 9175 Duffield N.G., Werner R.F.
 Local dynamics of meanfield quantum systems
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Dec 4, 91

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Abstract. In this paper we extend the theory of meanfielddynamical
semigroups to treat the irreversible meanfield dynamics of
quasilocal meanfield observables. These are observables which are
site averaged except within a region of tagged sites. In the
thermodynamic limit the tagged sites absorb the whole lattice, but
also become negligible in proportion to the bulk. We develop the
theory in detail for a class of interactions which contains the
meanfield versions of quantum lattice interactions with infinite
range. For this class we obtain an explicit form of the dynamics in
the thermodynamic limit. We show that the evolution of the bulk is
governed by a flow on the oneparticle state space, whereas the
evolution of local perturbations in the tagged region factorizes
over sites, and is governed by a cocycle of completely positive
maps. We obtain an Htheorem which suggests that local disturbances
typically become completely delocalized for large times, and we show
this to be true for a dense set of interactions. We characterize all
limiting evolutions for certain subclasses of interactions, and also
exhibit some possibilities beyond the class we study in detail: for
example, the limiting evolution of the bulk may exist, while the
local cocycle does not. In another case the bulk evolution is given
by a diffusion rather than a flow, and the local evolution no longer
factorizes over sites.
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