A modified truncation procedure for the BBGKY hierarchy

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Abstract
The approximations usually made to truncate the BBGKY hierarchy for a plasma are discussed; their failure at small inter-particle separations leads to divergence of the Balescu—Lenard collision integral. A number of authors have obtained convergent kinetic equations, often by rather complicated methods. It is shown here that, if the standard truncation procedure is modified in a way which makes it less obviously inconsistent for close approaches, the standard methods maybe closely followed in deriving a convergent collision integral which agrees to dominant order with the ‘cutoff’ Balescu—Lenard integral and with the other work on the problem. In fact, the kinetic equation obtained is identical with the Balescu—Lenard equation except that the Coulomb potential is replaced by another that is non-singular at the origin. A physical interpretation of this result is suggested.