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37 results on '"Claude-Alain Pillet"'

1. A Detailed Fluctuation Theorem for Heat Fluxes in Harmonic Networks out of Thermal Equilibrium

Author: Mayssa Hammami, Mondher Damak, Claude-Alain Pillet, Université de Sfax - University of Sfax, Département de Mathematiques [Sfax], Faculté des Sciences de Sfax, Université de Sfax - University of Sfax-Université de Sfax - University of Sfax, CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017)
Subjects: Physics, Thermal equilibrium, Entropy Production, Heat Fluxes, Fluctuation theorem, Entropy production, Large Deviations, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Statistical and Nonlinear Physics, Mechanics, Mathematical Physics (math-ph), Fluctuation Relations, 01 natural sciences, 010305 fluids & plasmas, 0103 physical sciences, Harmonic, Large deviations theory, Nonequilibrium Steady State, Nonequilibrium steady state, 010306 general physics, Rate function, Harmonic oscillator, Mathematical Physics
Abstract: International audience; We continue the investigation, started in [J. Stat. Phys. 166, 926-1015 (2017)], of a network of harmonic oscillators driven out of thermal equilibrium by heat reservoirs. We study the statistics of the fluctuations of the heat fluxes flowing between the network and the reservoirs in the nonequilibrium steady state and in the large time limit. We prove a large deviation principle for these fluctuations and derive the fluctuation relation satisfied by the associated rate function.
Published: 2020
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2. Large deviations and mixing for dissipative PDEs with unbounded random kicks

Author: Claude-Alain Pillet, Vahagn Nersesyan, Vojkan Jaksic, Armen Shirikyan, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), and CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Dissipative PDE's, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, General Physics and Astronomy, Markov process, 01 natural sciences, Measure (mathematics), large deviations, symbols.namesake, Mathematics - Analysis of PDEs, Mixing (mathematics), Exponential stability, 0103 physical sciences, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 35Q30, 35Q56, 76D05, 60B12, 60F10, 0101 mathematics, Navier–Stokes system, Mathematical Physics, Mathematics, Ginzburg–Landau equation, Markov chain, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], unbounded kicks, Bounded function, symbols, occupation measures, Large deviations theory, 010307 mathematical physics, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract: We study the problem of exponential mixing and large deviations for discrete-time Markov processes associated with a class of random dynamical systems. Under some dissipativity and regularisation hypotheses for the underlying deterministic dynamics and a non-degeneracy condition for the driving random force, we discuss the existence and uniqueness of a stationary measure and its exponential stability in the Kantorovich-Wasserstein metric. We next turn to the large deviation principle and establish its validity for the occupation measures of the Markov processes in question. The obtained results extend those established earlier for the case of the bounded noise and can be applied to the 2D Navier-Stokes system in a bounded domain and to the complex Ginzburg-Landau equation., 63 pages
Published: 2018
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3. Large Deviations and Fluctuation Theorem for Selectively Decoupled Measures on Shift Spaces

Author: Noé Cuneo, Armen Shirikyan, Vojkan Jaksic, Claude-Alain Pillet, Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011), and Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY)
Subjects: [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Quantum measurement, Dynamical Systems (math.DS), Mathematical proof, 01 natural sciences, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, Statistical physics, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Mathematics - Dynamical Systems, Mathematical Physics, Condensed Matter - Statistical Mechanics, Mathematics, Statistical Mechanics (cond-mat.stat-mech), Entropy production, Fluctuation theorem, Probability (math.PR), 010102 general mathematics, selectively decoupled measures, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Decoupling (cosmology), Multifractal system, Invariant (physics), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Large deviations, one-sided shift, Large deviations theory, 010307 mathematical physics, fluctuation theorem, Mathematics - Probability, Ruelle–Lanford functions
Abstract: International audience; We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes. We also prove the LDP for the entropy production of pairs of such measures and derive the related Fluctuation Relation. The proofs are based on Ruelle-Lanford functions, and the exposition is essentially self-contained.
Published: 2017
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4. Energy statistics in open harmonic networks

Author: Tristan Benoist, Claude-Alain Pillet, Vojkan Jaksic, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), ANR-11-IDEX-0002,UNITI,Université Fédérale de Toulouse(2011), ANR-14-CE25-0003,StoQ,Méthodes Stochastiques en Mécanique Quantique(2014), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Physics, [PHYS.PHYS.PHYS-CLASS-PH]Physics [physics]/Physics [physics]/Classical Physics [physics.class-ph], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Classical Physics (physics.class-ph), FOS: Physical sciences, Statistical and Nonlinear Physics, Physics - Classical Physics, Mathematical Physics (math-ph), 01 natural sciences, Variance-gamma distribution, symbols.namesake, Stochastic dynamics, 0103 physical sciences, symbols, Large deviations theory, 010307 mathematical physics, Statistical physics, Energy statistics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 010306 general physics, Hamiltonian (quantum mechanics), Rate function, Mathematical Physics
Abstract: We relate the large time asymptotics of the energy statistics in open harmonic networks to the variance-gamma distribution and prove a full Large Deviation Principle. We consider both Hamiltonian and stochastic dynamics, the later case including electronic RC networks. We compare our theoretical predictions with the experimental data obtained by Ciliberto et al. [Phys. Rev. Lett. 110, 180601 (2013)]., Comment: 15 pages, 2 figures
Published: 2017
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5. A mathematical account of the NEGF formalism

Author: Valeriu Moldoveanu, Horia D. Cornean, Claude-Alain Pillet, Department of Mathematical Sciences [Aalborg], Aalborg University [Denmark] (AAU), National Institute of Materials Physics (NIMP), Romanian Academy of Sciences, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, and Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Nuclear and High Energy Physics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 01 natural sciences, Many body, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mesoscale and Nanoscale Physics (cond-mat.mes-hall), 0103 physical sciences, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 010306 general physics, quantum transport, Mathematical Physics, Condensed Matter - Statistical Mechanics, Mathematical physics, Mathematics, nonequilibrium statistical mechanics, Quantum Physics, Statistical Mechanics (cond-mat.stat-mech), Condensed Matter - Mesoscale and Nanoscale Physics, Analytic continuation, 010102 general mathematics, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Formalism (philosophy of mathematics), nonequilibrium Green's functions, Keldysh formalism, Quantum Physics (quant-ph)
Abstract: The main goal of this paper is to put on solid mathematical grounds the so-called Non-Equilibrium Green's Function (NEGF) transport formalism for open systems. In particular, we derive the Jauho-Meir-Wingreen formula for the time-dependent current through an interacting sample coupled to non-interacting leads. Our proof is non-perturbative and uses neither complex-time Keldysh contours, nor Langreth rules of 'analytic continuation'. We also discuss other technical identities (Langreth, Keldysh) involving various many body Green's functions. Finally, we study the Dyson equation for the advanced/retarded interacting Green's function and we rigorously construct its (irreducible) self-energy, using the theory of Volterra operators., Annales Henri Poincar\'e 2017
Published: 2017
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6. Full statistics of erasure processes: Isothermal adiabatic theory and a statistical Landauer principle

Author: Tristan Benoist, Martin Fraas, Vojkan Jaksic, Claude-Alain Pillet, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), University of Munich, Ludwig-Maximilians-Universität München (LMU), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Quantum Physics, Statistical Mechanics (cond-mat.stat-mech), quantum information, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Mathematical Physics (math-ph), [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Quantum Physics (quant-ph), Mathematical Physics, Condensed Matter - Statistical Mechanics, quantum transport, nonequilibrium statistical mechanics
Abstract: We study driven finite quantum systems in contact with a thermal reservoir in the regime in which the system changes slowly in comparison to the equilibration time. The associated isothermal adiabatic theorem allows us to control the full statistics of energy transfers in quasi-static processes. Within this approach, we extend Landauer's Principle on the energetic cost of erasure processes to the level of the full statistics and elucidate the nature of the fluctuations breaking Landauer's bound., Comment: 24 pages, 4 figures; In the new version, Section 4 contains an extended discussion of the violation of Landauer's bound
Published: 2017

7. On entropy production of repeated quantum measurements I. General theory

Author: Claude-Alain Pillet, Tristan Benoist, Vojkan Jaksic, Yan Pautrat, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), ANR-12-IS01-0001,RMTQIT,Techniques de matrices aléatoires en théorie quantique de l'information(2012), ANR-14-CE25-0003,StoQ,Méthodes Stochastiques en Mécanique Quantique(2014), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Complex system, FOS: Physical sciences, Dynamical Systems (math.DS), 01 natural sciences, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Arrow of time, 0103 physical sciences, FOS: Mathematics, Statistical physics, 0101 mathematics, Mathematics - Dynamical Systems, 010306 general physics, Computer Science::Data Structures and Algorithms, Quantum, Mathematical Physics, Statistical hypothesis testing, Physics, Quantum Physics, Entropy production, Fluctuation theorem, 010102 general mathematics, Probability (math.PR), Statistical and Nonlinear Physics, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], General theory, Quantum Physics (quant-ph), Rate function, Mathematics - Probability
Abstract: International audience; We study entropy production (EP) in processes involving repeated quantum measurements of finite quantum systems. Adopting a dynamical system approach, we develop a thermodynamic formalism for the EP and study fine aspects of irreversibility related to the hypothesis testing of the arrow of time. Under a suitable chaoticity assumption, we establish a Large Deviation Principle and a Fluctuation Theorem for the EP.
Published: 2016
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8. Crystalline conductance and absolutely continuous spectrum of 1D samples

Author: Claude-Alain Pillet, Laurent Bruneau, Vojkan Jaksic, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Einstein Institute of Mathematics, The Hebrew University of Jerusalem (HUJ), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Nonequilibrium statistical mechanics, Materials science, Spectral theory, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Mathematics::Analysis of PDEs, FOS: Physical sciences, 01 natural sciences, Mathematics - Spectral Theory, Quantum transport, 0103 physical sciences, FOS: Mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Spectral Theory (math.SP), quantum transport, Mathematical Physics, nonequilibrium statistical mechanics, scattering theory, Quantum Physics, Condensed matter physics, 010102 general mathematics, Spectrum (functional analysis), Conductance, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), spectral theory, Limiting, Absolute continuity, Mathematics::Spectral Theory, Condensed Matter::Mesoscopic Systems and Quantum Hall Effect, 010307 mathematical physics, Scattering theory, Quantum Physics (quant-ph), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: International audience; We characterize the absolutely continuous spectrumof half-line one-dimensional Schrödinger operators in terms of thelimiting behavior of the Crystaline Landauer-Büttiker conductance of the associated finite samples.
Published: 2015
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9. Energy conservation, counting statistics, and return to equilibrium

Author: Claude-Alain Pillet, Annalisa Panati, Jane Panangaden, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
Subjects: return to equilibrium, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 01 natural sciences, symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Statistics, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Quantum, Real line, First law of thermodynamics, Mathematical Physics, quantum transport, Mathematics, Probability measure, nonequilibrium statistical mechanics, Thermal equilibrium, Quantum Physics, Thermal reservoir, 010102 general mathematics, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Operator algebra, counting statistics, symbols, 010307 mathematical physics, Hamiltonian (quantum mechanics), Quantum Physics (quant-ph)
Abstract: We study a microscopic Hamiltonian model describing an N-level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at temperature T. Assuming that the coupled system S+R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S->R and R->S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted $P_{S,\lambda,t}$, concentrated on the set of energy differences $\sigma(H_S)-\sigma(H_S)$ ($\sigma(H_S)$ is the spectrum of the Hamiltonian of S, $t$ is the length of the time interval during which the measurement of the energy transfer is performed, and $\lambda$ is the strength of the interaction between S and R). The second FCS, $P_{R,\lambda,t}$, describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit $t\rightarrow\infty$ of these two measures followed by the weak coupling limit $\lambda\rightarrow 0$ and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of $P_{R,\lambda,t}$ by quantum transfer operators.
Published: 2015
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10. On the steady state correlation functions of open interacting systems

Author: Claude-Alain Pillet, Horia D. Cornean, Valeriu Moldoveanu, Institut for Matematiske Fag, Aalborg university (MATH. DEPT., AALBORG UNIVERSITY), Aalborg University [Denmark] (AAU), National Institute of Materials Physics (NIMP), Romanian Academy of Sciences, Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
Subjects: Steady state (electronics), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Complex system, Hartree–Fock method, FOS: Physical sciences, 01 natural sciences, entropy production, nonequilibrium steady state, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mesoscale and Nanoscale Physics (cond-mat.mes-hall), 0103 physical sciences, Statistical physics, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 010306 general physics, Condensed Matter - Statistical Mechanics, Mathematical Physics, quantum transport, nonequilibrium statistical mechanics, Physics, Quantum Physics, Mesoscopic physics, Hartree-Fock approximation, Statistical Mechanics (cond-mat.stat-mech), Condensed Matter - Mesoscale and Nanoscale Physics, Entropy production, 010102 general mathematics, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), State (functional analysis), Fermion, Term (time), Green-Keldysh functions, interacting fermions, Quantum Physics (quant-ph)
Abstract: International audience; We address the existence of steady state Green-Keldysh correlation functions of interacting fermions in mesoscopic systems for both the partitioning and partition-free scenarios. Under some spectral assumptions on the non-interacting model and for sufficiently small interaction strength, we show that the system evolves to a NESS which does not depend on the profile of the time-dependent coupling strength/bias. For the partitioned setting we also show that the steady state is independent of the initial state of the inner sample. Closed formulae for the NESS two-point correlation functions (Green-Keldysh functions), in the form of a convergent expansion, are derived. In the partitioning approach, we show that the 0th order term in the interaction strength of the charge current leads to the Landauer-Büttiker formula, while the 1st order correction contains the mean-field (Hartree-Fock) results.
Published: 2014
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11. Entropic fluctuations in XY chains and reflectionless Jacobi matrices

Author: Vojkan Jaksic, Claude-Alain Pillet, Benjamin Landon, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
Subjects: Nuclear and High Energy Physics, Spectral theory, Gallavotti-Cohen fluctuation theorem, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Lambda, entropy production, 01 natural sciences, Mathematics - Spectral Theory, symbols.namesake, Open quantum system, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 0101 mathematics, 010306 general physics, Spectral Theory (math.SP), Mathematical Physics, nonequilibrium statistical mechanics, Mathematical physics, Thermal equilibrium, Physics, Quantum Physics, Evans-Searless fluctuation relation, 1D discrete Schrödinger operators, Entropy production, 010102 general mathematics, Sigma, Statistical and Nonlinear Physics, Observable, Mathematical Physics (math-ph), reflectionless Jacobi matrices, XY spin chain, symbols, Quantum Physics (quant-ph), Hamiltonian (quantum mechanics)
Abstract: We study entropic functionals/fluctuations of the XY chain with Hamiltonian $$\begin{array}{ll} \frac{1}{2} \sum\limits_{x \in \mathbb{Z}}J_x( \sigma_x^{(1)} \sigma_{x+1}^{(1)} +\sigma_x^{(2)} \sigma_{x+1}^{(2)}) + \lambda_x \sigma_x^{(3)}\end{array}$$ where initially the left (x ≤ 0)/right (x > 0) part of the chain is in thermal equilibrium at inverse temperature β l /β r . The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans–Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti–Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) of the energy/entropy flux associated with a repeated measurement protocol, the variational entropic functional (VAR) that arises as the quantization of the variational characterization of the classical Evans–Searles functional and a natural class of entropic functionals that interpolate between FCS and VAR. We compute these functionals in closed form in terms of the scattering data of the Jacobi matrix hu x = J x u x+1 + λ x u x + J x−1 u x−1 canonically associated with the XY chain. We show that all these functionals are identical if and only if h is reflectionless (we call this phenomenon entropic identity). If h is not reflectionless, then the ES and GC functionals remain equal but differ from the FCS, VAR and interpolating functionals. Furthermore, in the non-reflectionless case, the ES/GC functional does not vanish at α = 1 (i.e., the Kawasaki identity fails) and does not have the celebrated α ↔ 1 − α symmetry. The FCS, VAR and interpolating functionals always have this symmetry. In the Schrodinger case, where J x = J for all x, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schrodinger operators.
Published: 2012
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12. Entropic Functionals in Quantum Statistical Mechanics

Author: Claude-Alain Pillet, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Arne Jensen
Subjects: Physics, Nonequilibrium statistical mechanics, Quantitative Biology::Biomolecules, 010102 general mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], nonequilibrium steady states, 16. Peace & justice, 01 natural sciences, 3. Good health, Quantum transport, Classical mechanics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 010306 general physics, Quantum statistical mechanics, Quantum, quantum transport, entropic fluctuations, nonequilibrium statistical mechanics
Abstract: Published in the proceedings of the XVIIth International Congress on Mathematical Physics, Aalborg, Denmark, 6 - 11 August 2012. World Scientific, 2013. DOI 10.1142/9789814449243_0024; International audience; We describe quantum entropic functionals and outline a research program dealing with entropic fluctuations in non-equilibrium quantum statistical mechanics.
Published: 2012
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13. Temperature and Voltage Probes Far from Equilibrium

Author: Philippe Jacquet, Claude-Alain Pillet, Department of Physics, Kwansei Gakuin University, Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Nonequilibrium statistical mechanics, Iterative method, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 02 engineering and technology, 01 natural sciences, nonequilibrium steady state, Open quantum system, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mesoscale and Nanoscale Physics (cond-mat.mes-hall), 0103 physical sciences, Statistical physics, Autocatalytic reaction, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 010306 general physics, [PHYS.COND.CM-MSQHE]Physics [physics]/Condensed Matter [cond-mat]/Mesoscopic Systems and Quantum Hall Effect [cond-mat.mes-hall], Condensed Matter - Statistical Mechanics, Mathematical Physics, nonequilibrium statistical mechanics, Physics, Quantum Physics, Statistical Mechanics (cond-mat.stat-mech), Condensed Matter - Mesoscale and Nanoscale Physics, Landauer-Büttiker formalism, Mathematical Physics (math-ph), Condensed Matter::Mesoscopic Systems and Quantum Hall Effect, 021001 nanoscience & nanotechnology, Condensed Matter Physics, Electronic, Optical and Magnetic Materials, Nonequilibrium steady state, Quantum Physics (quant-ph), 0210 nano-technology, Voltage
Abstract: We consider an open system of non-interacting electrons consisting of a small sample connected to several reservoirs and temperature or voltage probes. We study the non-linear system of equations that determines the probe parameters. We show that it has a unique solution, which can be computed with a fast converging iterative algorithm. We illustrate our method with two well-known models: the three-terminal system and the open Aharovov-Bohm interferometer.
Published: 2011
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14. Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics

Author: Robert Seiringer, Claude-Alain Pillet, Vojkan Jaksic, Yoshiko Ogata, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Department of Mathematical Sciences, The University of Tokyo (UTokyo), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Pillet, Claude-Alain, and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
Subjects: FOS: Computer and information sciences, Computer Science - Information Theory, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 01 natural sciences, entropy production, quantum hypothesis testing, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Arrow of time, 0103 physical sciences, Statistical physics, 0101 mathematics, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], 010306 general physics, Quantum statistical mechanics, Quantum, Mathematical Physics, Statistical hypothesis testing, Mathematics, nonequilibrium statistical mechanics, Quantum Physics, Information Theory (cs.IT), 010102 general mathematics, Statistical and Nonlinear Physics, Statistical mechanics, Mathematical Physics (math-ph), [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], 16. Peace & justice, Mathematical theory, Large deviations theory, Quantum Physics (quant-ph), Rate function
Abstract: We extend the mathematical theory of quantum hypothesis testing to the general $W^*$-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time., Comment: 60 pages
Published: 2011

15. Entropic Fluctuations in Statistical Mechanics I. Classical Dynamical Systems

Author: Luc Rey-Bellet, Claude-Alain Pillet, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [University of Massachusetts], University of Massachusetts System (UMASS), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: [PHYS.PHYS.PHYS-CLASS-PH]Physics [physics]/Physics [physics]/Classical Physics [physics.class-ph], Dynamical systems theory, 82C05, 37A60, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], General Physics and Astronomy, FOS: Physical sciences, Dynamical Systems (math.DS), 01 natural sciences, entropy production, Hamiltonian system, linear response theory, Systems theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, Statistical physics, 0101 mathematics, Mathematics - Dynamical Systems, 010306 general physics, Quantum statistical mechanics, fluctuation theorems, Mathematical Physics, Condensed Matter - Statistical Mechanics, Mathematics, nonequilibrium statistical mechanics, Statistical Mechanics (cond-mat.stat-mech), Entropy (statistical thermodynamics), Applied Mathematics, 010102 general mathematics, Probability (math.PR), Statistical and Nonlinear Physics, Statistical mechanics, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Metric space, Mathematical structure, Mathematics - Probability
Abstract: Within the abstract framework of dynamical system theory we describe a general approach to the Transient (or Evans-Searles) and Steady State (or Gallavotti-Cohen) Fluctuation Theorems of non-equilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. Besides its conceptual simplicity, another advantage of our approach is its natural extension to quantum statistical mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms., Comment: 72 pages, revised version 12/10/2010, to be published in Nonlinearity
Published: 2011
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16. Thermal relaxation of a QED cavity

Author: Laurent Bruneau, Claude-Alain Pillet, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: Electromagnetic field, Rabi cycle, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Markov process, Thermal relaxation, 01 natural sciences, symbols.namesake, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, Metastability, 0103 physical sciences, one-atom maser, 0101 mathematics, cavity QED, 010306 general physics, Mathematical Physics, completely positive maps, Physics, Thermal equilibrium, Canonical ensemble, Rabi oscillations, 010102 general mathematics, Statistical and Nonlinear Physics, Invariant (physics), Quantum evolution, repeated interactions, Quantum electrodynamics, symbols
Abstract: International audience; We study repeated interactions of the quantized electromagnetic field in a cavity with single two-levels atoms. Using the Markovian nature of the resulting quantum evolution we study its large time asymptotics. We show that, whenever the atoms are distributed according to the canonical ensemble at temperature T>0 and some generic non-degeneracy condition is satisfied, the cavity field relaxes towards some invariant state. Under some more stringent non-resonance condition, this invariant state is thermal equilibrium at some renormalized temperature T*. Our result is non-perturbative in the strength of the atom-field coupling. The relaxation process is slow (non-exponential) due to the presence of infinitely many metastable states of the cavity field.
Published: 2009
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17. Central limit theorem for locally interacting Fermi gas

Author: Vojkan Jaksic, Yan Pautrat, Claude-Alain Pillet, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Pillet, Claude-Alain, Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: fluctuations, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Complex system, 01 natural sciences, linear response theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, 0103 physical sciences, NESS, 0101 mathematics, Special case, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Quantum, Mathematical Physics, Mathematics, Mathematical physics, Central limit theorem, 010102 general mathematics, nonequilibrium steady states, Statistical and Nonlinear Physics, Observable, [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], Reciprocity (electromagnetism), fluctuation-dissipation, 010307 mathematical physics, Fermi liquid theory, open system, Fermi gas, quantum nonequilibrium statistical mechanics
Abstract: International audience; 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 (mp_arc:07-1), complete the proof of the Fluctuation-Dissipation Theorem and the development of linear response theory for this class of models.
Published: 2009

18. Transport properties of quasi-free Fermions

Author: Yan Pautrat, Claude-Alain Pillet, Walter H. Aschbacher, Vojkan Jaksic, Zentrum Mathematik M5, Technische Universitaet Muenchen, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Non-equilibrium thermodynamics, 05.30 05.70, Electron, 01 natural sciences, entropy production, linear response theory, Kubo formula, Quantum mechanics, 0103 physical sciences, NESS, 0101 mathematics, steady-state, 010306 general physics, Quantum statistical mechanics, Mathematical Physics, nonequilibrium statistical mechanics, Physics, Steady state, Scattering, Entropy production, 010102 general mathematics, Statistical and Nonlinear Physics, Fermion, Condensed Matter::Mesoscopic Systems and Quantum Hall Effect, free Fermi gas, Nonlinear Sciences::Chaotic Dynamics, Onsager reciprocity relations
Abstract: International audience; Using the scattering approach to the construction of Non-Equilibrium Steady States proposed by Ruelle we study the transport properties of systems of independent electrons. We show that Landauer-Büttiker and Green-Kubo formulas hold under very general conditions.
Published: 2007
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19. The Green-Kubo formula for the spin-fermion system

Author: Yoshiko Ogata, Claude-Alain Pillet, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences, The University of Tokyo (UTokyo), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Inverse, 01 natural sciences, linear response, spin-fermion, Kubo formula, Quantum mechanics, 0103 physical sciences, NESS, Quantum system, steady state, 0101 mathematics, Mathematical Physics, nonequilibrium statistical mechanics, Thermal equilibrium, Physics, Entropy production, 010102 general mathematics, Statistical and Nonlinear Physics, Fermion, Green–Kubo relations, Onsager reciprocity relations, open systems, 010307 mathematical physics, Fermi gas
Abstract: The spin-fermion model describes a two level quantum system $$\mathcal{S}$$ (spin 1/2) coupled to finitely many free Fermi gas reservoirs $$\mathcal{R}_{j}$$ which are in thermal equilibrium at inverse temperatures β j . We consider non-equilibrium initial conditions where not all β j are the same. It is known that, at small coupling, the combined system $$\mathcal {S} + \sum_j \mathcal {R}_j$$ has a unique non-equilibrium steady state (NESS) characterized by strictly sitive entropy production. In this paper we study linear response in this NESS and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials.
Published: 2006
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20. Mathematical Theory of the Wigner-Weisskopf Atom

Author: Claude-Alain Pillet, Eugene Kritchevski, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), J. Derezinski, H. Siedentop, Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Spectral theory, rank one perturbations, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Non-equilibrium thermodynamics, 01 natural sciences, linear response, Open quantum system, Friedrichs model, Kubo formula, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, Atom (measure theory), 0103 physical sciences, NESS, steady state, 0101 mathematics, nonequilibrium statistical mechanics, Physics, Continuum (measurement), Entropy production, 010102 general mathematics, spectral theory, Second quantization, Mathematical theory, Onsager reciprocity relations, 010307 mathematical physics, Wigner-Weisskopf atom
Abstract: These notes provide an introduction to the spectral theory of the Friedrichs model. This model is used in quantum physics to describe the coupling of a discrete set of energy levels to a continuum of states. We restrict ourselves to the simplest possible case of a single energy level coupled to a continuum, the so-called Wigner-Weisskopf atom. We discuss both, perturbative and non-perturbative aspects. We also consider the fermionic second quantization of this model and discuss its nonequilibrium thermodynamics: steady states, steady currents and entropy production.
Published: 2006
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21. The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics

Author: Claude-Alain Pillet, Yoshiko Ogata, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences, The University of Tokyo (UTokyo), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
Subjects: 010102 general mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Statistical and Nonlinear Physics, Statistical mechanics, 16. Peace & justice, 01 natural sciences, Green–Kubo relations, linear response, Open quantum system, Onsager reciprocity relations, Kubo formula, Reciprocity (electromagnetism), 0103 physical sciences, NESS, Ergodic theory, steady state, 010307 mathematical physics, Onsager reciprocal relations, 0101 mathematics, Quantum statistical mechanics, Mathematical Physics, Mathematical physics, Mathematics, nonequilibrium statistical mechanics
Abstract: International audience; We study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Our derivation is axiomatic and the key assumptions concern ergodic properties of non-equilibrium steady states.
Published: 2006
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22. Perturbation theory of W*-dynamics, Liouvilleans and KMS states

Author: Jan Dereziński, Claude-Alain Pillet, Vojkan Jaksic, Department of Mathematical Methods in Physics [Warsaw] (KMMF), Faculty of Physics [Warsaw] (FUW), University of Warsaw (UW)-University of Warsaw (UW), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: Large class, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], W-algebra, Perturbation (astronomy), 01 natural sciences, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Quantum statistical mechanics, Mathematical Physics, Mathematical physics, Mathematics, perturbation theory, Mathematics::Operator Algebras, 010102 general mathematics, Mathematical analysis, Statistical and Nonlinear Physics, W*-dynamical systems, [PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph], quantum statistical mechanics, KMS states, Bounded function, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 010307 mathematical physics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: Given a W*-algebra [Formula: see text] with a W*-dynamics τ, we prove the existence of the perturbed W*-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If τ has a β-KMS state, and the perturbation satisfies some mild assumptions related to the Golden–Thompson inequality, we prove the existence of a β-KMS state for the perturbed W*-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.
Published: 2003
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23. A note on the entropy production formula

Author: Vojkan Jakšić, Claude-Alain Pillet, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, and Pillet, Claude-Alain
Subjects: Mathematics::Operator Algebras, 010102 general mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], relative entropy of states, [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], 16. Peace & justice, Physics::Classical Physics, 01 natural sciences, entropy production, Araki perturbation theory of KMS states, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, NESS, open systems, Computer Science::Programming Languages, steady state, 010307 mathematical physics, 0101 mathematics, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], nonequilibrium statistical mechanics
Abstract: International audience; We give an elementary derivation of the entropy production formula of [http://hal.archives-ouvertes.fr/hal-00005457] based on Araki Perturbation Theory of KMS states. Using this derivation we show that the entropy production of any normal, stationary state is zero.
Published: 2003

24. Non-Equilibrium Steady States of Finite Quantum Systems Coupled to Thermal Reservoirs

Author: Vojkan Jaksic, Claude-Alain Pillet, CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN), Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, entropy production, nonequilibrium steady state, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Quantum mechanics, 0103 physical sciences, Quantum system, 0101 mathematics, Quantum statistical mechanics, Quantum, Mathematical Physics, Thermal equilibrium, Physics, Quantum discord, Entropy production, 010102 general mathematics, Statistical and Nonlinear Physics, Statistical mechanics, [PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph], quantum statistical mechanics, Liouvillean, Quantum process, 010307 mathematical physics, open system, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: We study the non-equilibrium statistical mechanics of a 2-level quantum system, S, coupled to two independent free Fermi reservoirs R1, R2, which are in thermal equilibrium at inverse temperatures beta1 , beta2. We prove that, at small coupling, the combined quantum system S+R1+R2 has a unique non-equilibrium steady state (NESS) and that the approach to this NESS is exponentially fast. We show that the entropy production of the coupled system is strictly positive and relate this entropy production to the heat fluxes through the system. A part of our argument is general and deals with spectral theory of NESS. In the abstract setting of algebraic quantum statistical mechanics we introduce the new concept of C-Liouvillean, L, and relate the NESS to zero resonance eigenfunctions of L*. In the specific model S+R1+R2 we study the resonances of L* using the complex deformation technique developed previously by the authors.
Published: 2002
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25. On Entropy Production in Quantum Statistical Mechanics

Author: Vojkan Jaksic, Claude-Alain Pillet, CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: Physics, Entropy production, H-theorem, 010102 general mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Maximum entropy thermodynamics, Statistical and Nonlinear Physics, Entropy in thermodynamics and information theory, 01 natural sciences, Quantum relative entropy, Generalized relative entropy, nonequilibrium steady state, Entropy (classical thermodynamics), quantum statistical mechanics, 0103 physical sciences, C-star dynamical system, Statistical physics, 0101 mathematics, 010306 general physics, Mathematical Physics, Joint quantum entropy
Abstract: We propose a definition of entropy production in the framework of algebraic quantum statistical mechanics. We relate our definition to heat flows through the system. We also prove that entropy production is non-negative in natural non-equilibrium steady states.
Published: 2001
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26. Ergodic properties of classical dissipative systems I

Author: Claude-Alain Pillet, Vojkan Jaksic, Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: Dynamical systems theory, General Mathematics, return to equilibrium, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 01 natural sciences, Hamiltonian system, Langevin equation, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, 0101 mathematics, 010306 general physics, Brownian motion, Mathematics, ergodic theory, Thermal reservoir, Stochastic process, 010102 general mathematics, Mathematical analysis, Ornstein–Uhlenbeck process, thermal relaxation, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Dissipative system, open systems
Abstract: We consider a class of models in which a Hamiltonian system A, with a finite number of degrees of freedom, is brought into contact with an infinite heat reservoir B. We develop the formalism required to describe these models near thermal equilibrium. Using a combination of abstract spectral techniques and harmonic analysis we investigate the singular spectrum of the Liouvillean L of the coupled system A+B. We provide a natural set of conditions which ensure that the spectrum of L is purely absolutely continuous except for a simple eigenvalue at zero. It then follows from the spectral theory of dynamical systems (Koopmanism) that the system A+B is strongly mixing. From a probabilistic point of view, we study a new class of random processes on finite dimensional manifolds: non-Markovian Ornstein-Uhlenbeck processes. The paths of such a process are solutions of a random integro-differential equation with Gaussian noise which is a natural generalization of the well known Langevin equation. In this context we establish that, under appropriate conditions, the OU process is strongly mixing even when the Langevin equation has memory and is driven by non-white noise.
Published: 1998
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27. Spectral theory of thermal relaxation

Author: Vojkan Jaksic, Claude-Alain Pillet, Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: Canonical ensemble, Physics, Thermal equilibrium, Density matrix, Statistical ensemble, 010102 general mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Statistical and Nonlinear Physics, Statistical mechanics, 01 natural sciences, [PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph], 010305 fluids & plasmas, Microcanonical ensemble, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Open statistical ensemble, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Statistical physics, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Quantum statistical mechanics, Mathematical Physics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: International audience; We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system S, with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperature T. The zeroth law of thermodynamics postulates that, independently of its initial configuration, the system S approaches a unique stationary state as t-->infinity. By definition, this limiting state is the equilibrium state of S at temperature T. Statistical mechanics further identifies this state with the Gibbs canonical ensemble associated with S. For simple models we prove that the above picture is correct, provided the equilibrium state of the environment R is itself given by its canonical ensemble. In the quantum case we also obtain an exact formula for the thermal relaxation time.
Published: 1997
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28. On a model for quantum friction. II. Fermi's golden rule and dynamics at positive temperature

Author: Vojkan Jaksic, Claude-Alain Pillet, Pillet, Claude-Alain, Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: [PHYS.PHYS.PHYS-GEN-PH] Physics [physics]/Physics [physics]/General Physics [physics.gen-ph], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], return to equilibrium, 01 natural sciences, symbols.namesake, [PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, 0103 physical sciences, Master equation, [MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP], Fermi's golden rule, 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Quantum statistical mechanics, Mathematical Physics, Boson, Physics, 010102 general mathematics, Statistical and Nonlinear Physics, [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], [PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph], quantum statistical mechanics, Quantum harmonic oscillator, symbols, Dissipative system, spin boson, 010307 mathematical physics, Ground state, Hamiltonian (quantum mechanics), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: Euclid Identifier: euclid.cmp/1104286117; We investigate the dynamics of an TV-level system linearly coupled to a field of mass-less bosons at positive temperature. Using complex deformation techniques, we develop time-dependent perturbation theory and study spectral properties of the total Hamiltonian. We also calculate the lifetime of resonances to second order in the coupling.
Published: 1996

29. Spectral duality for planar billiards

Author: Jean-Pierre Eckmann, Claude-Alain Pillet, Département de Physique Théorique, Université de Genève = University of Geneva (UNIGE), Section de mathématiques [Genève], Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), University of Geneva [Switzerland], Université de Genève (UNIGE), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, and Pillet, Claude-Alain
Subjects: quantum billiard, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Duality (optimization), Boundary (topology), ddc:500.2, 81U20, 01 natural sciences, Combinatorics, Planar, Mathematics::K-Theory and Homology, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 47F05, ddc:510, 0101 mathematics, 35P25, Mathematical Physics, Eigenvalues and eigenvectors, [PHYS.COND.CM-MSQHE]Physics [physics]/Condensed Matter [cond-mat]/Mesoscopic Systems and Quantum Hall Effect [cond-mat.mes-hall], Complement (set theory), Physics, exterior scattering, Mathematics::Operator Algebras, 010102 general mathematics, Statistical and Nonlinear Physics, spectral duality, [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], Mathematics::Spectral Theory, Nonlinear Sciences - Chaotic Dynamics, 58G25, [PHYS.COND.CM-MSQHE] Physics [physics]/Condensed Matter [cond-mat]/Mesoscopic Systems and Quantum Hall Effect [cond-mat.mes-hall], 010101 applied mathematics, [NLIN.NLIN-CD] Nonlinear Sciences [physics]/Chaotic Dynamics [nlin.CD], 81Q10, Bounded function, [NLIN.NLIN-CD]Nonlinear Sciences [physics]/Chaotic Dynamics [nlin.CD], Piecewise, Chaotic Dynamics (nlin.CD), Energy (signal processing), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: For a bounded open domain $\Omega$ with connected complement in ${\bf R}^2$ and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\Delta_\Omega$ on $\Omega$ and the S-matrix on the complement $\Omega^c$. We show that the on-shell S-matrices ${\bf S}_k$ have eigenvalues converging to 1 as $k\uparrow k_0$ exactly when $-\Delta_\Omega$ has an eigenvalue at energy $k_0^2$. This includes multiplicities, and proves a weak form of ``transparency'' at $k=k_0$. We also show that stronger forms of transparency, such as ${\bf S}_{k_0}$ having an eigenvalue 1 are not expected to hold in general., Comment: 33 pages, Postscript, A4
Published: 1995

30. Selection of length distributions in living polymers

Author: Claude-Alain Pillet, A. C. Maggs, David Mukamel, Laboratoire de Physico-Chimie Théorique (LPCT), Ecole Superieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Complex Systems, Weizmann Institute of Science [Rehovot, Israël], Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: chemistry.chemical_classification, Steady state, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Thermodynamics, Polymer, 01 natural sciences, Micelle, [PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph], 010305 fluids & plasmas, Nonlinear system, chemistry, Steady state distribution, 0103 physical sciences, Statistical physics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 010306 general physics, Control parameters, Equations for a falling body, Selection (genetic algorithm), Mathematics
Abstract: International audience; The steady state distribution of polymer (or micelle) lengths under nonequilbrium conditions in which monomers are continuously extracted from a system is studied. The dynamical equations describing this process exhibit a one-parameter family of steady state distributions. A study of the dynamical equations suggests that they exhibit either a linear marginal or nonlinear marginal selection, depending on the control parameters of the model. The selection is explicitly demonstrated for a simplified linear version of the dynamical equations.
Published: 1994
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31. Stabilization of needle-crystals by the Gibbs-Thomson effect

Author: Claude-Alain Pillet, Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Département de Physique Théorique, University of Geneva [Switzerland], Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Université de Genève = University of Geneva (UNIGE), and Pillet, Claude-Alain
Subjects: [PHYS.PHYS.PHYS-CLASS-PH]Physics [physics]/Physics [physics]/Classical Physics [physics.class-ph], Current (mathematics), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Complex system, Pattern formation, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], dentritic growth, 01 natural sciences, 010305 fluids & plasmas, Discrete spectrum, pattern formation, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Statistical physics, Gibbs-Thomson, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 010306 general physics, Mathematical Physics, Mathematics, [MATH.MATH-QA] Mathematics [math]/Quantum Algebra [math.QA], needle crystal, Spectrum (functional analysis), Mathematical analysis, Statistical and Nonlinear Physics, [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], Nonlinear system, sidebranching, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], [PHYS.PHYS.PHYS-CLASS-PH] Physics [physics]/Physics [physics]/Classical Physics [physics.class-ph], Excitation
Abstract: http://projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.cmp/1104247981/body/pdf; We develop a scheme based on pseudo-differential operators to analyze the propagation of excitations in inhomogeneous extended systems. This method is used in a very specific situation, however we think that it has some generality and should apply to various other problems of current interest. We study the well known two-dimensional symmetric model of solidification introduced by Langer and Turski. Assuming the existence of Ivantsov-like steady-state solutions, we calculate their excitation spectrum. We show that there are no unstable propagating modes if the Gibbs-Thomson effect is taken into account. This proves that the growth of needle-crystals is stable with respect to side-branching.
Published: 1991

32. Open Quantum Systems II : The Markovian Approach

Author: Stéphane Attal, Alain Joye, claude-alain pillet, Stéphane Attal, Alain Joye, and claude-alain pillet
Subjects: Open systems (Physics), Quantum theory
Abstract: This volume is the second in a series of three volumes dedicated to the lecture notes of the summer school “Open Quantum Systems” which took place in the Institut Fourier in Grenoble, from June 16th to July 4th 2003. The contributions presented in thesevolumesarerevisedandexpandedversionsofthenotesprovidedtothestudents during the school. After the rst volume, developing the Hamiltonian approach of open quantum systems, this second volume is dedicated to the Markovian approach. The third volume presents both approaches, but at the recent research level. Open quantum systems A quantum open system is a quantum system which is interacting with another one. This is a general de nition, but in general, it is understood that one of the s- tems is rather “small” or “simple” compared to the other one which is supposed to be huge, to be the environment, a gas of particles, a beam of photons, a heat bath... The aim of quantum open system theory is to study the behaviour of this coupled system and in particular the dissipation of the small system in favour of the large one.
Published: 2006

33. Open Quantum Systems III : Recent Developments

Author: Stéphane Attal, Alain Joye, claude-alain pillet, Stéphane Attal, Alain Joye, and claude-alain pillet
Subjects: Distribution (Probability theory), Quantum theory, Open systems (Physics)
Abstract: This volume is the third and last of a series devoted to the lecture notes of the Grenoble Summer School on “Open Quantum Systems” which took place at the th th Institut Fourier from June 16 to July 4 2003. The contributions presented in this volumecorrespondtoexpanded versionsofthelecturenotesprovidedbytheauthors to the students of the Summer School. The corresponding lectures were scheduled in the last part of the School devoted to recent developments in the study of Open Quantum Systems. Whereas the rst two volumes were dedicated to a detailed exposition of the mathematical techniques and physical concepts relevant in the study of Open S- tems with noapriori pre-requisites, the contributions presented in this volume request from the reader some familiarity with these aspects. Indeed, the material presented here aims at leading the reader already acquainted with the basics in? quantum statistical mechanics, spectral theory of linear operators,C -dynamical systems, and quantum stochastic differential equations to the front of the current research done on various aspects of Open Quantum Systems. Nevertheless, pe- gogical efforts have been made by the various authors of these notes so that this volume should be essentially self-contained for a reader with minimal previous - posure to the themes listed above. In any case, the reader in need of complements can always turn to these rst two volumes. The topics covered in these lectures notes start with an introduction to n- equilibrium quantum statistical mechanics.
Published: 2006

34. Open Quantum Systems I : The Hamiltonian Approach

Author: Stéphane Attal, Alain Joye, claude-alain pillet, Stéphane Attal, Alain Joye, and claude-alain pillet
Subjects: Open systems (Physics), Quantum theory
Abstract: This is the?rst in a series of three volumes dedicated to the lecture notes of the Summer School ”Open Quantum Systems” which took place at the Institut Fourier in Grenoble from June 16th to July 4th 2003. The contributions presented in these volumes are revised and expanded versions of the notes provided to the students during the School. Closed vs. Open Systems By de?nition, the time evolution of a closed physical systemS is deterministic. It is usually described by a differential equation x? = X(x) on the manifold M of t t possible con?gurations of the system. If the initial con?guration x? M is known 0 then the solution of the corresponding initial value problem yields the con?guration x at any future time t. This applies to classical as well as to quantum systems. In the t classical case M is the phase space of the system and x describes the positions and t velocities of the various components (or degrees of freedom) ofS at time t. Inthe quantum case, according to the orthodoxinterpretation of quantum mechanics, M is a Hilbert space and x a unit vector – the wave function – describing the quantum t state of the system at time t. In both cases the knowledge of the state x allows t to predict the result of any measurement made onS at time t.
Published: 2006

35. Degenerate asymptotic perturbation theory

Author: Claude-Alain Pillet, Walter Hunziker, Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Intitut für Theoretische Physik, Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: Singular perturbation, Asymptotic analysis, Borel summability, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Holomorphic function, 01 natural sciences, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Perturbation theory, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics, Variable (mathematics), perturbation theory, asymptotic expansion, 010308 nuclear & particles physics, 010102 general mathematics, Mathematical analysis, Degenerate energy levels, quantum mechanics, Statistical and Nonlinear Physics, Mathematics::Logic, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Asymptotic expansion, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: http://projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.cmp/1103940281/body/pdf; International audience; Asymptotic Rayleigh-Schrodinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case. For this purpose we study the spectral properties of mxm-matrix functions A(kappa) of a complex variable kappa which have an asymptotic expansion $\sum_j A_j \kappa^j$ as kappa->0. We show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections of A(kappa) can be obtained from the set $\{A_j\}$ by analytic perturbation theory. Special attention is given to the case where A(kappa) is Borel-summable in some sector originating from kappa=0 with opening angle larger than pi. Here we prove that the asymptotic series describe individual eigenvalues and eigenprojections of A(kappa) which are shown to be holomorphic in S near kappa=0 and Borel summable if $A_j\ast=A_j$ for all j. We then fit these results into the scheme of Rayleigh-Schrodinger perturbation theory and we give some examples of asymptotic estimates for Schrdinger operators.
Published: 1983
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36. Quantum mechanics with stochastic time dependent potentials

Author: Claude-Alain Pillet, Centre de Physique Théorique - UMR 6207 (CPT), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), S. Albeverio, G. Casati, D. Merlini, and Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2
Subjects: Physics, Quantum dynamics, stochastic evolution, 010102 general mathematics, Stochastic interpretation, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], random potential, 01 natural sciences, Quantum mechanics, Quantum probability, Quantum stochastic calculus, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum process, 0103 physical sciences, Quantum operation, Statistical physics, 0101 mathematics, 010306 general physics, Quantum dissipation, Quantum statistical mechanics
Published: 1986
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37. Asymptotic completeness for a quantum particle in a Markovian short range potential

Author: Claude-Alain Pillet, Pillet, Claude-Alain, Centre de Physique Théorique - UMR 6207 (CPT), Centre National de la Recherche Scientifique (CNRS)-Université de Toulon (UTLN)-Université de Provence - Aix-Marseille 1-Université de la Méditerranée - Aix-Marseille 2, Intitut für Theoretische Physik, Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Complex system, Markov process, random potential, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, 81F05, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, Completeness (order theory), 0103 physical sciences, Bound state, [MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematical Physics, Physics, Range (particle radiation), scattering theory, [MATH.MATH-QA] Mathematics [math]/Quantum Algebra [math.QA], 010102 general mathematics, quantum mechanics, 82A31, Statistical and Nonlinear Physics, [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], Nonlinear system, symbols, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 82A05, disordered system, Scattering theory, 81C10, Energy (signal processing), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: http://projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.cmp/1104115334/body/pdf; International audience; Absence of bound states and asymptotic completeness are proven for a quantum particle in a time dependent random (Markovian) short range potential . Systems with confining potentials are also considered and unboundedness of the energy in time is shown.
Published: 1986

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