Non-commutative central limits

Goderis, D; Verbeure, André; Vets, P

doi:10.1007/bf00341282

Cited by 113 publications

(155 citation statements)

References 14 publications

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“…In this case the relevant concept is the CCR algebra over the symplectic space (C self , ς), often called the fluctuation algebra [GVV1]. The mathematical structure of the fluctuation algebra is discussed in many places in the literature, see e.g.…”

Section: Theorem 14mentioning

confidence: 99%

“…The mathematical structure of the fluctuation algebra is discussed in many places in the literature, see e.g. [GVV1]- [GVV6] and [MSTV,BR2,Pe,OP,De2] for general results about CCR algebras. For notational and reference purposes we recall a few basic facts.…”

Section: Theorem 14mentioning

confidence: 99%

“…The papers [GVV1]- [GVV6] aim at more physical applications: a satisfactory algebra of fluctuations is constructed for space fluctuations of local observables in a quantum spin system with a tranlation-invariant state. That state does not have to be a product state; however, the ergodic assumptions on that state are so strong that no nontrivial application was found beyond the product case.…”

Section: Theorem 14mentioning

confidence: 99%

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Central Limit Theorem for Locally Interacting Fermi Gas

Jakšić

Pautrat

Pillet

2008

Commun. Math. Phys.

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We consider a locally interacting Fermi gas in its natural non-equilibrium steady state and prove the Quantum Central Limit Theorem (QCLT) for a large class of observables. A special case of our results concerns finitely many free Fermi gas reservoirs coupled by local interactions. The QCLT for flux observables, together with the Green-Kubo formulas and the Onsager reciprocity relations previously established [JOP4], complete the proof of the Fluctuation-Dissipation Theorem and the development of linear response theory for this class of models.

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Section: Theorem 14mentioning

confidence: 99%

Section: Theorem 14mentioning

confidence: 99%

Section: Theorem 14mentioning

confidence: 99%

See 1 more Smart Citation

Central Limit Theorem for Locally Interacting Fermi Gas

Jakšić

Pautrat

Pillet

2008

Commun. Math. Phys.

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“…where a(f A ), a + (f A ) are "smeared" bosonic anihilation and creation operators [32,33]. The limit is understand in the sense of convergence of all correlation functions, where for Bose fields we choose a "vacuum" as a reference state.…”

Section: B Models Of Reservoirsmentioning

confidence: 99%

“…The limit is understand in the sense of convergence of all correlation functions, where for Bose fields we choose a "vacuum" as a reference state. The detailed structure of the Fock space for bosonic field is discussed in [10,32,33] and is not relevant here. One should think about bosonic fields as quantum counterparts of classical Gaussian random fields and the "vacuum" can represent any quasi-free state of bosons as for example arbitrary Gibbs state for noninteracting Bose gas.…”

Section: B Models Of Reservoirsmentioning

confidence: 99%

Invitation to Quantum Dynamical Semigroups

Alicki

2002

Dynamics of Dissipation

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The theory of quantum dynamical semigroups within the mathematically rigorous framework of completely positive dynamical maps is reviewed. First, the axiomatic approach which deals with phenomenological constructions and general mathematical structures is discussed. Then basic derivation schemes of the constructive approach including singular coupling, weak coupling and low density limits are presented in their higly simplified versions. Two-level system coupled to a heat bath, damped harmonic oscillator, models of decoherence, quantum Brownian particle and Bloch-Boltzmann equations are used as illustrations of the general theory. Physical and mathematical limitations of the quantum open system theory, the validity of Markovian approximation and alternative approaches are discussed also.

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Dissipative dynamics of quantum fluctuations

2015

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One way to look for complex behaviours in many-body quantum systems is to let the number N of degrees of freedom become large and focus upon collective observables. Mean-field quantities scaling as 1/N tend to commute, whence complexity at the quantum level can only be inherited from complexity at the classical level. Instead, fluctuations of microscopic observables scale as 1/ √ N and exhibit collective Bosonic features, typical of a mesoscopic regime half-way between the quantum one at the microscopic level and the classical one at the level of macroscopic averages. Here, we consider the mesoscopic behaviour emerging from an infinite quantum spin chain undergoing a microscopic dissipative, irreversible dynamics and from global states without long-range correlations and invariant under lattice translations and dynamics. We show that, from the fluctuations of one site spin observables whose linear span is mapped into itself by the dynamics, there emerge bosonic operators obeying a mesoscopic dissipative dynamics mapping Gaussian states into Gaussian states. Instead of just depleting quantum correlations because of decoherence effects, these maps can generate entanglement at the collective, mesoscopic level, a phenomenon with no classical analogue that embodies a peculiar complex behaviour at the interface between micro and macro regimes.

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Non-commutative central limits

Cited by 113 publications

References 14 publications

Central Limit Theorem for Locally Interacting Fermi Gas

Central Limit Theorem for Locally Interacting Fermi Gas

Invitation to Quantum Dynamical Semigroups

Dissipative dynamics of quantum fluctuations

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