $$\begin{eqnarray}\displaystyle \frac{1}{k^{\unicode[STIX]{x1D706}}u_{\unicode[STIX]{x1D706}}}=\text{i}\unicode[STIX]{x03C0}\left\{\unicode[STIX]{x1D6FF}(k^{\unicode[STIX]{x1D706}}u_{\unicode[STIX]{x1D706}})-\frac{\text{i}}{\unicode[STIX]{x03C0}}P.V.\left(\frac{1}{k^{\unicode[STIX]{x1D706}}u_{\unicode[STIX]{x1D706}}}\right)\right\}. & & \displaystyle\end{eqnarray}$$

This is inherited from the expression for $I_{0}$ , equation (3.8). Therefore, taking into account that

(4.3) $$\begin{eqnarray}\displaystyle k^{\unicode[STIX]{x1D706}}I_{\unicode[STIX]{x1D706}}=\frac{K_{2}(\unicode[STIX]{x1D6FC})}{K_{1}(\unicode[STIX]{x1D6FC})}, & & \displaystyle\end{eqnarray}$$

which is real, equation (3.36) can be written as:

(4.4) $$\begin{eqnarray}\displaystyle \frac{k^{2}+w_{p}^{2}k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}} & = & \displaystyle \frac{2ww_{p}^{2}(\text{Re}\,I_{0}+\text{i}\,\text{Im}\,I_{0})}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}+w_{p}^{2}\frac{\unicode[STIX]{x2202}(\text{Re}\,I_{0}+\text{i}\,\text{Im}\,I_{0})}{\unicode[STIX]{x2202}w}+\text{i}\unicode[STIX]{x1D70F}_{o}\frac{w_{p}^{2}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}\nonumber\\ \displaystyle & & \displaystyle \times \left(w+k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}(\text{Re}\,I_{0}+\text{i}\,\text{Im}\,I_{0})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}-k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)(\text{Re}\,I_{0}+\text{i}\,\text{Im}\,I_{0})\right).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Therefore, we must consider that the frequency as a function of the wavelength is complex. However, we can consider that the complex part is smaller with respect the real part ( $w=w_{o}+\text{i}\unicode[STIX]{x1D6FE}$ , $\unicode[STIX]{x1D6FE}\ll w_{o}$ ). Accordingly, let us first calculate the dispersion relation without considering the imaginary parts; that is, to obtain a relation which can be calculated by using a numerical approach without making any approximation but only taking into consideration at first glance that $\text{Im}\,I_{o}$ is not big enough to change the fundamental relation. Notice that the term multiplied by $\unicode[STIX]{x1D70F}_{o}$ is also neglected. Then,

(4.5) $$\begin{eqnarray}\displaystyle \frac{k^{2}+w_{p}^{2}k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}=\frac{2ww_{p}^{2}\text{Re}\,I_{o}}{k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}}+w_{p}^{2}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{o}}{\unicode[STIX]{x2202}w}. & & \displaystyle\end{eqnarray}$$

It has to be noticed that for $\unicode[STIX]{x1D707}>1$ , as will be demonstrated later in the paper, $\text{Im}\,I_{o}=0$ and in this region this approximation is exact without taking into account the terms multiplied by $\unicode[STIX]{x1D70F}_{o}$ . We have to analyse the function $\text{Re}\,I_{o}$ and use the expression deduced by Hakim & Mangeney (Reference Hakim and Mangeney1968)

(4.6a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}=m\unicode[STIX]{x1D709}=\frac{m}{\unicode[STIX]{x1D705}T_{o}},\quad \text{Re}\,\unicode[STIX]{x1D713}=\unicode[STIX]{x1D709}\text{Re}\,I_{o}=-\frac{\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}}{kK_{2}(\unicode[STIX]{x1D6FC})}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D6FC}}\left[\frac{1}{\unicode[STIX]{x1D6FC}}\int _{0}^{\infty }\frac{\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]}{(\unicode[STIX]{x1D707}^{2}-1)\cosh ^{2}\unicode[STIX]{x1D712}+1}\,\text{d}\unicode[STIX]{x1D712}\right]. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

From which

(4.7) $$\begin{eqnarray}\displaystyle \text{Re}\,I_{o} & = & \displaystyle -\frac{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}}{kK_{2}(\unicode[STIX]{x1D6FC})}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D6FC}}\left[\frac{1}{\unicode[STIX]{x1D6FC}}\int _{0}^{\infty }\frac{\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]}{(\unicode[STIX]{x1D707}^{2}-1)\cosh ^{2}\unicode[STIX]{x1D712}+1}\,\text{d}\unicode[STIX]{x1D712}\right]\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x1D707}}{kK_{2}(\unicode[STIX]{x1D6FC})}\int _{0}^{\infty }\frac{[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]}{(\unicode[STIX]{x1D707}^{2}-1)\cosh ^{2}\unicode[STIX]{x1D712}+1}\,\text{d}\unicode[STIX]{x1D712}.\end{eqnarray}$$

It has to be pointed out that this is an improper integral if $\unicode[STIX]{x1D707}<1$ and in this case it has to be calculated by using its principal value (the imaginary part $\text{Im}\,I_{o}$ is neglected as we mentioned above and $\text{Im}\,I_{o}=0$ for $\unicode[STIX]{x1D707}>1$ ),

(4.8) $$\begin{eqnarray}\displaystyle \text{Re}\,I_{o}=\left\{\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x1D707}}{kK_{2}(\unicode[STIX]{x1D6FC})}\int _{0}^{\infty }\frac{[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]}{(\unicode[STIX]{x1D707}^{2}-1)\cosh ^{2}\unicode[STIX]{x1D712}+1}\,\text{d}\unicode[STIX]{x1D712}\quad \text{for }\unicode[STIX]{x1D707}\geqslant 1\\[10.0pt] \displaystyle P.V\frac{\unicode[STIX]{x1D707}}{kK_{2}(\unicode[STIX]{x1D6FC})}\int _{0}^{\infty }\frac{[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]}{(\unicode[STIX]{x1D707}^{2}-1)\cosh ^{2}\unicode[STIX]{x1D712}+1}\,\text{d}\unicode[STIX]{x1D712}\quad \text{for }\unicode[STIX]{x1D707}<1\end{array}\right\}. & & \displaystyle\end{eqnarray}$$

A fundamental difference compared to the work done by Hakim & Mangeney (Reference Hakim and Mangeney1971) can be pointed out. Let us analyse the second line, that is for $\unicode[STIX]{x1D707}<1$

(4.9) $$\begin{eqnarray}\displaystyle \text{Re}\,I_{o}=\frac{\unicode[STIX]{x1D707}}{kK_{2}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FC}}\left\{\begin{array}{@{}c@{}}\displaystyle \lim _{\unicode[STIX]{x1D700}\rightarrow 0}\int _{0}^{\unicode[STIX]{x1D712}_{c}-\unicode[STIX]{x1D700}}\frac{[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]}{1-\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}}\,\text{d}\unicode[STIX]{x1D712}\\[10.0pt] \displaystyle +\,\lim _{\unicode[STIX]{x1D700}\rightarrow 0}\int _{\unicode[STIX]{x1D712}_{c}+\unicode[STIX]{x1D700}}^{\infty }\frac{[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]}{1-\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}}\,\text{d}\unicode[STIX]{x1D712}\end{array}\right\}\quad \text{for }\unicode[STIX]{x1D707}<1,\quad & & \displaystyle\end{eqnarray}$$

where

(4.10a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}^{2}=(1-\unicode[STIX]{x1D707}^{2})<1\quad \text{and}\quad \unicode[STIX]{x1D712}_{c}=\text{arccosh}\sqrt{\frac{1}{(1-\unicode[STIX]{x1D707}^{2})}}. & & \displaystyle\end{eqnarray}$$

Equation (4.9) can be expressed as

(4.11) $$\begin{eqnarray}\displaystyle \text{Re}\,I_{o}=\frac{\unicode[STIX]{x1D707}}{kK_{2}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FC}}[H_{1}+H_{2}],\quad \unicode[STIX]{x1D707}<1, & & \displaystyle\end{eqnarray}$$

where $H_{1}$ and $H_{2}$ represent the first and second integrals in (4.9), respectively. Now, in $H_{1}$ , we can make an expansion around $y=0$ , because

(4.12a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{c}=\text{arccosh}\sqrt{\frac{1}{(1-\unicode[STIX]{x1D707}^{2})}}>\unicode[STIX]{x1D712}\quad \Rightarrow \quad y=\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}=(1-\unicode[STIX]{x1D707}^{2})\cosh ^{2}\unicode[STIX]{x1D712}<1. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Then, we can use

(4.13) $$\begin{eqnarray}\displaystyle \frac{1}{1-y}=\mathop{\sum }_{j=0}^{\infty }y^{j}, & & \displaystyle\end{eqnarray}$$

to obtain

(4.14) $$\begin{eqnarray}\displaystyle H_{1}=\lim _{\unicode[STIX]{x1D700}\rightarrow 0}\int _{0}^{\unicode[STIX]{x1D712}_{c}-\unicode[STIX]{x1D700}}\mathop{\sum }_{j=0}^{\infty }(1-\unicode[STIX]{x1D707}^{2})^{j}\cosh ^{2j}\unicode[STIX]{x1D712}[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]\,\text{d}\unicode[STIX]{x1D712}. & & \displaystyle\end{eqnarray}$$

For $H_{2}$ , we have

(4.15a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{c}=\text{arccosh}\sqrt{\frac{1}{(1-\unicode[STIX]{x1D707}^{2})}}<\unicode[STIX]{x1D712}\quad \Rightarrow \quad y=\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}=(1-\unicode[STIX]{x1D707}^{2})\cosh ^{2}\unicode[STIX]{x1D712}>1. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Then, we can use

(4.16) $$\begin{eqnarray}\displaystyle \frac{1}{1-\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}-1}\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}\left(1-\displaystyle \frac{1}{\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}}\right)}\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}}\left(\frac{1}{1-\displaystyle \frac{1}{\unicode[STIX]{x1D70E}^{2}\cosh ^{2}\unicode[STIX]{x1D712}}}\right),\end{eqnarray}$$

and

(4.17) $$\begin{eqnarray}\displaystyle -\frac{1}{y\left(1-\displaystyle \frac{1}{y}\right)}=-\frac{1}{y}\mathop{\sum }_{j=0}^{\infty }\left(\frac{1}{y}\right)^{j}. & & \displaystyle\end{eqnarray}$$

Therefore, we express $H_{2}$ as

(4.18) $$\begin{eqnarray}\displaystyle H_{2} & = & \displaystyle -\lim _{\unicode[STIX]{x1D700}\rightarrow 0}\int _{\unicode[STIX]{x1D712}_{c}-\unicode[STIX]{x1D700}}^{\infty }\frac{1}{(1-\unicode[STIX]{x1D707}^{2})\cosh ^{2}\unicode[STIX]{x1D712}}\mathop{\sum }_{j=0}^{\infty }\left(\frac{1}{(1-\unicode[STIX]{x1D707}^{2})\cosh ^{2}\unicode[STIX]{x1D712}}\right)^{j}\nonumber\\ \displaystyle & & \displaystyle \times \,[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}]\,\text{d}\unicode[STIX]{x1D712}\nonumber\\ \displaystyle & = & \displaystyle -\lim _{\unicode[STIX]{x1D700}\rightarrow 0}\int _{\unicode[STIX]{x1D712}_{c}-\unicode[STIX]{x1D700}}^{\infty }\mathop{\sum }_{j=1}^{\infty }\left(\frac{1}{(1-\unicode[STIX]{x1D707}^{2})\cosh ^{2}\unicode[STIX]{x1D712}}\right)^{j}[\unicode[STIX]{x1D6FC}^{-1}+\cosh \unicode[STIX]{x1D712}]\exp [-\unicode[STIX]{x1D6FC}\cosh \unicode[STIX]{x1D712}].\qquad\end{eqnarray}$$

There is no way to identify this term with Bessel functions as Hakim and Mangeney did (Hakim & Mangeney Reference Hakim and Mangeney1971). It has to be noted that the summatory cannot be put out of the integral unless the series converges uniformly and this depends on the value of $\unicode[STIX]{x1D707}$ . Given the complexity of (4.14) and (4.18) that include integrals of infinite series, and our inability to find an analytical result, we must use numerical calculation. In this case we need to cut off the series at some point and to evaluate the error induced by using a finite series. However, this procedure does not provide additional value compared to the numerical calculation of the whole integral in (4.8). For the evaluation of these integrals we used NIntegrate built-in symbol of Wolfram Mathematic, and particularly, for $\unicode[STIX]{x1D707}<1$ we used the ‘Principal Value’ method of NIntegrate. Since we are interested in high temperatures, let us present graphically the results for different values of $\unicode[STIX]{x1D6FC}$ . In figures 1, 2, 3, 4 and 13, we present such dispersion relations for $\unicode[STIX]{x1D6FC}=100$ , $10$ , $1$ , $0.5$ and the extreme case of the Hagedorn temperature $\unicode[STIX]{x1D6FC}=0.0059$ respectively.

Figure 1.

Value of $w/w_{p}$ plotted against $k/w_{p}$ (red for the first branch $w_{1}/w_{p}$ and grey line for $w_{2}/w_{p}$ ), Zhang curve (green line for $w_{\text{Zhang}}/w_{p}$ ),  $\unicode[STIX]{x1D707}=1$ (blue line) and $V_{Te}$ (black line). For $\unicode[STIX]{x1D6FC}=100$ , the Zhang curve coincides with the first branch $w_{1}/w_{p}$ until the cutoff. It has to be highlighted that the second branch $w_{2}/w_{p}$ crosses the line which represents the thermal velocity ( $V_{T}$ ) and consequently below this point the second branch is represented by a grey dashed line. In this region the dispersion relation is not valid anymore.

Figure 2.

Same annotations than in figure 1. For $\unicode[STIX]{x1D6FC}=10$ , Zhang curve is close to the first branch $w_{1}/w_{p}$ until the cut-off. It has to be highlighted that the second branch $w_{2}/w_{p}$ crosses the line which represents the thermal velocity ( $V_{T}$ ) and consequently below this point the second branch is represented by a grey dashed line. In this region the dispersion relation is not anymore valid.

Figure 3.

Same annotations as figure 1. For $\unicode[STIX]{x1D6FC}=1$ , the Zhang curve differs from the first branch $w_{1}/w_{p}$ . It has to be highlighted that the second branch $w_{2}/w_{p}$ crosses the line which represents the thermal velocity ( $V_{T}$ ) close to the turning point (cutoff) and therefore, just a very small part of the second branch $w_{2}/w_{p}$ subsists. The other part of the second branch is represented by a grey dashed line. In this region the dispersion relation is not anymore valid.

Figure 4.

Same annotations as figure 1. For $\unicode[STIX]{x1D6FC}=0.5$ , the Zhang curve differs from the first branch $w_{1}/w_{p}$ . It has to be highlighted that the second branch $w_{2}/w_{p}$ crosses the line which represents the thermal velocity ( $V_{T}$ ) close to the turning point (cutoff) and therefore, just a very small part of the second branch $w_{2}/w_{p}$ subsists. The other part of the second branch is represented by a grey dashed line. In this region the dispersion relation is not anymore valid.

In figure 1, for $\unicode[STIX]{x1D6FC}=100$ , which is very close to a tokamak temperature ( $\unicode[STIX]{x1D6FC}\simeq 59$ ), we can notice that the dispersion relation gives a loop which represents one of the important results of the paper. The first part of the wave, $0<k/w_{p}<1$ , is equal to the classical result $w\simeq w_{p}$ . However, due to the correct expansions done in equations, (4.8), (4.9), (4.10), (4.14) and (4.18), after crossing the super-luminous region, the turning point appears and the loop is obtained contrary to other works. The main point is that, due to the existence of the turning point, it is not possible to find waves above a critical wavenumber and this is described by two branches: the first one is the red line in the figure 1 and it finishes at the turning point (or cutoff) where the second one begins which is represented by the grey line. It has to be highlighted that the Zhang dispersion relation (Zhang et al. Reference Zhang, Wu, Zhou, Zhu and He2013) described by the green line coincides with the red line until the effects of the real dispersion relations, which include the effects of the magnetic field and the $3$ - dimension case, are present. The Zhang dispersion relation is considered for a maximum wavenumber $k/w_{p}=3$ , arguing that it is sufficient for typical relativistic plasma oscillations and not because a cutoff is predicted as happens in our approach. It has to be noticed that part of the two branches belongs to the super-thermal wave region ( $V_{Te}<\unicode[STIX]{x1D707}<1$ ). It should also be noted that the black line, which represents the identity between the phase velocity and the thermal velocity, indicates that below ( $\unicode[STIX]{x1D707}<V_{Te}$ ) this line the dispersion relation is not valid anymore and consequently the grey line, which corresponds to the second branch of the loop, is dashed when it crosses the boundary of the thermal velocity. Indeed, for $\unicode[STIX]{x1D707}<V_{Te}$ or $\unicode[STIX]{x1D707}<V_{Ti}$ different physical arguments have to be included in the dispersion relation, obtaining distinct types of waves. It is also important that the thermal velocity has to be expressed by using the Synge relation and not the regular non-relativistic one; that is

(4.19) $$\begin{eqnarray}\displaystyle V_{Te}^{2}=1-\left[\frac{3}{\unicode[STIX]{x1D6FC}}+\text{Re}\,G(\unicode[STIX]{x1D6FC})\right]^{-2}, & & \displaystyle\end{eqnarray}$$

where

(4.20a,b ) $$\begin{eqnarray}\displaystyle G=k^{\unicode[STIX]{x1D708}}I_{\unicode[STIX]{x1D708}}+\text{i}\unicode[STIX]{x1D70F}_{o}k^{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D709}^{-1}\quad \Rightarrow \quad \text{Re}\,G(\unicode[STIX]{x1D6FC})=k^{\unicode[STIX]{x1D708}}I_{\unicode[STIX]{x1D708}}=\frac{K_{2}(\unicode[STIX]{x1D6FC})}{K_{1}(\unicode[STIX]{x1D6FC})}. & & \displaystyle\end{eqnarray}$$

Notice that when $\unicode[STIX]{x1D6FC}$ tends to infinity,

(4.21) $$\begin{eqnarray}\displaystyle V_{Te}^{2}=\frac{3}{\unicode[STIX]{x1D6FC}}=\frac{3\unicode[STIX]{x1D705}T}{m}, & & \displaystyle\end{eqnarray}$$

which corresponds to the non-relativistic case. Finally, it has to be pointed out that the wave has a cutoff (or turning point) at $k_{\max }\simeq 5.4w_{p}$ .

In figure 2, for $\unicode[STIX]{x1D6FC}=10$ , the loop still exists but a difference with the Zhang result begins to appear. The wave has a cutoff at $k_{\max }\simeq 1.9w_{p}$ . Moreover, there is a shift of the plasma frequency when the wavenumber tends to zero; that is

(4.22a,b ) $$\begin{eqnarray}\displaystyle w(k\rightarrow 0)\neq w_{p}\quad \text{and}\quad w(k\rightarrow 0)=\unicode[STIX]{x1D712}w_{p}. & & \displaystyle\end{eqnarray}$$

Zhang obtained a similar result

(4.23a,b ) $$\begin{eqnarray}\displaystyle w(k\rightarrow 0)\neq w_{p}\quad \text{and}\quad w(k\rightarrow 0)=\unicode[STIX]{x1D712}_{z}w_{p}, & & \displaystyle\end{eqnarray}$$

but

(4.24) $$\begin{eqnarray}\displaystyle 1>\unicode[STIX]{x1D712}_{z}>\unicode[STIX]{x1D712}. & & \displaystyle\end{eqnarray}$$

This effect is increased when $\unicode[STIX]{x1D6FC}$ drops (temperature grows) as we can see for $\unicode[STIX]{x1D6FC}=1$ , $\unicode[STIX]{x1D6FC}=0.5$ and the extreme case of the Hagedorn temperature ( $\unicode[STIX]{x1D6FC}=0.0059$ ). We call this effect the frequency plasma shift.

In figure 3, for $\unicode[STIX]{x1D6FC}=1$ , the shift effect is more important and the difference from the Zhang result is increased. However, since the thermal velocity line ( $\unicode[STIX]{x1D707}=V_{Te}$ ) approaches the line of the plane wave ( $\unicode[STIX]{x1D707}=1$ ), the second branch exists just in a small neighbourhood around the turning point. This effect is more important for decreasing $\unicode[STIX]{x1D6FC}$ , as can be seen in the cases for $\unicode[STIX]{x1D6FC}=0.5$ and the Hagedorn temperature. The wave has a cutoff at $k_{\max }\simeq 1.1w_{p}$ .

In figure 4, for $\unicode[STIX]{x1D6FC}=0.5$ , as is expected due to the relativistic effect, the difference from the Zhang result is accentuated. However, the second branch disappears completely due to coincidence between the thermal velocity line and the supra-luminous edge wave line ( $V_{T}\simeq \unicode[STIX]{x1D707}=1$ ). The wave has a cutoff at $k_{\max }\simeq 0.95w_{p}$ .

In figure 13, for $\unicode[STIX]{x1D6FC}=0.0059$ , the Hagedorn temperature, which represents the boundary where hadronic matter is no longer stable and which is the limit of the theory, the second branch cannot be considered due to the thermal velocity line equalling the super-luminous region wave line. However the plasma frequency continues to decrease and the cutoff wavenumber also decreases; that is

(4.25a,b ) $$\begin{eqnarray}\displaystyle w(k\rightarrow 0)\simeq 0.044\quad \text{and}\quad k_{\max }\simeq 0.1w_{p}. & & \displaystyle\end{eqnarray}$$

Although for $w>k$ and for not very low $\unicode[STIX]{x1D6FC}$ our results coincide with Hakim & Mangeney (Reference Hakim and Mangeney1968, Reference Hakim and Mangeney1971) and Zhang et al. (Reference Zhang, Wu, Zhou, Zhu and He2013), for $w<k$ the existence of the loop was not predicted by Hakim and Mangeney, neither by Zhang. In the case of Hakim and Mangeney the differences came from the expansion done in an incorrect form and for Zhang due to the $1$ -dimension character of his model which cannot consider the effects of the magnetic field generated by high temperatures. Moreover, as the temperature grows, the turning or cutoff point tends to the point $(0,0)$ and the wave does not exist. Although, the Hagedorn temperature represents the lower limit for $\unicode[STIX]{x1D6FC}$ , it has to be noticed that when dealing with high temperatures care must be taken since the density must be such that there is no degeneracy. This point is considered later in the article when the RRFD is analysed.

4.2 Landau damping

Let us now consider the imaginary parts of (3.36) or (4.4). First, we will not take into account the terms containing $\unicode[STIX]{x1D70F}_{o}$ . We need now to consider that

(4.26a,b ) $$\begin{eqnarray}\displaystyle w=w_{o}+\text{i}\unicode[STIX]{x1D6FE}\quad k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}=(w_{o}^{2}+2\text{i}\unicode[STIX]{x1D6FE}w-k^{2}), & & \displaystyle\end{eqnarray}$$

where $w_{o}$ and $\unicode[STIX]{x1D6FE}$ are real. It can be supposed that $w_{o}\gg \unicode[STIX]{x1D6FE}$ . Therefore, by neglecting terms of second power in $\unicode[STIX]{x1D6FE}$ ( $\unicode[STIX]{x1D6FE}^{2}\simeq 0$ ), and making an expansion in $i\unicode[STIX]{x1D6FE}$ , we have

(4.27) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\text{Re}\,I_{0}(w_{o}+\text{i}\unicode[STIX]{x1D6FE})=[\text{Re}\,I_{0}]_{w=w_{o}}+\text{i}\unicode[STIX]{x1D6FE}\left[\displaystyle \frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right]_{w=w_{o}}\quad \text{and}\\[11.0pt] \text{Im}\,I_{0}(w_{o}+\text{i}\unicode[STIX]{x1D6FE})=[\text{Im}\,I_{0}]_{w=w_{o}}+\text{i}\unicode[STIX]{x1D6FE}\left[\displaystyle \frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right]_{w=w_{o}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

In the following we will drop the parenthesis notation; that is for example $[\text{Re}\,I_{0}]_{w=w_{o}}=\text{Re}\,I_{0}$ , etc. We arrive at

(4.28) $$\begin{eqnarray}\displaystyle k^{2}+w_{p}^{2}k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}} & = & \displaystyle 2w_{o}w_{p}^{2}\left[\left(\text{Re}\,I_{0}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right)+\text{i}\left[\text{Im}\,I_{0}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right]\right]+w_{p}^{2}k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle \times \left(\left(\frac{\unicode[STIX]{x2202}\text{Re}\,I_{o}}{\unicode[STIX]{x2202}w_{o}}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right)+\text{i}\left(\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right)\right).\end{eqnarray}$$

Therefore, we have

(4.29) $$\begin{eqnarray}\displaystyle k^{2}+w_{p}^{2}k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}} & = & \displaystyle 2(w_{o}+\text{i}\unicode[STIX]{x1D6FE})w_{p}^{2}\left[\text{Re}\,I_{0}-\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{i}\left[\text{Im}\,I_{0}+\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right]\right]\nonumber\\ \displaystyle & & \displaystyle +\,w_{p}^{2}[w_{o}^{2}+2\text{i}\unicode[STIX]{x1D6FE}w_{o}-k^{2}]\nonumber\\ \displaystyle & & \displaystyle \times \left(\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}-\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}+\text{i}\left[\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right]\right).\end{eqnarray}$$

The real part of this equation corrects the dispersion relation and it is:

(4.30) $$\begin{eqnarray}\displaystyle k^{2}+w_{p}^{2}k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}} & = & \displaystyle 2w_{p}^{2}\left(w_{o}\text{Re}\,I_{0}-\unicode[STIX]{x1D6FE}w_{o}\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}-\unicode[STIX]{x1D6FE}\text{Im}\,I_{0}\right)\nonumber\\ \displaystyle & & \displaystyle +\,w_{p}^{2}[w_{o}^{2}-k^{2}]\left[\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}-\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right]+w_{p}^{2}\left[-2w_{o}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right],\qquad\end{eqnarray}$$

which can be solved numerically by firstly analysing the behaviour of both $\text{Re}\,I_{o}$ and $\text{Im}\,I_{0}$ , and knowing the value of $\unicode[STIX]{x1D6FE}$ . However, since $\unicode[STIX]{x1D6FE}\ll w_{o}$ the differences between (4.30) and (4.5) are negligible. This fact can be shown by solving (4.30) for different values of $\unicode[STIX]{x1D6FC}$ and comparing the results with those already obtained in § 4.1. However, for the imaginary part, the result is important as it leads to a damping.

For the imaginary part, we have

(4.31) $$\begin{eqnarray}\displaystyle 0 & = & \displaystyle 2\left(w_{o}\left[\text{Im}\,I_{0}+\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right]+\unicode[STIX]{x1D6FE}\text{Re}\,I_{0}\right)\nonumber\\ \displaystyle & & \displaystyle +\,[w_{o}^{2}-k^{2}]\left[\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right]+2\unicode[STIX]{x1D6FE}w_{o}\frac{\unicode[STIX]{x2202}(\text{Re}\,I_{0})}{\unicode[STIX]{x2202}w_{o}}.\end{eqnarray}$$

We obtain

(4.32) $$\begin{eqnarray}\displaystyle & & \displaystyle 2w_{o}\text{Im}\,I_{0}+[w_{o}^{2}-k^{2}]\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}=-2w_{o}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}-2\unicode[STIX]{x1D6FE}\text{Re}\,I_{0}\nonumber\\ \displaystyle & & \displaystyle \quad -\,[w_{o}^{2}-k^{2}]\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}-2\unicode[STIX]{x1D6FE}w_{o}\frac{\unicode[STIX]{x2202}(\text{Re}\,I_{0})}{\unicode[STIX]{x2202}w_{o}}.\end{eqnarray}$$

Finally, The LD can be represented by the Landau factor $\unicode[STIX]{x1D6FE}_{L}$ ,

(4.33) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{L}=-\frac{2w_{o}\text{Im}\,I_{0}+[w_{o}^{2}-k^{2}]\displaystyle \frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}}{2\left[\displaystyle 2w_{o}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{Re}\,I_{0}+\frac{1}{2}[w_{o}^{2}-k^{2}]\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right]}. & & \displaystyle\end{eqnarray}$$

Before continuing, we need to analyse the value of $\text{Im}\,I_{o}$ (we have already analysed $\text{Re}\,I_{o}$ and see Hakim & Mangeney (Reference Hakim and Mangeney1968, Reference Hakim and Mangeney1971) for $\text{Im}\,I_{o}$ ),

(4.34) $$\begin{eqnarray}\displaystyle \text{Im}\,I_{o}=\left\{\begin{array}{@{}c@{}}0\quad \text{for }\unicode[STIX]{x1D707}>1\\[5.0pt] \displaystyle \frac{\unicode[STIX]{x03C0}}{2kK_{2}(\unicode[STIX]{x1D6FC})}\exp \left\{\frac{-\unicode[STIX]{x1D6FC}}{(1-\unicode[STIX]{x1D707}^{2})^{1/2}}\right\}\left[\frac{1}{\unicode[STIX]{x1D6FC}}+\frac{1}{(1-\unicode[STIX]{x1D707}^{2})^{1/2}}\right]\quad \text{for }\unicode[STIX]{x1D707}<1\end{array}\right\}. & & \displaystyle\end{eqnarray}$$

Therefore,

(4.35) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{L}=\left\{\begin{array}{@{}c@{}}0\quad \text{for }\unicode[STIX]{x1D707}>1\\[5.0pt] \displaystyle -\frac{2w_{o}\text{Im}\,I_{0}+[w_{o}^{2}-k^{2}]\displaystyle \frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}}{2\left[\displaystyle 2w_{o}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{Re}\,I_{0}+\frac{1}{2}[w_{o}^{2}-k^{2}]\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right]}\quad \text{for }\unicode[STIX]{x1D707}<1\end{array}\right\}. & & \displaystyle\end{eqnarray}$$

This means that Landau damping does not exist for $\unicode[STIX]{x1D707}>1$ ; that is: there is no LD in the super-luminous region (see figures 5, 7, 9 and 11).

To understand the effect of the LD, it is necessary to compare the Landau decay time $T_{d}$ with the period $T=2\unicode[STIX]{x03C0}/w_{o}$ obtained from (4.5). This is

(4.36a,b ) $$\begin{eqnarray}\displaystyle T=\frac{2\unicode[STIX]{x03C0}}{w_{o}}\quad \text{and}\quad T_{d}=\frac{1}{\unicode[STIX]{x1D6FE}}, & & \displaystyle\end{eqnarray}$$

where the wave is damped in the cases where the ratio

(4.37) $$\begin{eqnarray}\displaystyle R_{L}=\frac{T_{d}}{T}<1. & & \displaystyle\end{eqnarray}$$

Do not confuse $R_{L}$ with the logarithmic damping decrement used in astrophysics, it is related to the inverse of it.

Figure 5.

For $\unicode[STIX]{x1D6FC}=100$ , the blue line represents the Landau damping in the first branch ( $\unicode[STIX]{x1D6FE}_{L}/w_{p}$ ). For $k<2.5w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}\simeq 0$ and for $k>2.5w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}>0$ . The red line represents the Landau damping in the second branch ( $\unicode[STIX]{x1D6FE}_{L}/w_{p}$ ). For $k>5w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}<0$ .

Figure 6.

For $\unicode[STIX]{x1D6FC}=100$ , the red line represents a stable first branch since $R_{L1}>1$ when $k<4.7w_{p}$ . The dashed red line represents a damped first branch (severe Landau damping) since $0<R_{L1}<1$ when $k>4.7w_{p}$ . The thick red line represents an unstable second branch (possible anti-damping) since $0>R_{L2}>-1$ when $k>5w_{p}$ .

Figure 7.

For $\unicode[STIX]{x1D6FC}=10$ , the blue line represents the Landau damping in the first branch ( $\unicode[STIX]{x1D6FE}_{L}/w_{p}$ ). For $k<1.25w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}\simeq 0$ and for $k>1.25w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}>0$ . The red line represents the Landau damping in the second branch ( $\unicode[STIX]{x1D6FE}_{L}/w_{p}$ ). For $k>1.8w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}<0$ .

Figure 8.

For $\unicode[STIX]{x1D6FC}=10$ , the red line represents a stable first branch since $R_{L1}>1$ when $k<1.8w_{p}$ . The dashed red line represents a damped first branch (severe Landau damping) since $0<R_{L1}<1$ when $k>1.8w_{p}$ . The thick red line represents an unstable second branch (possible anti-damping) since $0>R_{L2}>-1$ when $k>1.8w_{p}$ .

Figure 9.

For $\unicode[STIX]{x1D6FC}=1$ , the blue line represents the Landau damping in the first branch ( $\unicode[STIX]{x1D6FE}_{L}/w_{p}$ ). For $k<1.04w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}\simeq 0$ and for $k>1.04w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}>0$ . The red line represents the Landau damping in the second branch ( $\unicode[STIX]{x1D6FE}_{L2}/w_{p}$ ). For $k>1.04w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}<0$ .

Figure 10.

For $\unicode[STIX]{x1D6FC}=1$ , the red line represents a stable first branch since $R_{L1}>1$ when $k<1.04w_{p}$ . There is not a dashed red line representing a damped wave since $0<R_{L1}<1$ practically coincides with the turning point at $k\simeq 1.04w_{p}$ . The thick red line represents an unstable second branch, $0>R_{L2}>-1$ (possible anti-damping), but it exists in a small neighbourhood around $k\simeq 1.04w_{p}$ .

Figure 11.

For $\unicode[STIX]{x1D6FC}=0.5$ , the blue line represents the Landau damping in the first branch ( $\unicode[STIX]{x1D6FE}_{L}/w_{p}$ ). For $k<0.9w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}\simeq 0$ and for $k>0.9w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}>0$ . The red line represents the Landau damping in the second branch ( $\unicode[STIX]{x1D6FE}_{L}/w_{p}$ ). For $k>0.9w_{p}$ , $\unicode[STIX]{x1D6FE}_{L}/w_{p}<0$ .

Figure 12.

For $\unicode[STIX]{x1D6FC}=0.5$ , the red line represents a stable first branch since $R_{L1}>1$ when $k<0.9w_{p}$ . There is not a dashed red line representing a damped wave since $0<R_{L1}<1$ practically coincides with the turning point at $k\simeq 0.9w_{p}$ . The thick red line represents an unstable second branch, $0>R_{L2}>-1$ (possible anti-damping), but it exists in a small neighbourhood around $k\simeq 0.9w_{p}$ .

Let us describe the properties of the waves for the different values of $R_{L}$ . It has to be noted that the definition of the Fourier transform implies that a positive $\unicode[STIX]{x1D6FE}$ corresponds to damped oscillations while a negative value corresponds to unstable oscillations (resonance). This means that for:

$R_{L}>1$ the wave is practically not damped due to the LD and we will call it a stable wave;

$0<R_{L}<1$ the wave is severely damped and it decays in a time less than its period and we will call it a damped wave;

$0>R_{L}>-1$ the wave suffers a resonance and it is unstable and we will call it an unstable wave (below in the paper it will be called a bipolar wave due to the change of local temperature generated by itself).

4.2.2 Analysis of the stability of the second branch

It has to be noted that the second branch is not completely present in the dispersion relation curve for values of $\unicode[STIX]{x1D6FC}=100$ and $\unicode[STIX]{x1D6FC}=10$ (see figures 6 and 8). Indeed, the second branches cross the thermal velocity line ( $\unicode[STIX]{x1D707}=V_{\unicode[STIX]{x1D719}}=V_{Te}$ ) and from this point the dispersion relation is not valid. For each value of $\unicode[STIX]{x1D6FC}$ , we have

(i) $\unicode[STIX]{x1D6FC}=100$ and $5w_{p}>k>k_{T}$ , where $k_{T}$ is the wavenumber corresponding to the turning point, $0>R_{L}>-1$ and the wave is unstable (possible anti-damping);

(ii) $\unicode[STIX]{x1D6FC}=10$ and $1.8w_{p}>k>k_{T}$ , where $k_{T}$ is the wavenumber corresponding to the turning point, $0>R_{L}>-1$ and the wave is unstable (possible anti-damping);

(iii) $\unicode[STIX]{x1D6FC}=1$ , the second branch exists in a small neighbourhood around $k\simeq 1.04w_{p}$ where $0>R_{L}>-1$ and the wave is unstable (possible anti-damping, see figure 10);

(iv) $\unicode[STIX]{x1D6FC}=0.5$ , the second branch exists in a small neighbourhood around $k\simeq 0.9w_{p}$ where $0>R_{L}>-1$ and the wave is unstable (possible anti-damping, see figure 12).

Figure 13.

The wave at the Hagedorn temperature $\unicode[STIX]{x1D6FC}=0.0059$ and $w_{p}\simeq 5.085\times 10^{22}~\text{rad}~\text{sec}^{-1}$ . The red line represents the first branch $w_{1}/w_{p}$ . The second branch does not exist since it is below the line $\unicode[STIX]{x1D707}=V_{T}$ . The turning point corresponds to a wavenumber $k=0.12w_{p}$ .

The interesting point consists of noticing that in the second branch there is a possible anti-damping. However, the existence of this apparent resonance can be objected to for the following reasons: when a perturbation is made and a resonance may appear, the motion of the particles increases the temperature slightly. Therefore, we can consider that the wave exists in the plasma with a new $\unicode[STIX]{x1D6FC}$ which has decayed a little. Finally, the wavenumber at this increased temperature may give a frequency corresponding to the first branch which is damped by the LD effect in this region or disappears due to the cutoff. That is, the wave is alternatively anti-damped and damped or disappears, and a pulse can be detected. Therefore, those waves can be called bipolar waves.

4.3 Radiation reaction force damping

In order to calculate the RRFD, let us now consider the term which only contains $\unicode[STIX]{x1D70F}_{o}$ multiplied by $k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}$ in order to calculate the imaginary part due to the radiation reaction force; that is,

(4.38) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}=\text{i}\unicode[STIX]{x1D70F}_{o}w_{p}^{2}\left(w+k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}(\text{Re}\,I_{0}+\text{i}\,\text{Im}\,I_{0})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}-k^{\unicode[STIX]{x1D6FC}}k_{\unicode[STIX]{x1D6FC}}\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)(\text{Re}\,I_{0}+\text{i}\,\text{Im}\,I_{0})\right).\quad & & \displaystyle\end{eqnarray}$$

By using (4.26) and (4.27), we have

(4.39) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}=\text{i}\unicode[STIX]{x1D70F}_{o}w_{p}^{2}\left(\begin{array}{@{}c@{}}\displaystyle w_{o}+\text{i}\unicode[STIX]{x1D6FE}+(w_{o}^{2}+2\text{i}\unicode[STIX]{x1D6FE}w_{o}-k^{2})\\[7.0pt] \displaystyle \left(\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\unicode[STIX]{x2202}w}+\text{i}\left(\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\unicode[STIX]{x2202}w}\right)\right)\\[10.0pt] \displaystyle -(w_{o}^{2}+2\text{i}\unicode[STIX]{x1D6FE}w_{o}-k^{2})\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\\[10.0pt] \displaystyle \left(\text{Re}\,I_{0}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{i}\left[\text{Im}\,I_{0}+\text{i}\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right]\right)\end{array}\right). & & \displaystyle\end{eqnarray}$$

The real part generated by this term can be neglected due to the value of $\unicode[STIX]{x1D70F}_{o}$ . Therefore, let us put our attention just in the imaginary part.

(4.40) $$\begin{eqnarray}\displaystyle \text{i}\,\text{Im}\,\unicode[STIX]{x1D6E4}=\text{i}\unicode[STIX]{x1D70F}_{o}w_{p}^{2}\left[\begin{array}{@{}c@{}}\displaystyle w_{o}+(w_{o}^{2}-k^{2})\left(\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\unicode[STIX]{x2202}w}\right)\\[10.0pt] \displaystyle -\,2\unicode[STIX]{x1D6FE}w_{o}\left(\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\right)\\[10.0pt] \displaystyle -(w_{o}^{2}-k^{2})\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\left(\text{Re}\,I_{0}-\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right)\\[10.0pt] \displaystyle -\,2\unicode[STIX]{x1D6FE}w_{o}\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\text{Im}\,I_{0}\end{array}\right]. & & \displaystyle\end{eqnarray}$$

This must equal the denominator of the Landau damping, equation (4.33); that is

(4.41) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D70F}_{o}w_{p}^{2}\left[\begin{array}{@{}c@{}}\displaystyle w_{o}+(w_{o}^{2}-k^{2})\left(\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}^{2}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\unicode[STIX]{x2202}w}\right)-2\unicode[STIX]{x1D6FE}w_{o}\left(\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\right)\\[12.0pt] \displaystyle -(w_{o}^{2}-k^{2})\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\left(\text{Re}\,I_{0}-\unicode[STIX]{x1D6FE}\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}\right)-2\unicode[STIX]{x1D6FE}w_{o}\left[\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\text{Im}\,I_{0}\right]\end{array}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =-2w_{p}^{2}\unicode[STIX]{x1D6FE}\left[2w_{o}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{Re}\,I_{0}+\frac{1}{2}[w_{o}^{2}-k^{2}]\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right],\end{eqnarray}$$

which can be solved and the RRFD can be represented by the RRFD factor $\unicode[STIX]{x1D6FE}_{R}$

(4.42) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{R}=-\frac{\unicode[STIX]{x1D70F}_{o}\left[w_{o}+(w_{o}^{2}-k^{2})\displaystyle \frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}-(w_{o}^{2}-k^{2})\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\text{Re}\,I_{0}\right]}{\left[\begin{array}{@{}c@{}}\displaystyle 2\left[2w_{o}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{Re}\,I_{0}+\frac{1}{2}[w_{o}^{2}-k^{2}]\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right]\\[10.0pt] -\unicode[STIX]{x1D70F}_{o}\left[\begin{array}{@{}c@{}}\displaystyle (w_{o}^{2}-k^{2})\left(\frac{\unicode[STIX]{x2202}^{2}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}\unicode[STIX]{x2202}w}\right)+2w_{o}\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\left(\text{Im}\,I_{0}+\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\right)\\[10.0pt] \displaystyle -\frac{\unicode[STIX]{x2202}\text{Im}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}(w_{o}^{2}-k^{2})\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)\end{array}\right]\end{array}\right]}.\quad & & \displaystyle\end{eqnarray}$$

In the denominator the term with $\unicode[STIX]{x1D70F}_{o}$ may be neglected for some particular values of $w_{o}$ leading to a simple expression, see equation (4.43).

However, this damping due to the radiation is in general very small, unlike other works (Hakim & Mangeney Reference Hakim and Mangeney1968, Reference Hakim and Mangeney1971), unless the plasma frequency is very large. Indeed, for many authors the RRFD is increased for high temperatures ( $\unicode[STIX]{x1D6FC}\ll 1$ ) but, in these cases, the cutoff appears for $k\leqslant w_{p}$ . That is, for high temperatures ( $\unicode[STIX]{x1D6FC}\ll 1$ ), the wave exists just in the first branch where the LD vanishes. Moreover, for $k\leqslant w_{p}$ the RRFD is very small in general unless the value of $w_{p}$ is such that $\unicode[STIX]{x1D70F}_{o}w_{p}\simeq 1$ and not just for $\unicode[STIX]{x1D6FC}\ll 1$ . Let us see now why it is necessary that $\unicode[STIX]{x1D70F}_{o}w_{p}\simeq 1$ in order that the RRFD is important. To show this fact, let us consider $\unicode[STIX]{x1D6FC}\gg 1$ (low temperatures). For our purposes we can put $\unicode[STIX]{x1D6FC}=\infty$ . In this case $\text{Im}\,I_{o}\simeq 0$ and (4.42) turns out to be

(4.43) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{R}=-\frac{\unicode[STIX]{x1D70F}_{o}\left[\displaystyle w_{o}+(w_{o}^{2}-k^{2})\left(\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\right)-(w_{o}^{2}-k^{2})\left(k^{\unicode[STIX]{x1D6FC}}I_{\unicode[STIX]{x1D6FC}}+\frac{3}{\unicode[STIX]{x1D6FC}}\right)(\text{Re}\,I_{0})\right]}{2\left[\displaystyle 2w_{o}\frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}+\text{Re}\,I_{0}+\frac{1}{2}[w_{o}^{2}-k^{2}]\frac{\unicode[STIX]{x2202}^{2}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}^{2}}\right]}. & & \displaystyle\end{eqnarray}$$

Additionally, for $\unicode[STIX]{x1D6FC}=\infty$ , we have

(4.44a-d ) $$\begin{eqnarray}\displaystyle I_{o}=\frac{1}{w_{o}},\quad \frac{\unicode[STIX]{x2202}\text{Re}\,I_{0}}{\unicode[STIX]{x2202}w_{o}}=-\frac{1}{w^{2}},\quad \frac{K_{1}(\infty )}{K_{2}(\infty )}=\frac{1}{n_{o}}=1\quad \text{and}\quad w_{o}=w_{p}. & & \displaystyle\end{eqnarray}$$

Using these approximations, we arrive at

(4.45) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{R}=\frac{\unicode[STIX]{x1D70F}_{o}w_{p}^{2}}{2}. & & \displaystyle\end{eqnarray}$$

Therefore, since $\unicode[STIX]{x1D70F}_{o}\simeq 6.26\times 10^{-24}\sec$ , the plasma frequency must be big enough in order that the damping can be appreciable. Since the RRFD is important only when its decay time $T_{R}=1/\unicode[STIX]{x1D6FE}_{R}$ is smaller than the corresponding period of the wave; then

(4.46) $$\begin{eqnarray}\displaystyle R_{R}=\frac{T_{R}}{T}<1. & & \displaystyle\end{eqnarray}$$

In this case

(4.47) $$\begin{eqnarray}\displaystyle R_{R}=\frac{1}{\unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}_{o}w_{p}}<1. & & \displaystyle\end{eqnarray}$$

Therefore, in order that the RRFD is important we need

(4.48) $$\begin{eqnarray}\displaystyle w_{p}>\frac{1}{\unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}_{o}}. & & \displaystyle\end{eqnarray}$$

Consequently, it is necessary that for a system of electrons,

(4.49) $$\begin{eqnarray}\displaystyle w_{p}\simeq 5.085\times 10^{22}~\text{rad}~\text{sec}^{-1}. & & \displaystyle\end{eqnarray}$$

However, with this kind of frequency we risk having a degenerate plasma and the applied theory is meaningless. Indeed (in international system of units (SI)),

(4.50) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{o}w_{p}=\unicode[STIX]{x1D70F}_{o}\sqrt{\frac{ne^{2}}{m_{e}\unicode[STIX]{x1D700}_{o}}}\simeq \frac{1}{\unicode[STIX]{x03C0}}, & & \displaystyle\end{eqnarray}$$

where $n$ , $e$ , $m_{e}$ and $\unicode[STIX]{x1D700}_{o}$ are the density of electrons, the charge of the electron, the electron mass and the dielectric constant in a vacuum, respectively. Accordingly,

(4.51) $$\begin{eqnarray}\displaystyle n=\frac{m_{e}\unicode[STIX]{x1D700}_{o}}{\unicode[STIX]{x03C0}^{2}\unicode[STIX]{x1D70F}_{o}^{2}e^{2}}\simeq 0.8124\times 10^{42}. & & \displaystyle\end{eqnarray}$$

However, following Phillips (Reference Phillips, Ellison, Mandl and Sandiford1998), Brown (Reference Brown2005) the limit between the degenerate and the non-degenerate cases is represented by equation

(4.52) $$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D705}T}{mc^{2}}>\left[\left(\frac{n}{n_{o}}\right)^{2/3}+1\right]^{1/2}, & & \displaystyle\end{eqnarray}$$

where $n_{o}=5.88\times 10^{35}$ particles per cubic metre. This means that for the limit case of the Hagedorn temperature ( $T=1.005\times 10^{12}$ or $\unicode[STIX]{x1D6FC}=0.0059$ ), the permitted limit for the density of particles per cubic metre is such that

(4.53) $$\begin{eqnarray}\displaystyle \left[\left(\frac{n_{\max }}{5.88\times 10^{35}}\right)^{2/3}+1\right]^{1/2}=169.5, & & \displaystyle\end{eqnarray}$$

which corresponds to $n_{\max }=2.8633\times 10^{42}$ particles per cubic metre, Therefore,

(4.54) $$\begin{eqnarray}\displaystyle n=\frac{m_{e}\unicode[STIX]{x1D700}_{o}}{\unicode[STIX]{x03C0}^{2}\unicode[STIX]{x1D70F}_{o}^{2}e^{2}}\simeq 0.8124\times 10^{42}<n_{\max }. & & \displaystyle\end{eqnarray}$$

In astrophysical systems it may be possible to find such conditions.

Figure 14.

For $\unicode[STIX]{x1D6FC}=0.0059$ and $w_{p}\simeq 5.085\times 10^{22}~\text{rad}~\text{sec}^{-1}$ , $\unicode[STIX]{x1D6FE}_{L1}/w_{p}=0$ and $\unicode[STIX]{x1D6FE}_{R1}/w_{p}$ is represented by the blue line.

Figure 15.

For $\unicode[STIX]{x1D6FC}=0.0059$ and $w_{p}\simeq 5.085\times 10^{22}~\text{rad}~\text{sec}^{-1}$ . The dashed red line represents the damped first branch since $0<R_{R1}<1$ for $k<0.12w_{p}$ . The red line corresponds to a stable wave in the first branch since $R_{R1}>1$ when $k>0.12w_{p}$ .

Figure 16.

Final wave for $\unicode[STIX]{x1D6FC}=0.0059$ and $w_{p}\simeq 5.085\times 10^{22}~\text{rad}~\text{sec}^{-1}$ . The wave is stable just when $0.12w_{p}>k>0.14w_{p}$ .

Figure 17.

Final wave for $\unicode[STIX]{x1D6FC}=100$ . The red line represents the stable first branch; the dashed red line represents the damped first branch wave (severe Landau damping); the thick red line represents the unstable second branch (possible anti-damping).

We can now evaluate the RRFD by using $w_{p}\simeq 5.085\times 10^{22}~\text{rad}~\text{sec}^{-1}$ , and $\bar{\unicode[STIX]{x1D6FE}}_{R}=\unicode[STIX]{x1D6FE}_{R}/w_{p}$ and the value of $R_{R}$ (see figures 14 and 15). In these conditions, the RRFD cancels the wave for $\unicode[STIX]{x1D6FC}=0.0059$ (see figure 15) from $k=0$ until $k=0.12w_{p}$ and permits the existence of the wave from $k=0.12w_{p}$ until the cutoff point (see figure 16). However, since in these circumstances the thermal velocity $V_{Te}$ is very close to the phase velocity $V_{\unicode[STIX]{x1D719}}=w/ck=\unicode[STIX]{x1D707}=1$ , we cannot ensure the existence of this part of the wave.

It has to be noticed that for $\unicode[STIX]{x1D6FC}=100$ , $10$ , $1$ and $0.5$ , by using (4.52) and (4.53), it is necessary to have very small $n_{\max }$ and consequently $\unicode[STIX]{x1D70F}_{o}w_{p}\ll 1$ and the RRFD is neglected.