Our starting point is the Vlasov equation for an unmagnetized, collisionless
Since we are dealing with small amplitude waves, it is appropriate to
linearize the Vlasov equation. Suppose that the electron
distribution function is written
Let us now follow the standard procedure for analyzing small amplitude
waves, by assuming that all perturbed quantities vary with
Equations (989) and (990) reduce to
We can interpret Eq. (993) as the dispersion relation for electrostatic plasma waves, relating the wave-vector, , to the frequency, . However, in doing so, we run up against a serious problem, since the integral has a singularity in velocity space, where , and is, therefore, not properly defined.
The way around this problem was first pointed out by Landau in a very
influential paper which laid the basis of much subsequent
research on plasma oscillations and instabilities. Landau showed that,
instead of simply assuming that varies in time as
the problem must be regarded as an initial value problem in which
is given at and found at later times.
We may still Fourier analyze with respect to , so we write
In order to solve Eqs. (995) and (996) as an initial value problem, we
introduce the Laplace transform of with respect to :
Noting that the Laplace transform of
(as is easily shown by integration by parts), we can Laplace transform Eqs. (995)
and (996) to obtain
The Laplace transform of the distribution function is written
Having found the Laplace transforms of the electric field and the perturbed
distribution function, we must now invert them to obtain
and as functions of time. The inverse Laplace transform
of the distribution function is given by
Rather than trying to obtain a general expression for , from Eqs. (1004) and (1005), we shall concentrate on the behaviour of the perturbed distribution function at large times. Looking at Fig. 31, we note that if has only a finite number of simple poles in the region , then we may deform the contour as shown in Fig. 32, with a loop around each of the singularities. A pole at gives a contribution going as , whilst the vertical part of the contour goes as . For sufficiently long times this latter contribution is negligible, and the behaviour is dominated by contributions from the poles furthest to the right.
Equations (1001)-(1004) all involve integrals of the form
If is sufficiently well-behaved that it can be continued off the real axis as an analytic function of a complex variable then the continuation of (1006) as the singularity crosses the real axis in the complex -plane, from the upper to the lower half-plane, is obtained by letting the singularity take the contour with it, as shown in Fig. 33.
Note that the ability to deform the contour into that of Fig. 32, and find a dominant contribution to and from a few poles, depends on and having smooth enough velocity dependences that the integrals appearing in Eqs. (1001)-(1004) can be continued sufficiently far into the left-hand half of the complex -plane.
If we consider the electric field given by the inversion of Eq. (1001), we see that its behaviour at large times is dominated by the zero of which lies furthest to the right in the complex -plane. According to Eqs. (1004) and (1005), has a similar contribution, as well as a contribution going as . Thus, for sufficiently long times after the initiation of the wave, the electric field depends only on the positions of the roots of in the complex -plane. The distribution function has a corresponding contribution from the poles, as well as a component going as . For large times, the latter component of the distribution function is a rapidly oscillating function of velocity, and its contribution to the charge density, obtained by integrating over , is negligible.
As we have already noted, the function is equivalent to the left-hand side of Eq. (993), provided that is replaced by . Thus, the dispersion relation, (993), obtained via Fourier transformation of the Vlasov equation, gives the correct behaviour at large times as long as the singular integral is treated correctly. Adapting the procedure which we found using the variable , we see that the integral is defined as it is written for , and analytically continued, by deforming the contour of integration in the -plane (as shown in Fig. 33), into the region . The simplest way to remember how to do the analytic continuation is to note that the integral is continued from the part of the -plane corresponding to growing perturbations, to that corresponding to damped perturbations. Once we know this rule, we can obtain kinetic dispersion relations in a fairly direct manner via Fourier transformation of the Vlasov equation, and there is no need to attempt the more complicated Laplace transform solution.
In Sect. 4, where we investigated the cold-plasma dispersion relation, we found that for any given there were a finite number of values of , say , , , and a general solution was a linear superposition of functions varying in time as , , etc. This set of values of is called the spectrum, and the cold-plasma equations yield a discrete spectrum. On the other hand, in the kinetic problem we obtain contributions to the distribution function going as , with taking any real value. All of the mathematical difficulties of the kinetic problem arise from the existence of this continuous spectrum. At short times, the behaviour is very complicated, and depends on the details of the initial perturbation. It is only asymptotically that a mode varying as is obtained, with determined by a dispersion relation which is solely a function of the unperturbed state. As we have seen, the emergence of such a mode depends on the initial velocity disturbance being sufficiently smooth.
Suppose, for the sake of simplicity, that the background plasma state is a
Maxwellian distribution. Working in terms of , rather than , the kinetic dispersion
relation for electrostatic waves takes the form
Suppose that is sufficiently small that
range of where
is non-negligible. It follows
that we can expand the denominator of the principal part integral in a
If we compare the above results with those for a cold-plasma, where the dispersion relation for an electrostatic plasma wave was found to be simply , we see, firstly, that now depends on , according to Eq. (1012), so that in a warm plasma the electrostatic plasma wave is a propagating mode, with a non-zero group velocity. Secondly, we now have an imaginary part to , given by Eq. (1015), corresponding, since it is negative, to the damping of the wave in time. This damping is generally known as Landau damping. If (i.e., if the wave-length is much larger than the Debye length) then the imaginary part of is small compared to the real part, and the wave is only lightly damped. However, as the wave-length becomes comparable to the Debye length, the imaginary part of becomes comparable to the real part, and the damping becomes strong. Admittedly, the approximate solution given above is not very accurate in the short wave-length case, but it is sufficient to indicate the existence of very strong damping.
There are no dissipative effects included in the collisionless Vlasov equation. Thus, it can easily be verified that if the particle velocities are reversed at any time then the solution up to that point is simply reversed in time. At first sight, this reversible behaviour does not seem to be consistent with the fact that an initial perturbation dies out. However, we should note that it is only the electric field which decays. The distribution function contains an undamped term going as . Furthermore, the decay of the electric field depends on there being a sufficiently smooth initial perturbation in velocity space. The presence of the term means that as time advances the velocity space dependence of the perturbation becomes more and more convoluted. It follows that if we reverse the velocities after some time then we are not starting with a smooth distribution. Under these circumstances, there is no contradiction in the fact that under time reversal the electric field will grow initially, until the smooth initial state is recreated, and subsequently decay away.