LOCAL DYNAMICS OF MEAN-FIELD QUANTUM-SYSTEMS
|MECHANICS; Physics; Physics, Multidisciplinary; RELATIVE ENTROPY; SEMIGROUPS; VARIATIONAL EXPRESSION
|BIRKHAUSER VERLAG AG
|HELVETICA PHYSICA ACTA
In this paper we extend the theory of mean-field-dynamical semigroups given in [DW1,Du1] to treat the irreversible mean-field dynamics of quasi-local mean-field 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 mean-field 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 one-particle 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 H-theorem 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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checked on Mar 5, 2024