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Article Dans Une Revue Physica A: Statistical Mechanics and its Applications Année : 1995

Analysis of Certain Binary Collision Approximation Closures of the BBGKY Hierarchy

Résumé

Boercker and Dufty (BD) introduce a binary collision approximation into the BBGKY hierarchy in order to describe the low-density behaviour of a quantum gas. Their approach involves the three-particle reduced density operator being approximated in a particular manner that favours correlation between two of the particles, while ignoring the correlation with the third. The tradition of previous derivations, on the other hand, has been to neglect the three-particle term altogether and assume a generalized form of Boltzmann's Stosszahlansatz. Both formalisms reach the same final result: the Waldmann-Snider equation (WS), a quantum version of the Boltzmann equation. We compare the two derivations in two ways: (a) by finding iterated series solutions of the BBGKY hierarchy and of the Boltzmann equation, which demonstrates what kind of terms are absent in the WS and BD approaches in comparison with exact dynamics; (b) by computing an exact correction to the WS equation, which vanishes in the BD scheme provided the mean time of duration of a collision is much shorter than the mean free time between collisions. This correction is shown to be related to the standard three-body collision integral arising in the theory of the density corrections to the Boltzmann equation. We also comment on the related work of Klimontovich, who introduces an approximation analogous to that of Boercker and Dufty.
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Dates et versions

hal-00001461 , version 1 (15-04-2004)

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  • HAL Id : hal-00001461 , version 1

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Robert F. Snider, William J. Mullin, Franck Laloë. Analysis of Certain Binary Collision Approximation Closures of the BBGKY Hierarchy. Physica A: Statistical Mechanics and its Applications, 1995, 218, pp.155. ⟨hal-00001461⟩
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