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Electrostatic Instabilities of a Uniform Non‐Maxwellian Plasma

### Abstract

A stability criterion is obtained starting from Vlasov's collision‐free kinetic equations. Possible instabilities propagating parallel to an arbitrary unit vector **e** are related to a function , where *g*_{i} (**v**) is the normalized unperturbed distribution function, and the plasma frequency, for the *j*th type of particle. By using a method related to the Nyquist criterion, it is shown that plasma oscillations growing exponentially with time are possible if and only if *F*(*u*) has a minimum at a value *u* = ξ such that . A study of the initial‐value problem confirms that the plasma is normally stable if no exponentially growing modes exist; but there is an exceptional class of distribution functions (recognizable by means of an extension of the above criterion) for which linearized stability theory breaks down. The method is applied to several examples, of which the most important is a model of a current‐carrying plasma with Maxwell distributions at different temperatures for electrons and ions. The meaning of the mathematical assumptions made is carefully discussed.

© 1960 The American Institute of Physics

Received 09 October 1959
Published online 22 November 2004

/content/aip/journal/pof1/3/2/10.1063/1.1706024

1.

1.A. Vlasov, Zhur. Eksp. i Teoret. Fiz. 8, 291 (1938).

2.

2.See I. B. Bernstein, Phys. Rev. 109, 10 (1958).

3.

3.O. Buneman, Phys. Rev. Letters 1, 8 (1958).

4.

4.P. L. Auer, Phys. Rev. Letters 1, 411 (1958).

5.

5.F. D. Kahn, Astrophys. J. 129, 468 (1959).

6.

6.S. G. Mikhlin, Integral Equations (Pergamon Press, London, England, 1957), pp. 115–116. Theorems 1 and 3 of this reference show that is bounded and continuous, since the boundedness of imply a Lipschitz condition on

7.

7.This curve is closely related to the Nyquist diagram of servomechanism theory, which has been applied for plasmas by E. G. Harris [Phys. Rev. Letters 2, 34 (1959)].

7.N. G. Van Kampen [Physica 21, 949 (1955)] has used a similar method.

8.

8.E. T. Copson, Theory of Functions of a Complex Variable (Oxford University Press, Oxford, England, 1935), p. 119.

9.

9.W. K. Hayman (private communication, 1959).

10.

10.S. Tamor (unpublished) and others have obtained similar criteria.

11.

11.E. C. Titchmarsh, Theory of Fourier Integrals, (Oxford University Press, Oxford, England, 1937), pp. 311–312.

12.

12.L. Landau, J. Phys. (U.S.S.R.) 10, 25 (1946).

13.

13.N. G. van Kampen [Physica 21, 949 (1955)] also avoids the use of analytic continuation.

14.

14.See reference 11, pp. 115–116, Theorem (84), with the following modifications:

15.

15.R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain (American Mathematical Society Colloquim Publications, New York, 1934), Vol. XIX, Sec. 18.

16.

16.See reference 11, p. 11 (Theorem 1).

17.

17.D. Bohm and E. P. Gross, Phys. Rev. 75, 1864 (1949).

18.

18.I. Bernstein, J. Greene, and M. Kruskal, Phys. Rev. 108, 546 (1957).

19.

19.P. Berz [Proc. Phys. Soc. B69, 939 (1956)] proved this result by a different method for the case

19.P. L. Auer [Phys. Rev. Letters 1, 411 (1958)] extended Berz’ result to unsymmetrical

20.

20.N. G. van Kampen (see the work cited in footnote 13) has obtained the same result.

21.

21.R. Q. Twiss, Phys. Rev. 88, 1392 (1952).

22.

22.M. Sumi, J. Phys. Soc. Japan 13, 1476 (1958).

23.

23.Numerical data used in plotting Figs. 1 and 6 were taken from a paper by Harris [Astrophys. J. 108, 112 (1948)].

24.

24.S. Tamor (unpublished) has shown for the special case that the critical value of is without imposing the conditions (29)–(31).

25.

25.I. Bernstein (unpublished) also obtained this result for

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