Physics

Yu. L. Klimontovich

On the Statistical Theory of Nonequilibrium Processes in a Plasma (Allowance for the Nonlinear Interaction of Waves in Kinetic Equations)*

(Presented by Academician N. N. Bogolyubov, 6 III 1964)

Kinetic equations for a plasma with allowance for the nonlinear interaction of waves are considered for two limiting cases.

1. The mean fields are equal to zero. Under this condition the change in the distribution functions of charged particles \(f_a\) is completely determined by the correlations of the random deviations of the phase density \(\delta N_a\) and of the electric and magnetic fields \(\delta \mathbf{E}\), \(\delta \mathbf{B}\).

2. Correlations are completely neglected. Under this condition the Vlasov equations are valid. A quasilinear approximation for these equations is considered.

The present work differs from works \((^{1-4})\) in the following way. Nonlinear equations are considered not for plasmons, but for the complex amplitudes of the field, from which equations for the real amplitudes and phases are obtained. This makes it possible to determine the conditions under which the assumption of rapidly varying phases, adopted in \((^{1-4})\), is valid. The influence of changes in the functions \(f_a\) on the spectrum is taken into account. It is shown that, when the nonlinear interaction of three, four, and a larger number of waves is taken into account, there is no closed system of equations for the functions \(f_a\) and second correlations (plasmons). Instead, there is a closed system of equations for the functions \(f_a\) and several correlation functions, the number of which depends on the number of interacting waves. Perturbation theory in the field is not used. A small number of charged particles interacting with the field is assumed. This ensures the slowness of the field variation. In this respect the approach is analogous to that used in nonlinear optics \((^5)\).

Let us consider the first case. From the equation for the random deviation of the phase density \((^6)\) \(\delta N_a\) we find

\[ \delta N_a=\delta N_a^{\mathrm{l}}+\delta N_a^{\mathrm{nl}}, \tag{1} \]

where \(\delta N_a^{\mathrm{l}}\) is the part of \(\delta N_a\) depending linearly on \(\delta \mathbf{E}\), and \(\delta N_a^{\mathrm{nl}}\) nonlinearly:

\[ \begin{aligned} \delta N_a^{\mathrm{l}}(\mu t,\mu \mathbf{q},\omega,\mathbf{k},\mathbf{p}) &=\gamma_i^a(\mu t,\mu \mathbf{q},\omega,\mathbf{k})\, \delta E_i(\mu t,\mu \mathbf{q},\omega,\mathbf{k}) \\ &\quad+\frac{\partial}{\partial \omega}\frac{1}{\omega-\mathbf{k}\mathbf{v}}\, \frac{\partial}{\partial \mu t} \left(\delta \mathbf{E}\,\frac{\partial f_a}{\partial \mathbf{p}}\right) \\ &\quad+\frac{1}{2}\left[ \left(\frac{\partial \omega}{\partial t}\frac{\partial}{\partial \omega} +\frac{\partial \mathbf{k}}{\partial t}\frac{\partial}{\partial \mathbf{k}}\right) \frac{\partial}{\partial \omega}\frac{1}{\omega-\mathbf{k}\mathbf{v}} \right]\delta \mathbf{E}\,\frac{\partial f_a}{\partial \mathbf{p}} \\ &\quad-\frac{\partial}{\partial \mathbf{k}}\frac{1}{\omega-\mathbf{k}\mathbf{v}}\, \frac{\partial}{\partial \mu \mathbf{q}} \left(\delta \mathbf{E}\,\frac{\partial f_a}{\partial \mathbf{p}}\right) \\ &\quad-\frac{1}{2}\left[ \left(\frac{\partial \omega}{\partial q_i}\frac{\partial}{\partial \omega} +\frac{\partial k_j}{\partial q_i}\frac{\partial}{\partial k_j}\right) \frac{\partial}{\partial k_i}\frac{1}{\omega-\mathbf{k}\mathbf{v}} \right]\delta \mathbf{E}\,\frac{\partial f_a}{\partial \mathbf{p}} . \end{aligned} \tag{2} \]

\[ \begin{aligned} \delta N_a^{\mathrm{nl}} &=\frac{1}{(2\pi)^4}\int d\Omega' d\Omega''\, \delta(\Omega-\Omega'-\Omega'')\, \gamma_{ijk}^a(\Omega,\Omega'')\, \delta\!\left(\delta E_j(\Omega')\,\delta E_k(\Omega'')\right) \\ &\quad+\frac{1}{(2\pi)^8}\int d\Omega' d\Omega'' d\Omega'''\, \delta(\Omega-\Omega'-\Omega''-\Omega''')\, \gamma_{ijkl}^a(\Omega,\Omega''+\Omega''',\Omega''') \\ &\qquad\qquad\times \delta\!\left[\delta E_j(\Omega')\, \delta\!\left(\delta E_k(\Omega'')\,\delta E_l(\Omega''')\right)\right]. \end{aligned} \tag{3} \]

Here \(\Omega\) is the set \(\omega,\mathbf{k}\); \(d\Omega=d\omega\,d\mathbf{k}\), and the following notation has been used for the tensors

\[ \gamma_j^a(\Omega)= -\frac{i e_a n_a}{\omega-\mathbf{k}\mathbf{v}}\, \frac{\partial f_a}{\partial p_j}, \qquad \gamma_{jk}^a(\Omega,\Omega')= -\frac{e_a^2 n_a}{\omega-\mathbf{k}\mathbf{v}}\, \frac{\partial}{\partial p_j}\frac{1}{\omega'-\mathbf{k}'\mathbf{v}}\, \frac{\partial f_a}{\partial p_k}, \]

* Report at the Second All-Union Congress of Mechanics.

\[ \gamma^{a}_{jkl}(\Omega,\Omega',\Omega'')= \frac{i e_a^3 n_a}{\omega-\mathbf{k}\mathbf{v}}\, \frac{\partial}{\partial p_j}\, \frac{1}{\omega'-\mathbf{k}'\mathbf{v}}\, \frac{\partial}{\partial p_k}\, \frac{1}{\omega''-\mathbf{k}''\mathbf{v}}\, \frac{\partial f_a}{\partial p_l}, \]

\(\mu\) is a small parameter, \(f_a=f_a(\mu t,\mu\mathbf{q},|\mathbf{p}|)\).

To obtain equations for the functions \(\delta \mathbf{E}\), \(\delta \mathbf{B}\), it is necessary to know the expressions for the Fourier components of the electric induction vector \(\delta \mathbf{D}\) and its derivatives. Using expressions (2), (3), we obtain, for example, for a Coulomb plasma,

\[ \left(\frac{\partial \delta \mathbf{D}}{\partial t}\right)_{\mu t,\mu q,\omega,\mathbf{k}} = -i\omega\delta\mathbf{E} + \frac{\partial \delta\mathbf{E}}{\partial \mu t} + 4\pi\sum_a e_a\int \mathbf{v}\,\delta N_a\,d\mathbf{p} =0 \]

or

\[ \frac{\partial \delta\mathbf{E}}{\partial \mu t} + \mathbf{v}_{\mathrm{gr}}\frac{\partial \delta\mathbf{E}}{\partial \mu\mathbf{q}} = -(\gamma+\hat{\Gamma})\delta\mathbf{E} + i\omega\delta\mathbf{D}^{\mathrm{nl}} \Big/ \frac{\partial \omega\varepsilon^{(0)'}}{\partial \omega}. \tag{4} \]

Here the following notation has been introduced:

\[ \hat{\Gamma}=\Gamma_{ij} = \left[ \frac{\partial}{\partial\mu t}\frac{\partial\omega\varepsilon_{ij}^{(0)'}}{\partial\omega} + \frac{1}{2} \left( \frac{\partial\omega}{\partial t}\frac{\partial}{\partial\omega} + \frac{\partial k}{\partial t}\frac{\partial}{\partial k} \right) \frac{\partial\omega\varepsilon_{ij}^{(0)'}}{\partial\omega} \right. \]

\[ \left. - \frac{\partial}{\partial\mu\mathbf{q}} \frac{\partial\omega\varepsilon_{ij}^{(0)'}}{\partial k} - \frac{1}{2} \left( \frac{\partial\omega}{\partial q}\frac{\partial}{\partial\omega} + \frac{\partial k_l}{\partial q}\frac{\partial}{\partial k_l} \right) \frac{\partial\omega\varepsilon_{ij}^{(0)'}}{\partial k} \right] \Big/ \frac{\partial\omega\varepsilon^{(0)'}}{\partial\omega}, \tag{5} \]

\[ \delta\mathbf{D}^{\mathrm{nl}} = \frac{4\pi i}{\omega} \sum_a e_a\int \mathbf{v}\,\delta N_a^{\mathrm{nl}}\,d\mathbf{p}, \qquad \varepsilon^{(0)'}= \frac{k_i\varepsilon_{ij}^{(0)'}k_j}{k^2}. \]

The tensor \(\varepsilon_{ij}(\mu t,\mu\mathbf{q},\omega,\mathbf{k})\) is determined by the expression

\[ \varepsilon_{ij} = \delta_{ij} + i\sum_a\frac{4\pi e_a}{\omega} \int v_i\gamma_j^a\,d\mathbf{p} = \varepsilon_{ij}^{(0)'}+\varepsilon_{ij}''. \]

The expression for the vector \(\delta\mathbf{D}^{\mathrm{nl}}\) contains the tensors

\[ \chi_{ijk} = \sum_a\frac{4\pi}{\omega}e_a i \int v_i\gamma_{jk}^a\,d\mathbf{p}, \qquad \theta_{ijkl} = \sum_a\frac{4\pi e_a i}{\omega} \int v_i\gamma_{jkl}^a\,d\mathbf{p}. \]

In deriving equation (4), terms of order \(\mu\) were equated (the nonlinear terms were also regarded as being of order \(\mu\)). The zeroth-order equation in \(\mu\) is the eikonal equation \(\psi\). For example, for a Coulomb plasma it has the form

\[ \varepsilon^{(0)'}(\omega,\mathbf{k},\mu t,\mu\mathbf{q})=0, \qquad \omega=-\frac{\partial\psi}{\partial t}, \qquad \mathbf{k}=\frac{\partial\psi}{\partial\mathbf{q}}. \tag{6} \]

Expression (5) determines an additional decrement caused by the nonstationarity and inhomogeneity of the medium. The derivatives with respect to \(\omega,\mathbf{k}\) with respect to \(\mu t,\mu\mathbf{q}\) that enter into it can in many particular cases be found without determining the eikonal. For example, in the absence of spatial dispersion,

\[ \frac{\partial\omega}{\partial t} = - \frac{\partial\varepsilon}{\partial\mu t} \Big/ \frac{\partial\varepsilon}{\partial\omega}, \qquad \frac{\partial\omega}{\partial q} = - \frac{\partial\varepsilon}{\partial\mu q} \Big/ \frac{\partial\varepsilon}{\partial\omega}. \]

Another example is provided by the case of one-dimensional spatial inhomogeneity for a stationary plasma (see the review \((^7)\)). Let us consider equations for the spectral functions of the field when the interaction of a finite number of natural waves is taken into account. As an example, consider a three-wave interaction. We represent \(\delta\mathbf{E}\) in the form

\[ \delta\mathbf{E}(\mu t,\mu\mathbf{q},\omega,\mathbf{k}) = (2\pi)^4 \sum_{1\leq \alpha \leq 3} \mathbf{E}_{\alpha}(\mu t,\mu\mathbf{q})\, \delta(\omega-\omega_{\alpha})\, \delta(\mathbf{k}-\mathbf{k}_{\alpha}); \]

\(\omega_\alpha, k_\alpha\) are connected by the dispersion relation and satisfy the conservation laws for wave energy and momentum. For the stationary and homogeneous case, with three-wave interaction, using equations (4) we obtain a closed system of equations for the double and triple spectral functions if, in \(\delta D^{\mathrm{nl}}\), the first vanishing term is retained. This system of equations has the stationary solution

\[ (EEE)_{\alpha\beta\gamma} = -\frac{\varepsilon''(\Omega_\alpha)\varepsilon''(\Omega_\beta)\varepsilon''(\Omega_\gamma)} {\chi''_{\alpha\beta\gamma}\chi''_{\beta\gamma\alpha}\chi''_{\gamma\alpha\beta}}, \tag{7} \]

\[ (EE)_\alpha = -\frac{\chi''_{\alpha\beta\gamma}}{\varepsilon''(\Omega_\alpha)}(EEE)_{\alpha\beta\gamma}, \qquad \Delta\varepsilon'(\Omega_\alpha) = \frac{\chi'_{\alpha\beta\gamma}}{\chi''_{\alpha\beta\gamma}}\varepsilon''(\Omega_\alpha). \tag{8} \]

Here \(\Delta\varepsilon'\) is the change of \(\varepsilon'\) due to nonlinear interaction,

\[ \chi_{\alpha\beta\gamma} = k^0_{i\alpha} \bigl( \bar{k}^0_{i\beta}k^0_{k\gamma}\chi_{ijk}(\Omega_\alpha,\Omega_\gamma) + k^0_{j\gamma}k^0_{k\beta}\chi_{ijk}(\Omega_\alpha,\Omega_\beta) \bigr), \qquad k^0=\frac{k}{|k|}. \tag{9} \]

If \(\Delta\varepsilon' \ll \varepsilon''\), then in the nonstationary but homogeneous case the equations for the spectral functions have the form*

\[ \frac{\partial (EE)_\alpha}{\partial t} = -2\gamma_{\mathrm{eff}}(EE)_\alpha - 2\chi_{\alpha\beta\gamma}(EEE)_{\alpha\beta\gamma} \bigg/ \frac{\partial \varepsilon^{(0)'}}{\partial\omega_\alpha}, \qquad \gamma_{\mathrm{eff}}=\gamma+\Gamma, \tag{10} \]

\[ \frac{\partial (EEE)_{\alpha\beta\gamma}}{\partial t} = -\bigl(\gamma_{\mathrm{eff}}(\Omega_\alpha)+\gamma_{\mathrm{eff}}(\Omega_\beta)+\gamma_{\mathrm{eff}}(\Omega_\gamma)\bigr) (EEE)_{\alpha\beta\gamma} \]

\[ - \bigl( \chi_{\alpha\beta\gamma}(EE)_\beta(EE)_\gamma + \chi_{\beta\gamma\alpha}(EE)_\gamma(EE)_\alpha + \chi_{\gamma\alpha\beta}(EE)_\alpha(EE)_\beta \bigr) \bigg/ \frac{\partial \varepsilon^{(0)'}}{\partial\omega_\alpha}. \tag{11} \]

The stationary solution of these equations coincides with (8), (9). In the general case, equations (10), (11) are not closed, since \(\gamma, \chi\) depend on the functions \(f_a\). In the homogeneous case the equations for \(f_a\) can be written in the form

\[ \frac{\partial f_a}{\partial t}=I_a^{\mathrm{st}}+I_a^{\mathrm{n}}, \]

where \(I_a^{\mathrm{st}}\) is the collision integral, and \(I_a^{\mathrm{n}}\) takes into account the contribution of the waves.

The expression for \(I_a^{\mathrm{n}}\) is obtained with the aid of expression (2). It depends, in turn, on the spectral functions \((EE)_\alpha, (EEE)_{\alpha\beta\gamma}\). In the quasistationary approximation for the field, the spectral functions are eliminated from the equation for \(f_a\) by means of the solutions (7), (8), in which \(\varepsilon''\) must be replaced by \(\varepsilon''_{\mathrm{eff}}\). As a result one obtains a closed system of kinetic equations for the functions \(f_a\), taking into account the nonlinear interaction of waves. It is essential that now \(I_a^{\mathrm{n}}\) no longer depends explicitly on the charge. The interaction enters only through the values of the roots of the dispersion equation. Consequently, the relaxation time of the function \(f_a\) also does not explicitly depend on the charge.

By an analogous procedure, equations are obtained in the quasilinear approximation. In contrast to what was considered above, the quasilinear approximation takes into account coherent interaction of waves. In the general case, equations are obtained that describe both coherent and incoherent interaction of waves. Such equations make it possible, in particular, to describe the process of appearance and development of a mean field excited by a random field.

Moscow State University
named after M. V. Lomonosov

Received
5 III 1964

REFERENCES

1. B. B. Kadomtsev, V. I. Petviashvili, ZhETF, 43, 963 (1962).
2. V. P. Silin, Prikl. mekh. i tekhn. fiz., No. 1 (1964).
3. A. A. Vedenov, in: Problems of Plasma Theory, vol. 3, 1963, p. 203.
4. V. I. Karpman, DAN, 152, 587 (1963).
5. S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics, Inst. of Scientific Information, 1964.
6. Yu. L. Klimontovich, Statistical Theory of Nonequilibrium Processes in Plasma, Moscow, 1964.
7. A. A. Rukhadze, V. P. Silin, UFN, 82, No. 3 (1964).

* No summation over repeated indices is performed.