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Reviews in Mathematical Physics

MEAN-FIELD DYNAMICAL SEMIGROUPS ON C*-ALGEBRAS

Nick G. Duffield, Reinhard F. Werner

Published Sep 1, 1992• 18 citations
Publisher

Abstract

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].

DOI
10.1142/s0129055x92000108

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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].
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
This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .
An infinite quantum system with correctly defined dynamics τQ as an automorphism group of a C*-algebra 𝒞 of observables is determined by any continuous unitary representation U(G) of a connected Lie group G, as well as by an arbitrary differentiable real function Q on the dual space 𝔤* to the Lie algebra 𝔤 of G with the canonically defined Poisson flow φQ on 𝔤*. For specific choices of Q and G, the system can be obtained as the thermodynamic limit of a net of finite lattice systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)]. A simple nontrivial model of this type is the quasispin BCS model of superconductivity in the strong coupling limit, or a corresponding model of the Josephson junction. A peculiar feature of the considered models is τQ noninvariance of the usually considered C*-algebra 𝒜 of quasilocal observables, as well as an important role of classical dynamics of a set of macroscopic (intensive) observables Qφ in the description of τQ. The work is restricted here to norm-continuous representations U(G), in which case 𝒞 is isomorphic to the tensor product 𝒜⊗𝒩, where 𝒩 is the commutative algebra of classical (intensive) observables of the considered infinite quantum system.
This paper is devoted to the aspects of derivation of dynamics equation of quantum system under a stochastic dynamics. We study the conditions, under which a sequence of random variations of wave function can approximate a random diffusion process in a Hilbert space. A random variation $|\psi_0\rangle \mapsto G_{t_N}\ldots G_{t_1}|\psi_0\rangle =|\psi_{t_N}\rangle $ is associated with a transform of distribution of vector $|\psi\rangle $, as well as with the variation of its characteristic functional φ(v)=Eexp⁡(iRe⁡⟨v∣ψ⟩).\varphi(v)=\mathbb{E}\exp(i\operatorname{Re}\langle v|\psi\rangle ).φ(v)=Eexp(iRe⟨v∣ψ⟩). For a continuous random walk we study the approximation of a Markov semigroup by the Markov operators of discrete random walk. We pay a special attention to the cases, when the derivative of random operator F′(0)F'(0)F′(0) is an unbounded operator. However, we restrict the consideration to the case when the Markov operators of random walks with the operators GtG_tGt​ mutually commute. The characteristic functional is transformed by the Markov operator of adjoint process, and in contrast to the dynamics of wave function, it has a deterministic nature that allows us to rely on the developed theory of semigroups in Banach spaces. The most illustrative examples are the process of continuous measurement, that is, the measurement of trajectories of some observable, and the random control.
. 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...
Following the work of P. Bona [J. Math. Phys. 29, 2223 (1988)], a class of infinite mean-field quantum lattice systems is considered within the framework of algebraic quantum mechanics, and the following question is discussed: Which are the possible dynamical descriptions of the system, i.e., in which representations π of the system’s quasilocal C*-algebra 𝒜 do the local time evolutions converge in the thermodynamic limit to an automorphism group of a certain enlarged C*-algebra 𝒞π containing π(𝒜) and (some of) the classical observables of the system in π. To this end we associate a ‘‘classical state space’’ Eπ to any representation π (which is a subset of the phase-space E⊇RL defined by Bona for the largest possible representation), and show how it is characterized by the π-normal states, thereby obtaining a way to determine Eπ explicitly in many cases. The answer to the question posed then reads: The limiting (Heisenberg) dynamics exists in π on 𝒞π if and only if the set Eπ is invariant under the flow φQt on E which describes the time evolution of classical observables. Conversely, to any invariant, closed subset B of E, there exists a dynamical description of the quantum system with classical state space B.