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It can be shown, using the Liouville theorem, that a system performing Hamiltonian evolution does not increase its entropy (e.g., see What maximizes entropy?.) BBGKY hierarchy seems to be just an alternative way of describing Hamiltonian evolution, so I expect that it produces the same prediction...

On the other hand, Boltzmann H-theorem does predict entropy increase on the basis of the Boltzmann equation, obtained by truncating BBGKY hierarchy.

The question is essentially: is the H-theorem a consequence of truncating the hierarchy of equations, or is there some difference between the Liouville and BBGKY description of Hamiltonian evolution, which results in entropy increasing (in BBGKY)?

Tobias Fünke
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Roger V.
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The Liouville equation and the BBGKY hierarchy are strictly equivalent. You therefore do not get an increase in entropy in either case. When you close the BBGKY at first order to get the Boltzmann equation, you invoke Stosszahlansatz which breaks time reversal symmetry hence the H-theorem. You can find a detailed discussion in Dorfman's An Introduction to Chaos in Nonequilibrium Statistical Mechanics for example.

Roger V.
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LPZ
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