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Quantum Interacting Particle Systems : Lecture Notes of the Volterra-CIRM International School, Trento, Italy, 23-29 September 2000.

By: Contributor(s): Series: Qp-Pq: Quantum Probability and White Noise Analysis SerPublisher: Singapore : World Scientific Publishing Co Pte Ltd, 2002Copyright date: ©2002Description: 1 online resource (357 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9789812776853
Subject(s): Genre/Form: Additional physical formats: Print version:: Quantum Interacting Particle Systems : Lecture Notes of the Volterra-CIRM International School, Trento, Italy, 23-29 September 2000DDC classification:
  • 530.12
LOC classification:
  • QC174.7.P7Q33 2002
Online resources:
Contents:
Intro -- Contents -- Preface -- Chapter 1 Lectures on Quantum Interacting Particle Systems -- 1.0 Introduction -- 1.1 Basic ideas of the stochastic limit -- 1.1.1 Quantum dynamics and flows -- 1.1.2 The stochastic golden rule: general scheme -- 1.1.3 Discrete spectrum systems -- 1.1.4 Quantum fields and white noises -- 1.1.5 Dipole interaction Hamiltonians -- 1.1.6 The stochastic golden rule -- 1.1.7 The Langevin equation -- 1.1.8 The quantum Feynman-Kac formula: master equation -- 1.1.9 Subalgebras invariant under the generator -- 1.1.10 Structure of the invariant states -- 1.1.11 The Langevin equation: generic systems -- 1.1.12 Appendix: Two-level system and Boltzmannian (or free) statistics -- 1.1.13 Appendix: Structure of discrete spectrum quantum dynamical systems -- 1.1.14 Appendix: Spectral theory for Heisenberg evolutions -- 1.1.15 The Langevin equation for the density matrix -- 1.1.16 The master equation for reduced density matrix -- 1.1.17 Structure of the invariant states -- 1.1.18 Evolution of the diagonal and off-diagonal elements -- 1.1.19 Stationary states of the reduced evolution -- 1.1.20 Evolution of density matrix in the generic case -- 1.1.21 Decoherence: vanishing of off-diagonal terms -- 1.1.22 Discussion of decoherence -- 1.1.23 Classical detailed balance -- 1.1.24 A Lyapunov function for the reduced dynamics -- 1.1.25 Non-detailed balance stationary state for a system driven by master equation -- 1.1.26 Example of non-Gibbs detailed balance stationary state -- 1.1.27 The simplest non-generic master equation -- 1.1.28 Quantum detailed balance -- 1.1.29 Time reversal and invariant states -- 1.1.30 Quantum detailed balance in the sense of Frigerio-Gorini-Kossakowski-Verri -- 1.2 Equilibrium and nonequilibrium spin-boson models -- 1.2.1 The stochastic limit of the spin-boson model -- 1.2.2 The Langevin equation.
1.2.3 Evolution of the density matrix: master equation -- 1.2.4 Decoherence for one spin -- 1.2.5 Classical subprocesses and Abelian subalgebras -- 1.3 Dynamics of quantum spin systems -- 1.3.1 Introduction -- 1.3.2 A model describing electrical conductivity in lattice systems -- 1.3.3 The stochastic limit of the model -- 1.3.4 The Langevin equation -- 1.3.5 The open Ising model -- 1.3.6 The Langevin equation for quantum spin systems -- 1.3.7 Spin systems with finite range interaction -- 1.3.8 Quasigeneric spin systems -- 1.3.9 Quantum extension of the Glauber dynamics -- 1.3.10 Master equation: generic case -- 1.3.11 Evolution in the diagonal subalgebra: the classical Glauber dynamics -- 1.3.12 Dependence of decoherence on the volume -- 1.4 Structure of the Markov flows -- 1.4.1 Introduction -- 1.4.2 Markov flows -- 1.4.3 Evolutions associated to Markov flows -- 1.4.4 Semigroups associated to Markovian cocycles -- 1.4.5 The extended semigroup of a flow -- 1.4.6 Vector flows -- 1.4.7 Existence of infinite volume dynamics -- 1.5 Existence of infinite volume flows for quantum spin systems -- 1.5.1 Form of the generator -- 1.5.2 Matsui's Lemma -- 1.5.3 Quasigeneric case -- 1.5.4 Non-quasigeneric case -- 1.5.5 Thermodynamic limit -- 1.5.6 The homomorphic property for the flow -- 1.5.7 Appendix: Convergence of the iterated series in dimension 1 -- 1.6 Appendix: Basic notions on semigroups and dissipations -- 1.6.1 Semigroups in Banach spaces -- 1.6.2 Semigroups with dissipative generators -- 1.6.3 Limits of semigroups -- 1.6.4 Dissipative operators in C*-algebras -- 1.7 Bibliography -- Chapter 2 Lectures on the Qualitative Analysis of Quantum Markov Semigroups -- 2.1 Quantum Markov semigroups and master equations -- 2.1.1 Preliminaries -- 2.1.2 Examples of master equations -- 2.1.3 The minimal quantum dynamical semigroup.
2.2 The existence of stationary states -- 2.2.1 Tightness -- 2.2.2 A general result -- 2.2.3 Conditions on the generator -- 2.2.4 Applications -- 2.3 Faithful stationary states and irreducibility -- 2.3.1 The support of an invariant state -- 2.3.2 Subharmonic projections -- 2.3.3 Applications -- 2.4 The convergence towards the equilibrium -- 2.4.1 A result due to Frigerio and Verri -- 2.4.2 Quantum Markovian cocycles -- 2.4.3 Main results -- 2.4.4 Applications -- 2.5 Bibliography -- Chapter 3 Analysis of Classical and Quantum Interacting Particle Systems -- 3.0 Introduction -- 3.1 Construction and ergodicity of jump type dynamics - Lecture I -- 3.1.1 Introduction -- 3.1.2 Semiclassical dissipative dynamics at finite temperatures for quantum spin system on a lattice -- 3.1.3 Strong ergodicity -- 3.1.4 Construction of jump type dynamics -- 3.1.5 Regularity properties of the elementary Markov generators -- 3.1.6 Equivalence of quadratic forms and L2 ergodicity -- 3.2 Hypercontractivity and strong ergodicity for classical systems - Lecture II -- 3.2.1 Introduction -- 3.2.2 Hypercontractivity and logarithmic Sobolev inequality -- 3.2.3 Properties of the logarithmic Sobolev inequality -- 3.2.4 Strong ergodicity via hypercontractivity -- 3.2.5 Equivalence theorem -- 3.3 Logarithmic Sobolev inequality in non-commutative setting - Lecture III -- 3.3.1 Introduction -- 3.3.2 Non-commutative Lp spaces -- 3.3.3 Hypercontractive semigroup in non-commutative Lp spaces -- 3.3.4 Entropy and infinitesimal description of hypercontractivity -- 3.4 Particle structure and hypercontractivity - Lecture IV -- 3.4.1 Introduction -- 3.4.2 The general strategy -- 3.4.3 Gaussian bounds in product spaces and hypercontractivity -- 3.4.4 Hypercontractivity in non-commutative product spaces -- 3.5 Appendix I -- 3.6 Appendix II -- 3.7 Bibliography.
Summary: The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, …) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra, produces precisely those interacting particle systems studied in classical statistical mechanics. Moreover, in many interesting cases, the underlying classical process "drives" the quantum one, at least as far as ergodicity or convergence to equilibrium are concerned. Thus many deep results concerning classical systems can be directly applied to carry information on the corresponding quantum system. The thermodynamic limit itself is obtained thanks to a technique (the four-semigroup method, new even in the classical case) which reduces the infinitesimal structure of a stochastic flow to that of four semigroups canonically associated to it (Chap. 1). Simple and effective methods to analyze qualitatively the ergodic behavior of quantum Markov semigroups are discussed in Chap. 2. Powerful estimates used to control the infinite volume limit, ergodic behavior and the spectral gap (Gaussian, exponential and hypercontractive bounds, classical and quantum logarithmic Sobolev inequalities, …) are discussed in Chap. 3. Contents: Lectures on Quantum Interacting Particle Systems (L Accardi & S Kozyrev): Basic Ideas of the Stochastic Limit; Equilibrium and Nonequilibrium Spin-Boson Models; Dynamics of Quantum Spin Systems; Structure of the Markov Flows; Existence ofSummary: Infinite Volume Flows for Quantum Spin Systems; Appendix: Basic Notions on Semigroups and Dissipations; Lectures on the Qualitative Analysis of Quantum Markov Semigroups (F Fagnola & R Rebolledo): Quantum Markov Semigroups and Master Equations; The Existence of Stationary States; Faithful Stationary States and Irreducibility; The Convergence Towards the Equilibrium; Analysis of Classical and Quantum Interacting Particle Systems (B Zegarlinski): Construction and Ergodicity of Jump Type Dynamics - Lecture I; Hypercontractivity and Strong Ergodicity for Classical Systems - Lecture II; Logarithmic Sobolev Inequality in Non-Commutative Setting - Lecture III; Particle Structure and Hypercontractivity - Lecture IV. Readership: Researchers in probability & statistics, stochastic, mathematical physics, quantum physics.
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Intro -- Contents -- Preface -- Chapter 1 Lectures on Quantum Interacting Particle Systems -- 1.0 Introduction -- 1.1 Basic ideas of the stochastic limit -- 1.1.1 Quantum dynamics and flows -- 1.1.2 The stochastic golden rule: general scheme -- 1.1.3 Discrete spectrum systems -- 1.1.4 Quantum fields and white noises -- 1.1.5 Dipole interaction Hamiltonians -- 1.1.6 The stochastic golden rule -- 1.1.7 The Langevin equation -- 1.1.8 The quantum Feynman-Kac formula: master equation -- 1.1.9 Subalgebras invariant under the generator -- 1.1.10 Structure of the invariant states -- 1.1.11 The Langevin equation: generic systems -- 1.1.12 Appendix: Two-level system and Boltzmannian (or free) statistics -- 1.1.13 Appendix: Structure of discrete spectrum quantum dynamical systems -- 1.1.14 Appendix: Spectral theory for Heisenberg evolutions -- 1.1.15 The Langevin equation for the density matrix -- 1.1.16 The master equation for reduced density matrix -- 1.1.17 Structure of the invariant states -- 1.1.18 Evolution of the diagonal and off-diagonal elements -- 1.1.19 Stationary states of the reduced evolution -- 1.1.20 Evolution of density matrix in the generic case -- 1.1.21 Decoherence: vanishing of off-diagonal terms -- 1.1.22 Discussion of decoherence -- 1.1.23 Classical detailed balance -- 1.1.24 A Lyapunov function for the reduced dynamics -- 1.1.25 Non-detailed balance stationary state for a system driven by master equation -- 1.1.26 Example of non-Gibbs detailed balance stationary state -- 1.1.27 The simplest non-generic master equation -- 1.1.28 Quantum detailed balance -- 1.1.29 Time reversal and invariant states -- 1.1.30 Quantum detailed balance in the sense of Frigerio-Gorini-Kossakowski-Verri -- 1.2 Equilibrium and nonequilibrium spin-boson models -- 1.2.1 The stochastic limit of the spin-boson model -- 1.2.2 The Langevin equation.

1.2.3 Evolution of the density matrix: master equation -- 1.2.4 Decoherence for one spin -- 1.2.5 Classical subprocesses and Abelian subalgebras -- 1.3 Dynamics of quantum spin systems -- 1.3.1 Introduction -- 1.3.2 A model describing electrical conductivity in lattice systems -- 1.3.3 The stochastic limit of the model -- 1.3.4 The Langevin equation -- 1.3.5 The open Ising model -- 1.3.6 The Langevin equation for quantum spin systems -- 1.3.7 Spin systems with finite range interaction -- 1.3.8 Quasigeneric spin systems -- 1.3.9 Quantum extension of the Glauber dynamics -- 1.3.10 Master equation: generic case -- 1.3.11 Evolution in the diagonal subalgebra: the classical Glauber dynamics -- 1.3.12 Dependence of decoherence on the volume -- 1.4 Structure of the Markov flows -- 1.4.1 Introduction -- 1.4.2 Markov flows -- 1.4.3 Evolutions associated to Markov flows -- 1.4.4 Semigroups associated to Markovian cocycles -- 1.4.5 The extended semigroup of a flow -- 1.4.6 Vector flows -- 1.4.7 Existence of infinite volume dynamics -- 1.5 Existence of infinite volume flows for quantum spin systems -- 1.5.1 Form of the generator -- 1.5.2 Matsui's Lemma -- 1.5.3 Quasigeneric case -- 1.5.4 Non-quasigeneric case -- 1.5.5 Thermodynamic limit -- 1.5.6 The homomorphic property for the flow -- 1.5.7 Appendix: Convergence of the iterated series in dimension 1 -- 1.6 Appendix: Basic notions on semigroups and dissipations -- 1.6.1 Semigroups in Banach spaces -- 1.6.2 Semigroups with dissipative generators -- 1.6.3 Limits of semigroups -- 1.6.4 Dissipative operators in C*-algebras -- 1.7 Bibliography -- Chapter 2 Lectures on the Qualitative Analysis of Quantum Markov Semigroups -- 2.1 Quantum Markov semigroups and master equations -- 2.1.1 Preliminaries -- 2.1.2 Examples of master equations -- 2.1.3 The minimal quantum dynamical semigroup.

2.2 The existence of stationary states -- 2.2.1 Tightness -- 2.2.2 A general result -- 2.2.3 Conditions on the generator -- 2.2.4 Applications -- 2.3 Faithful stationary states and irreducibility -- 2.3.1 The support of an invariant state -- 2.3.2 Subharmonic projections -- 2.3.3 Applications -- 2.4 The convergence towards the equilibrium -- 2.4.1 A result due to Frigerio and Verri -- 2.4.2 Quantum Markovian cocycles -- 2.4.3 Main results -- 2.4.4 Applications -- 2.5 Bibliography -- Chapter 3 Analysis of Classical and Quantum Interacting Particle Systems -- 3.0 Introduction -- 3.1 Construction and ergodicity of jump type dynamics - Lecture I -- 3.1.1 Introduction -- 3.1.2 Semiclassical dissipative dynamics at finite temperatures for quantum spin system on a lattice -- 3.1.3 Strong ergodicity -- 3.1.4 Construction of jump type dynamics -- 3.1.5 Regularity properties of the elementary Markov generators -- 3.1.6 Equivalence of quadratic forms and L2 ergodicity -- 3.2 Hypercontractivity and strong ergodicity for classical systems - Lecture II -- 3.2.1 Introduction -- 3.2.2 Hypercontractivity and logarithmic Sobolev inequality -- 3.2.3 Properties of the logarithmic Sobolev inequality -- 3.2.4 Strong ergodicity via hypercontractivity -- 3.2.5 Equivalence theorem -- 3.3 Logarithmic Sobolev inequality in non-commutative setting - Lecture III -- 3.3.1 Introduction -- 3.3.2 Non-commutative Lp spaces -- 3.3.3 Hypercontractive semigroup in non-commutative Lp spaces -- 3.3.4 Entropy and infinitesimal description of hypercontractivity -- 3.4 Particle structure and hypercontractivity - Lecture IV -- 3.4.1 Introduction -- 3.4.2 The general strategy -- 3.4.3 Gaussian bounds in product spaces and hypercontractivity -- 3.4.4 Hypercontractivity in non-commutative product spaces -- 3.5 Appendix I -- 3.6 Appendix II -- 3.7 Bibliography.

The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, …) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra, produces precisely those interacting particle systems studied in classical statistical mechanics. Moreover, in many interesting cases, the underlying classical process "drives" the quantum one, at least as far as ergodicity or convergence to equilibrium are concerned. Thus many deep results concerning classical systems can be directly applied to carry information on the corresponding quantum system. The thermodynamic limit itself is obtained thanks to a technique (the four-semigroup method, new even in the classical case) which reduces the infinitesimal structure of a stochastic flow to that of four semigroups canonically associated to it (Chap. 1). Simple and effective methods to analyze qualitatively the ergodic behavior of quantum Markov semigroups are discussed in Chap. 2. Powerful estimates used to control the infinite volume limit, ergodic behavior and the spectral gap (Gaussian, exponential and hypercontractive bounds, classical and quantum logarithmic Sobolev inequalities, …) are discussed in Chap. 3. Contents: Lectures on Quantum Interacting Particle Systems (L Accardi & S Kozyrev): Basic Ideas of the Stochastic Limit; Equilibrium and Nonequilibrium Spin-Boson Models; Dynamics of Quantum Spin Systems; Structure of the Markov Flows; Existence of

Infinite Volume Flows for Quantum Spin Systems; Appendix: Basic Notions on Semigroups and Dissipations; Lectures on the Qualitative Analysis of Quantum Markov Semigroups (F Fagnola & R Rebolledo): Quantum Markov Semigroups and Master Equations; The Existence of Stationary States; Faithful Stationary States and Irreducibility; The Convergence Towards the Equilibrium; Analysis of Classical and Quantum Interacting Particle Systems (B Zegarlinski): Construction and Ergodicity of Jump Type Dynamics - Lecture I; Hypercontractivity and Strong Ergodicity for Classical Systems - Lecture II; Logarithmic Sobolev Inequality in Non-Commutative Setting - Lecture III; Particle Structure and Hypercontractivity - Lecture IV. Readership: Researchers in probability & statistics, stochastic, mathematical physics, quantum physics.

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