Generating Functional for the BBGKY Hierarchy and the N-Identical-Body Problem
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We formulate the N-body problem for identical particles by identifying a generating functional for the reduced n-particle probability density functions, n ≤ N. The evolution equation for the generating functional follows by substituting the corresponding representation of the Liouville density P N , called the de Finetti representation, into Liouville’s equation. We illustrate this process in model problems, which indicate that there is a difficulty when the flux of the generator depends on its properties away from the constraint plane of fixed normalisation. The evolution equation for the generating functional then depends crucially on N, the actual number of particles, which in real problems is seldom known exactly. For fixed transition rates, and for bosons satisfying detailed balance, the difficulty does not arise, and the de Finetti representation provides useful insights. For the general case of classical statistical mechanics, the difficulty forces us to conclude that the de Finetti generator is not a useful alternative to the BBGKY hierarchy.
KeywordsPhase Space Evolution Equation Detailed Balance Closure Scheme Classical Statistical Mechanic
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