Quasilinear diffusion equation

Remark. The quasilinear diffusion equation (2.4) has an interesting transformation property. Write it in the form (VIII.1.11) taking 6= 1 for convenience,... 

Since the observed low frequency noise is broad band and the wave intensity is not very strong, it is reasonable to represent the perpendicular heating due to electromagnetic ion cyclotron resonance by the quasilinear diffusion operator. In the absence of an external DC electric field, the ion evolution equation along a geomagnetic field line is [10] ... 

With these scalings, the quasilinear ion evolution equation (11) may be re-expressed as a convective-diffusion equation ... 

In the case of the quasilinear Fokker-Planck equation (2.4), the free energy U defined in terms of the stationary solution by (2.6) is identical with the potential in the deterministic equation (5.2). That identity is often taken for granted when time-dependent solutions have to be constructed for systems of which only the equilibrium distribution is known. We shall now show, however, that it holds only for systems of diffusion type whose Fokker-Planck equation is quasilinear, i.e., of the form (2.4). 

So far we studied instabilities in discrete one-step processes. Now consider diffusion processes as in XI. For simplicity we restrict ourselves to diffusion in an external potential U(x), as described by the quasilinear Fokker-Planck equation (XI.2.4) ... 

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