By Johan Grasman

Half A: The Fokker-Planck Equation. half B: Asymptotic answer of the go out challenge. half C: functions

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**Asymptotic methods for the Fokker-Planck equation and the exit problem in applications**

Half A: The Fokker-Planck Equation. half B: Asymptotic resolution of the go out challenge. half C: functions

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**Sample text**

9) shows that the growth rate is less than the bandwidth of the growing waves provided the condition njno ^(Vb/vb)3 is satisfied, where factors of order unity are ignored. 17), confirming that the kinetic version takes over from the reactive version for nxln0 x(Vb/vb)3. 7). 10) 2vb However because the growth rate depends on k (apart from the weak dependence through a)t(k)) only in the form k-v6, it is almost independent of the component of k orthogonal to V;,. 8) may be replaced by at k x oipl(vb cos 6) — Vb.

Let k = kM be a solution of the dispersion equation for a mode M. 26) becomes ^ M * L . 27) Let us denote by M + and M — poles in the upper half fc-plane and in the lower half fc-plane respectively. dze""g(z) is well behaved for Im(fcz)>0. Thus for positive z outside the source region Interpretation of complex solutions (a) (b) 41 Im k (1) Recj Rek —»• Fig. 3 As Im co is reduced from F to 0 forfixedRe co (Fig. 3(a)) the solutions kM(co) move along paths in the complex-fc plane (Fig. 3(b)). Solutions such as (1) correspond to evanescent modes, and solutions such as (2) which cross the Re k axis corresponding to amplifying modes.

In the interplanetary medium, the beams tend to be very weak and long times are available for them to relax. Consequently the kinetic version and not the reactive version of the instability is likely to be the more relevant in applications outside the laboratory. 5 The Buneman instability An instability which is closely related to the weak-beam instability was pointed out by Buneman (1958, 1959). A current with current density J implies a relative motion of the electrons and the ions at a drift speed »„ = — • (3-19) nee The longitudinal part of the dielectric tensor in the rest frame of the electrons is K > , k) = 1 - ^ - , *"* , 2 , (3-20) (eo-k-v,;)2 a2 where the electrons are assumed to be cold.

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