Singular Limits of Solutions of Boltzmanns Equations
Singular Limits of Solutions of Boltzmanns Equations
Harold Grad
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One can generalize the Boltzmann equation to obtain a sequence of two-point, three-point, etc. Kinetic equations for a rarefied gas without invoking molecular chaos. (eg. [17, 21]). The closure procedure is not trivial since generalizations of Boltzmann 's chaos rule are not unique. For example, one can introduce a (2 ) closure in an equation governing df /9t such that the ordinary Boltzmann equation is exactly satisfied by f = /f even (2) . . when f is not chaotic; (this unduly restrictive pro...perty is enjoyed by many higher order kinetic equations, eg. See [21, 22, 23, 24]) It is possible however, to formulate a set of desireable criteria which make the closure essentially unique. * It is useful to make a rough distinction between correlations (referring to two or more positions and one time) and fluctuations (two or more times and any number of positions) . To extend the Boltzmann equation to include long range non-chaos, or instability, or turbulence requires correlations. To calculate the scattering 'C.
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