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The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The kinetic equation is given by the formula:

$\left[{\partial _t}{{\rm{f}}_p}(t) = - {F_n}\frac{{\partial {{\rm{f}}_p}(t)}}{{\partial {p_n}}} + {I_p}({{\rm{f}}_{p'}}(t))\right]$, (${F_n} = - e{E_n}$, $\int {{d^3}p{{\rm{f}}_p}(t) = n} $), (1)

where ${E_n}$ is external electric field, $ - e$ is charge of an electron, $n$ is electron density. Perturbation theory is created in terms of spectral theory of operator of collision integral ${\bf{K}}$, which could be defined as ${\bf{K}}{a_p} = - w_p^{ - 1}{I_p}({w_{p'}}{a_{p'}})$ (${w_p}$ is Maxwell distribution; ${I_p}({w_{p'}}) = 0$). Linear operator ${\bf{K}}$ is a symmetric and positively defined one. Complete orthonormal system of its own functions ${g_{ip}}$ could be used to find solutions of kinetic equation (1) as series by mods:

${{\rm{f}}_p} = {w_p}(1 + {g_p})$, ${g_p} = \sum\limits_i {{c_i}{g_{ip}}} $. (2)

We use irreducible polynomials as our own functions. The scalar ${A_p}$ and vector ${B_p}{p_l}$ eigenfunctions and corresponding eigenvalues ${\lambda _{\,T}}$, ${\lambda _u}$ play a decisive role among its own functions

${\bf{K}}{A_p} = {\lambda _{\,T}}{A_p}$, ${\bf{K}}{B_p}{p_l} = {\lambda _u}{B_p}{p_l}$ ($\langle {A_p}{\varepsilon _p}\rangle \equiv 3n/2$, $\langle {B_p}{\varepsilon _p}\rangle \equiv 3n/2$). (3)

It is convenient to investigate the relaxation processes in the system in terms of average electron energy $\varepsilon $ and momentum ${\pi _l}$ densities. It is established that their evolution is exact described at all times by scalar and vector modes

$\left[\varepsilon = {\varepsilon _0} + {c_T}3n/2\right]$, $\left[{\pi _l} = mn{c_{{u_l}}}\right]$, (${\varepsilon _0} \equiv 3nT/2$), (4)

where ${c_T}$, ${c_{{u_l}}}$ – coefficients in series (2) with its eigenfunctions ${A_p}$, ${B_p}{p_l}$ ($m$ – electron mass, ${T_0}$ – ion system temperature). It is proved that quantities $\varepsilon $ і ${\pi _l}$ at all times and for an arbitrary external electric field ${E_n}$ satisfy the equation:

$\left[{\partial _t}{\pi _l} = n{F_l} - {\lambda _u}{\pi _l}\right]$, $\left[{\partial _t}\varepsilon = \frac{1}{m}{\pi _l}{F_l} - {\lambda _{\,T}}(\varepsilon - {\varepsilon _0})\right]$. (5)

The results (4), (5) were found by using irreducible tensors as eigenfunctions of the operator ${\bf{K}}$. Formulas (5) show that eigenvalues ${\lambda _{\,T}}$, ${\lambda _u}$ describe relaxation process in the absence of external electric field

$\varepsilon \to \varepsilon_0$, when ${t\gg\tau_{T}}$ and $\pi _l \to 0$, when ${t\gg\tau_{u}}$, (${\tau _T} \equiv 1/{\lambda _{\,T}}$, ${\tau _u} \equiv 1/{\lambda _u}$). (6)

In terms of temperature $T$ and velocity ${u_l}$ of electron system

$\varepsilon \equiv 3nT/2 + mn{u^2}/2$, ${\pi _l} \equiv mn{u_l}$ (7)

equations (5) take the form

${\partial _t}T = - {\lambda _T}(T - {T_0}) + (2{\lambda _u} - {\lambda _T})m{u^2}/3$, ${\partial _t}{u_n} = - {\lambda _u}{u_n} + \frac{1}{m}{F_n}$ (8)

These equations are exact and valid at all times and arbitrary electric field. The first one does not contains the electric field. At equilibrium equation (8) gives

${u_l}(t)\mathop = \limits_{t > > {\tau _T},{\tau _u}} u_l^{eq}$, $u_l^{eq} = - \nu {E_l}$, $\nu \equiv \frac{e}{{m{\lambda _u}}}$; $j_l^{eq} = \sigma {E_l}$, $\sigma \equiv \frac{{{e^2}n}}{{{\lambda _u}}}$;

$T(t)\mathop = \limits_{t > > {\tau _T},{\tau _u}}{T^{eq}}$, ${T^{eq}} = {T_0} + \Delta T$, $\Delta T \equiv \frac{{{e^2}(2{\lambda _u} - {\lambda _T})}}{{3m{\lambda _T}\lambda _u^2}}{E^2}$ (9)

The expression for the mobility of electrons $\nu$ and the plasma conductivity $\sigma $ in (9) are exact. The last formula accurately describes the effect of temperature differences between the electron and ion components of the plasma in equilibrium in the presence of an electric field. This effect was previously discussed in [2] as an approximate result and without accuracy control.

[1] Sokolovsky А.I., Sokolovsky S.A., Нrinishyn O.A. On relaxation processes in a completely ionized plasma. East European Journal of Physics. Vol. 3 (2020). P. 19-30; doi.org / 10.26565/2312-4334-2020-3-03.

[2] Smirnov В.М. Kinetika elektronov v gazakh i kondensirovannykh sistemakh. UFN Vol. 172 (12) (2002). – P. 1411-1445 (in Russian).