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Quantum Markovian semigroups on quantum spin systems: Glauber dynamics
최베니,고철기,박용문 대한수학회 2008 대한수학회지 Vol.45 No.4
We study a class of KMS-symmetric quantum Markovian semigroups on a quantum spin system (A, γ, ω), where A is a quasi-local algebra, γ is a strongly continuous one parameter group of *-automorphisms of A and ω is a Gibbs state on A. The semigroups can be considered as the extension of semigroups on the nontrivial abelian subalgebra. Let H be a Hilbert space corresponding to the GNS representation constructed from ω. Using the general construction method of Dirichlet form developed in [8], we construct the symmetric Markovian semigroup {Tt}t≥0 on H. The semigroup {Tt}t≥0 acts separately on two subspaces Hd and Hod of H, where Hd is the diagonal subspace and Hod is the off-diagonal subspace, H = Hd Hod. The restriction of the semigroup {Tt}t≥0 on Hd is Glauber dynamics, and for any η∈ Hod, Ttη decays to zero exponentially fast as t approaches to the infinity. We study a class of KMS-symmetric quantum Markovian semigroups on a quantum spin system (A, γ, ω), where A is a quasi-local algebra, γ is a strongly continuous one parameter group of *-automorphisms of A and ω is a Gibbs state on A. The semigroups can be considered as the extension of semigroups on the nontrivial abelian subalgebra. Let H be a Hilbert space corresponding to the GNS representation constructed from ω. Using the general construction method of Dirichlet form developed in [8], we construct the symmetric Markovian semigroup {Tt}t≥0 on H. The semigroup {Tt}t≥0 acts separately on two subspaces Hd and Hod of H, where Hd is the diagonal subspace and Hod is the off-diagonal subspace, H = Hd Hod. The restriction of the semigroup {Tt}t≥0 on Hd is Glauber dynamics, and for any η∈ Hod, Ttη decays to zero exponentially fast as t approaches to the infinity.
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Quantum dynamical semigroup and its asymptotic behaviors
최베니 대한수학회 2004 대한수학회보 Vol.41 No.1
In this study we consider quantum dynamical semigroup with anormal faithful invariant state. A quantum dynamical semigroupalpha = {alpha _t }_{t geq 0} is a class of linear normalidentity-preserving mappings on a von Neumann algebra mathcal M with semigroup property and some positivity condition. Weinvestigate the asymptotic behaviors of the semigroup such asergodicity or mixing properties in terms of their eigenvaluesunder the assumption that the semigroup satisfies positivity. Thisextends the result of cite{Wa} which is obtained under theassumption that the semigroup satisfy 2-positivity.
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