UDC 533.916

PHYSICS

R. V. POLOVIN, V. V. ROZHKOV

ON THE QUESTION OF THE KINETIC DESCRIPTION OF RAPID PROCESSES IN A PLASMA

(Presented by Academician L. I. Sedov on 8 IV 1969)

The state of a plasma is usually described by the distribution function $f(t,\mathbf r,\mathbf v)$, which is the probability density of the random coordinates $\mathbf r(t)$ and velocity $\mathbf v(t)$ of a particle. The distribution function depends on time $t$ as on a parameter and satisfies a kinetic equation of the Fokker—Planck type

\[ \frac{\partial f}{\partial t} = - v_i \frac{\partial f}{\partial x_i} - \frac{\partial}{\partial v_i}(A_i f) + \frac{1}{2}\frac{\partial^2}{\partial v_i \partial v_j}(D_{ij} f). \tag{1} \]

On the other hand, it is known from the theory of random processes [1] that an equation of type (1) exists only for Markov random processes $\{\mathbf r(t),\mathbf v(t)\}$, for which the rate of change of the distribution function $\partial f/\partial t$ at time $t$ depends only on the value of $f$ at this time and does not depend on the preceding history, i.e., on the values of $f$ at preceding instants of time. In particular, the process $\{\mathbf r(t),\mathbf v(t)\}$ will be non-Markovian if $\{\mathbf r(t),\mathbf v(t)\}$ is the solution of a system of differential equations whose coefficients are random functions with a nonzero correlation time.

In the case of a plasma the state of a particle with mass $m$ and charge $e$ is determined by the equations

\[ \frac{d\mathbf r}{dt}=\mathbf v,\qquad \frac{d\mathbf v}{dt}=\frac{e}{m}\mathbf E(\mathbf r,t), \tag{2} \]

where the electric field $\mathbf E$ is a random process with a correlation function that may be approximated by the exponential

\[ \langle E_i[\mathbf r(t),t]E_j[\mathbf r(t+\tau),t+\tau]\rangle = \delta_{ij}\sigma^2 e^{-\nu|\tau|}. \tag{3} \]

The correlation time $1/\nu$ usually has the order of the period of plasma oscillations.

To reveal the main features of this random process, we shall consider the simplest case of one-dimensional homogeneous (i.e., independent of $\mathbf r$) motion. Then equations (2), (3) are simplified:

\[ \dot v=\frac{e}{m}E, \tag{4} \]

\[ \langle E(t)E(t+\tau)\rangle=\sigma^2 e^{-\nu|\tau|}. \tag{5} \]

The process $v(t)$ is non-Markovian, but it can be augmented to a Markovian one [1]. To do this, using the Wiener—Khinchin theorem, we transform relation (5) to the form

\[ \dot E+\nu E=u(t), \tag{6} \]

where $u(t)$ is white noise having zero correlation time

\[ \langle u(t)u(t+\tau)\rangle=2\sigma^2\nu\delta(\tau). \tag{7} \]

The two-dimensional random process \(\{v(t), E(t)\}\), described by equations (4), (6), is Markovian, and its distribution function \(f(v,E)\) satisfies the Fokker—Planck equation (1)

\[ \frac{\partial f}{\partial t} = -\frac{e}{m}E\frac{\partial f}{\partial v} + \nu\frac{\partial(fE)}{\partial E} + \sigma^{2}\nu\frac{\partial^{2}f}{\partial E^{2}} . \tag{8} \]

If the electric field had no “memory,” i.e., if it itself were white noise, then the distribution function would depend only on the velocity: \(f(v)\). (The distribution function has the same form after the lapse of a time much longer than the correlation time. Such a consideration is used in the theory of stochastic acceleration of particles \({}^{(2,3)}\).) The simplest “memory” of the type (5) leads to the distribution function \(f(v,E)\), or, by virtue of (4), \(f(v,\dot v)\). More complicated correlation functions of the electric field will lead to a dependence of the distribution functions on higher derivatives of the velocity:

\[ f(v);\ f(v,\dot v);\ f(v,\dot v,\ddot v);\ f(v,\dot v,\ddot v,\dddot v),\ldots \tag{9} \]

Such a chain of one-particle distribution functions, depending on the velocity \(v\) and its derivatives (this possibility was first pointed out by Vlasov \({}^{(4)}\)), is analogous to Bogolyubov’s chain \({}^{(5)}\) \(f(v_1), f(v_1,v_2), f(v_1,v_2,v_3), \ldots\) of many-particle distribution functions depending only on the velocity. The chain (9) may prove preferable in the case of strongly turbulent plasma, when the correlations of the distribution functions \(g(v_1,v_2)=f(v_1,v_2)-f(v_1,v_2)\) are not small \({}^{(6)}\). (At present, a quantitative theory of strongly turbulent plasma has not yet been created \({}^{(7)}\).)

It can be shown that in a quiescent plasma the velocity of particles is a Markov process, if one neglects terms that are small in comparison with the Coulomb logarithm.

The authors express their gratitude to A. I. Akhiezer and L. I. Sedov for valuable discussions.

Physical-Technical Institute
Academy of Sciences of the Ukrainian SSR
Kharkov

Received
26 II 1969

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