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PHYSICS
Yu. L. KLIMONTOVICH and V. P. SILIN
ON FLUCTUATIONS IN A COLLISIONLESS PLASMA
(Presented by Academician M. A. Leontovich on 17 I 1962)
The theory of fluctuations in a nonequilibrium plasma is at present attracting the attention of many investigators. Because, in a rather wide range of frequencies and wave numbers, the relaxation time of the field* is shorter than the plasma relaxation time, one can construct a theory of fluctuations by specifying one or another form of the particle distribution function and neglecting its time dependence. In such a theory the field fluctuations are wholly and completely determined by the state of the plasma particles. If there is a transparency region in the spectrum of field oscillations, then it follows from what has been said that the absence of interaction with radiation in such a region does not permit one to take into account the corresponding contribution from fluctuations, since the field in the transparency region will no longer be determined by the particle distribution.
The present communication is devoted to the theory of fluctuations of distribution functions in a collisionless plasma, when not only the Coulomb interaction of the particles is important, but the transverse electromagnetic field must also be taken into account. We point out that the corresponding theory only for fluctuations of the electromagnetic field was developed in work (¹).
The method for determining the spectral functions used in the present work differs substantially from the method employed in works (², ³) for the theory of fluctuations in a plasma that takes into account only the Coulomb interaction. In contrast to these works, the space-time spectral distribution functions of the particles are found without preliminary determination of the simultaneous correlation functions. This makes it possible to simplify substantially the solution of the problem, especially when the transverse field is included. Such a method gives a microscopic justification of the method proposed in (⁴, ⁵) for calculating fluctuations by introducing a random source into the kinetic equation.
As the starting point we take the system of equations for the microscopic phase densities
\(N_a(\mathbf r,\mathbf p,t)=\sum_i \delta(\mathbf r-\mathbf r_i(t))\,\delta(\mathbf p-\mathbf p_i(t))\) and the microscopic fields \(\mathbf E,\mathbf B\), the usefulness of which was shown by one of the authors (⁶). Following that work, we write the system of equations for the functions \(N_a,\mathbf E,\mathbf B\) in the form
\[ \frac{\partial N_a}{\partial t} +\mathbf v_a\frac{\partial N_a}{\partial \mathbf r_a} +e_a\left(\mathbf E+\frac{1}{c}[\mathbf v_a\mathbf B]\right) \frac{\partial N_a}{\partial \mathbf p_a}=0, \tag{1} \]
\[ \operatorname{rot}\mathbf B= \frac{1}{c}\frac{\partial\mathbf E}{\partial t} +\frac{4\pi}{c}\sum_b e_b\int \mathbf v_b N_b\,d\mathbf p_b, \qquad \operatorname{div}\mathbf B=0, \tag{2} \]
\[ \operatorname{div}\mathbf E= 4\pi\sum_b e_b\int N_b\,d\mathbf p_b, \qquad \operatorname{rot}\mathbf E= -\frac{1}{c}\frac{\partial\mathbf B}{\partial t}. \]
The system of equations (1), (2) only in appearance coincides with the equations with a self-consistent field (see (⁶)). The difference consists in the fact that this system for microscopic functions is exact and therefore can serve as the starting point also for calculating fluctuations.
* In this case one may speak of field oscillations, or photons, only if the corresponding damping decrements are much smaller than the frequency.
Under ordinary conditions for a collisionless plasma, when many particles are contained in a sphere of Debye radius, for deviations of the functions \(N_a\), \(\mathbf E\), \(\mathbf B\) from the mean values \(\langle N_a\rangle=n_a f_a\), \(\langle \mathbf E\rangle\), \(\langle \mathbf B\rangle\), one may use the linear equations \((^3)\). The solutions of such equations, \(\delta N_a\), \(\delta\mathbf E\), and \(\delta\mathbf B\), can be expressed in terms of the values of these functions at the initial time \(t=0\). Thus, using the one-sided Fourier transform in time
\[ \delta N_a(\omega,\mathbf k,\mathbf p_a) = \int_0^\infty dt\int d\mathbf r_a\, e^{i(\omega t-\mathbf k\mathbf r_a)-\Delta t} \delta N_a(\mathbf r_a,\mathbf p_a,t) \tag{3} \]
(and analogously for \(\delta\mathbf E\) and \(\delta\mathbf B\)), for the spatially isotropic case we have
\[ \delta N_a(\omega,\mathbf k,\mathbf p_a) = \frac{i}{\omega+i\Delta-\mathbf k\mathbf v_a} \left\{ \delta N_a(\mathbf k,\mathbf p_a,t=0) - e_a n_a f'_a\,\mathbf v_a\delta\mathbf E(\omega,\mathbf k) \right\}; \tag{4} \]
\[ \begin{aligned} \delta\mathbf E(\omega,\mathbf k) &= \frac{ i\omega[\mathbf k[\delta\mathbf E(\mathbf k,t=0)]] - ic[\mathbf k\delta\mathbf B(\mathbf k,t=0)]\,k^2 }{ k^2\left[(\omega+i\Delta)^2\varepsilon^{tr}(\omega+i\Delta,\mathbf k)-c^2k^2\right] } \\ &\quad + 4\pi\sum_a e_a\int d\mathbf p_a\, \frac{\delta N_a(\mathbf k,\mathbf p,t=0)} {\omega+i\Delta-\mathbf k\mathbf v_a} \left\{ \frac{\omega[\mathbf k[\mathbf v_a\mathbf k]]} {k^2\left[(\omega+i\Delta)^2\varepsilon^{tr}(\omega+i\Delta,\mathbf k)-c^2k^2\right]} + \frac{\mathbf k}{k^2\varepsilon^l(\omega+i\Delta,\mathbf k)} \right\}. \end{aligned} \tag{5} \]
Here \(f'_a\) is the derivative of the distribution function with respect to the particle energy, and \(\varepsilon^l\) and \(\varepsilon^{tr}\) are the longitudinal and transverse dielectric permittivities of the plasma.
To determine the spectral correlation functions we shall use formulas of the form
\[ (2\pi)^4\delta(\mathbf k-\mathbf k')(\delta N_a(\mathbf p_a)\delta N_b(\mathbf p_b))_{\omega,\mathbf k} = \lim_{\Delta\to0}2\Delta \left\langle \delta N_a(\omega,\mathbf k,\mathbf p_a) \delta N_b^*(\omega,\mathbf k,\mathbf p_b) \right\rangle, \tag{6} \]
where \(\langle\ \rangle\) denotes averaging over the ensemble. As in papers \((^2,^3)\), the spectral functions are then expressed in terms of the equal-time functions
\[ \left\langle \delta N_a(\mathbf k,\mathbf p_a,t=0) \delta N_b^*(\mathbf k,\mathbf p_b,t=0) \right\rangle = \]
\[ = (2\pi)^3\delta(\mathbf k-\mathbf k') \left\{ \delta_{ab}\delta(\mathbf p_a-\mathbf p_b)n_af_a + n_an_b g_{ab}(\mathbf k,\mathbf p_a,\mathbf p_b) \right\}. \]
Here \(g_{ab}\) is the correlation function of a pair of particles. However, an extremely important simplifying circumstance is the fact that, in the limit \(\Delta=0\) in the region of plasma opacity, the terms containing the correlation function make no contribution to the spectral functions defined by formula (6). The same holds for correlations of the fields, as well as of the fields and particles. Then from (4)—(5) we obtain:
\[ \begin{aligned} (E_iE_j)_{\omega,\mathbf k} &= \sum_a \left( \frac{4\pi e_a}{k^2} \right)^2 n_a\int d\mathbf p_a f_a\, \delta(\omega-\mathbf k\mathbf v_a) \\ &\quad\times \left\{ \frac{k_i k_j}{|\varepsilon^l(\omega,\mathbf k)|^2} + \frac{\omega^2[\mathbf k\mathbf v_a]^2(k^2\delta_{ij}-k_i k_j)} {2|\omega^2\varepsilon^{tr}(\omega,\mathbf k)-c^2k^2|^2} \right\}, \end{aligned} \tag{7} \]
\[ \begin{aligned} (\delta N_a\delta N_b)_{\omega,\mathbf k} &= \delta_{ab}\delta(\mathbf p_a-\mathbf p_b) \delta(\omega-\mathbf k\mathbf v_a)n_af_a \\ &\quad + e_ae_b n_an_b \frac{v_a^i v_b^j(E_iE_j)_{\omega,\mathbf k}} {(\omega+i0-\mathbf k\mathbf v_a)(\omega-i0-\mathbf k\mathbf v_b)} f'_a f'_b \\ &\quad - \frac{4\pi}{k^2}e_ae_b n_a f_a n_b f'_b \frac{\delta(\omega-\mathbf k\mathbf v_a)} {\omega-i0-\mathbf v\mathbf k_b} \left\{ \frac{\mathbf k\mathbf v_b}{\varepsilon^l(\omega-i0,\mathbf k)} + \frac{[\mathbf k\mathbf v_a][\mathbf k\mathbf v_b]\omega} {(\omega-i0)^2\varepsilon^{tr}(\omega-i0,\mathbf k)-c^2k^2} \right\} \\ &\quad - \frac{4\pi}{k^2}e_ae_b n_a f'_a n_b f_b \frac{\delta(\omega-\mathbf k\mathbf v_b)} {\omega+i0-\mathbf k\mathbf v_a} \left\{ \frac{\mathbf k\mathbf v_a}{\varepsilon^l(\omega+i0,\mathbf k)} + \frac{[\mathbf k\mathbf v_a][\mathbf k\mathbf v_b]\omega} {(\omega+i0)^2\varepsilon^{tr}(\omega+i0,\mathbf k)-c^2k^2} \right\}. \end{aligned} \tag{8} \]
An analogous consideration for a plasma with an isotropic distribution of particles in momenta, situated in a constant magnetic field, gives:
\[ (E_iE_j)_{\omega,\mathbf{k}}= A^{-1}_{ir}(\omega,\mathbf{k})A^{*-1}_{jl}(\omega,\mathbf{k}) \sum_a\left(\frac{4\pi e_a\omega}{c^2}\right)^2 \int d\mathbf{p}_a f_a \times \]
\[ \times \sum_{n=-\infty}^{\infty} \delta(\omega-n\Omega_a-k_z v^z)\,\mathscr{T}_{rl}; \tag{9} \]
\[ (\delta N_a(\mathbf{p}_a)\delta N_b(\mathbf{p}_b))_{\omega,\mathbf{k}} = e_ae_bn_an_b \Bigg\{ (E_iE_j)_{\omega,\mathbf{k}}f'_a f'_b W^i_a(\omega,\mathbf{k},\mathbf{p}_a)W^{*j}_b(\omega,\mathbf{k},\mathbf{p}_b) - \]
\[ - f'_b f_a\frac{4\pi i\omega}{c^2} A^{*-1}_{ir}(\omega,\mathbf{k}) W^{*i}_b(\omega,\mathbf{k},\mathbf{p}_b) Y^r_a(\omega,\mathbf{k},\mathbf{p}_a) + \]
\[ + f'_a f_b\frac{4\pi i\omega}{c^2} A^{-1}_{ir}W^i_aY^{*r}_b + n_a f_a\delta_{ab} \lim_{\Delta\to0}2\Delta \int_0^\infty dt\int_0^\infty dt'\, \exp[i\omega(t-t')] \times \]
\[ \times \delta\!\left(\mathbf{P}_a(0,t,\mathbf{p}_a)- \mathbf{P}_b(0,t',\mathbf{p}_b)\right) \exp\!\left[ i\mathbf{k}\left(\mathbf{R}_a(0,t,\mathbf{p}_a,0) -\mathbf{R}_b(0,t',\mathbf{p}_b,0)\right) \right] \Bigg\}, \]
\[ W^i_a(\omega,\mathbf{k},\mathbf{p}_a) = \int_{-\infty}^{0}dt\, \exp\!\left[-i(\omega-i\Delta)t +i\mathbf{k}\mathbf{R}_a(t,0,\mathbf{p}_a,0)\right] V^i_a(t,0,\mathbf{p}_a), \]
\[ Y^i_a(\omega,\mathbf{k},\mathbf{p}_a) = \int_{-\infty}^{\infty}d\tau\, \exp\!\left[-i\omega\tau +i\mathbf{k}\mathbf{R}_a(\tau,0,\mathbf{p}_a,0)\right] V^i_a(t,0,\mathbf{p}_a), \tag{10} \]
\[ A_{ij}(\omega,\mathbf{k}) = (\omega/c)^2\varepsilon_{ij}(\omega,\mathbf{k}) -\delta_{ij}k^2+k_ik_j . \]
Here \(\mathbf{P}_a(t,t',\mathbf{p}_a)\), \(\mathbf{V}_a(t,t',\mathbf{p}_a)\), and \(\mathbf{R}_a(t,t',\mathbf{p}_a,\mathbf{r}_a)\) are, respectively, the momentum, velocity, and coordinate of a particle at time \(t\), regarded as functions of the corresponding values at time \(t'\). When the transverse-field effect is neglected, formulas (8)—(10) correspond to those obtained in \((^3)\).
The method used substantially simplifies the derivation of the kinetic equation for a relativistic plasma. For this purpose it is sufficient to find the correlation of the Lorentz force and \(\delta N_a\), which, according to (1), determines the collision integral. Formulas (4)—(7) lead to the collision integral obtained in \((^{7,8})\).
In the case of a plasma situated in a constant magnetic field and for an isotropic distribution of particles in momenta, we easily obtain the following expression for the collision integral:
\[ I_a[f_a] = \frac{\partial}{\partial p_a^i} \int\frac{d\omega}{2\pi}\frac{d\mathbf{k}}{(2\pi)^3} e_a^2 \left\{ \delta_{il} + \frac{1}{\omega} (k^l v_a^i-\delta_{il}\mathbf{k}\mathbf{V}_a) \right\} A^{-1}_{lr} \times \]
\[ \times \left\{ f_a W^{*j}_a(\omega,\mathbf{k},\mathbf{p}_a) A^{*-1}_{js}(\omega,\mathbf{k}) \sum_b \left(\frac{4\pi\omega e_b}{c^2}\right)^2 n_b \int d\mathbf{p}_b f_b v_b^rY_b^s(\omega,\mathbf{k},\mathbf{p}_b) + \right. \]
\[ \left. + f_a\frac{4\pi i\omega^2}{c^2} Y^{*r}_a(\omega,\mathbf{k},\mathbf{p}_a) \right\}. \tag{11} \]
Formulas (8) and (10) make it possible to find the equal-time correlation functions for an opaque plasma
\[ g_{ab}(\mathbf{k},\mathbf{p}_a,\mathbf{p}_b) = \frac{1}{n_an_b} \left\{ \int d\omega\, (\delta N_a(\mathbf{p}_a)\delta N_b(\mathbf{p}_b))_{\omega,\mathbf{k}} - \delta_{ab}\delta(\mathbf{p}_a-\mathbf{p}_b)n_af_a \right\}. \]
In this case, for a plasma without strong fields, according to (8) we have
$$ \begin{aligned} g_{ab}(\mathbf{k},\mathbf{p}_a,\mathbf{p}_b)={}& \frac{e_a e_b}{\mathbf{k}\mathbf{v}_a-\mathbf{k}\mathbf{v}_b-i\Delta} \Biggl\{ f'_a f_b\,\frac{4\pi}{k^2} \left( \frac{\mathbf{k}\mathbf{v}_a}{\varepsilon^l(\mathbf{k}\mathbf{v}_b+i0,\mathbf{k})} + \frac{(\mathbf{k}\mathbf{v}_b)[\mathbf{k}\mathbf{v}_a][\mathbf{k}\mathbf{v}_b]} {(\mathbf{k}\mathbf{v}_b+i0)^2\varepsilon^{tr}(\mathbf{k}\mathbf{v}_b-i0,\mathbf{k})-c^2k^2} \right) \\ &\quad - f_a f'_b\,\frac{4\pi}{k^2} \left( \frac{\mathbf{k}\mathbf{v}_b}{\varepsilon^l(\mathbf{k}\mathbf{v}_a-i0,\mathbf{k})} + \frac{(\mathbf{k}\mathbf{v}_a)[\mathbf{k}\mathbf{v}_a][\mathbf{k}\mathbf{v}_b]} {(\mathbf{k}\mathbf{v}_a-i0)^2\varepsilon^{tr}(\mathbf{k}\mathbf{v}_a-i0,\mathbf{k})-c^2k^2} \right) \\ &\quad +\left(\frac{4\pi}{k^2}\right)^2 f'_a f'_b \sum_\gamma e_\gamma^2 n_\gamma \int d\mathbf{p}_\gamma f_\gamma \left( \frac{1}{\mathbf{k}\mathbf{v}_b-\mathbf{k}\mathbf{v}_\gamma+i0} - \frac{1}{\mathbf{k}\mathbf{v}_a-\mathbf{k}\mathbf{v}_\gamma-i0} \right) \\ &\quad\times \left( \frac{(\mathbf{k}\mathbf{v}_a)(\mathbf{k}\mathbf{v}_b)} {|\varepsilon^l(\mathbf{k}\mathbf{v}_\gamma,\mathbf{k})|^2} + \frac{[\mathbf{k}\mathbf{v}_a][\mathbf{k}\mathbf{v}_b](\mathbf{k}\mathbf{v}_\gamma)^2[\mathbf{k}\mathbf{v}_\gamma]^2} {|(\mathbf{k}\mathbf{v}_\gamma)^2\varepsilon^{tr}(\mathbf{k}\mathbf{v}_\gamma,\mathbf{k})-c^2k^2|^2} \right) \Biggr\}. \end{aligned} \tag{12} $$
In the particular case of a Maxwellian distribution of particles with equal temperatures, from this we have
$$ g_{ab}=-\frac{4\pi e_a e_b f_a f_b}{k^2\varkappa T+\sum_\gamma 4\pi e_\gamma^2 n_\gamma}, \tag{13} $$
which corresponds to the Debye shielding.
Moscow State University
named after M. V. Lomonosov
Received
24 X 1961
CITED LITERATURE
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