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Local correlation functions for mean-field dynamical semigroups on C*-algebras

Nick G. Duffield

Published Jul 5, 1991• 3 citations
Publisher

Abstract

We study the dynamics of local correlation functions in dissipative mean-field systems. This is done by extending the abstract notion of a mean-field dynamical semigroups on C*-algebra given in [1], from an evolution on site-averaged observables, to one on a class of local observables. Conditions are established under which this generalized mean-field dynamics factorises, in the thermodynamic limit, into contributions from disjoints for single site observables in this limit. We demonstrate that these conditions are satisfied for a large class of model systems

DOI
10.5169/seals-116315

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We study the dynamics of local correlation functions in dissipative mean-field systems. This is done by extending the abstract notion of a mean-field dynamical semigroups on C*-algebra given in [1], from an evolution on site-averaged observables, to one on a class of local observables. Conditions are established under which this generalized mean-field dynamics factorises, in the thermodynamic limit, into contributions from disjoints for single site observables in this limit. We demonstrate that these conditions are satisfied for a large class of model systems
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We study a notion of the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra [Formula: see text] with itself. In analogy with the theory of semigroups on Banach spaces we give abstract conditions for the existence of these limits. These conditions are verified in the case of semigroups whose generators are determined by the successive resymmetrizations of a fixed operator, as well as generators which can be approximated by generators of this type. This includes the time evolutions of the mean-field versions of quantum lattice systems. In these cases the limiting dynamical semigroup is given by a continuous flow on the state space of [Formula: see text]. For a class of such flows we show stability by constructing a Liapunov function. We also give examples where the limiting evolution is given by a diffusion, rather than a flow on the state space of [Formula: see text].
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