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The nature of the Spectrum and the Runaway Phenomenon

Very little is known about the nature of the spectrum of for collision operators corresponding to real interactions.       The main features of the spectrum of , but not all of its detailed properties are known for the hard sphere potential, and for potentials [Grad [1963], Dorfman [1963], Kuscer and Williams [1967], Yan and Wannier [1968] and Pao [1974]] . The spectrum for Maxwell molecules ( potential) is the only potential for which the spectrum is known completely [see preceding refs]. Almost nothing is known for the potentials having an attractive component.

We have chosen a discrete spectrum because our exactly solvable model has this structure, and we wish to understand the features of this model in the first instance, and also because the mathematics of discrete spectra is vastly simpler to that of continuous spectra. (This is why the theory of compact operators, and of bounded self-adjoint operators is so much more developed than the case of general linear operators.) However, the existence of runaway   [Howorka et al. [1979], Moruzzi and Kondo [1980]] shows that there are circumstances where this assumption fails. In this phenomenon, there are regions of the parameter for which the transport coefficients are not    well defined. The motion of the centroid is reminiscent of acceleration rather that that of a steady drift velocity. The arguments in the previous section rule out the possibility of runaway arising when the spectrum is discrete. Consequently, we can say that a necessary condition for runaway is that the spectrum must have continuous or residual components.

Cavalleri and Paveri-Fontana [1972] give as sufficient condition for runaway to occur that the integral of the velocity dependent    collision frequency should exist. This collision frequency is defined by

where is the background gas distribution function, and is the total scattering cross section. Since necessity and sufficiency are often closely connected conditions in mathematics, one may speculate that there is a strong connection between the asymptotic form of for large c, and the structure of the spectrum of .



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Next: Klein-Kramers Model Up: The Linear Boltzmann Previous: Formal Solution of



Russell Standish
Thu May 18 11:43:52 EST 1995