Author: "Claude-Alain Pillet" - Searchworks@Jio Institute Digital Library Search Results

Your search keyword '"Claude-Alain Pillet"' showing total 67 results

Start Over Author "Claude-Alain Pillet"

67 results on '"Claude-Alain Pillet"'

1. Markovian Repeated Interaction Quantum Systems

Author: Jean-François Bougron, Alain Joye, Claude-Alain Pillet, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Université de Cergy Pontoise (UCP), Université Paris-Seine, 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), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), CPT - E5 Physique statistique et systèmes complexes, and ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017)
Subjects: fluctuation relations, Quantum Physics, Statistical Mechanics (cond-mat.stat-mech), fluctuation, Markov chain, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), entropy production, dynamical system, Open quantum systems, linear response, quantum, thermodynamics, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], adiabatic, spectral, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], [PHYS.COND]Physics [physics]/Condensed Matter [cond-mat], entropy, Quantum Physics (quant-ph), Mathematical Physics, Condensed Matter - Statistical Mechanics, nonequilibrium statistical mechanics
Abstract: International audience; We study a class of dynamical semigroups $(\mathbb{L}^n)_{n\in\mathbb{N}}$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system $(\mathcal{L}_{\omega_n}\circ\cdots\circ\mathcal{L}_{\omega_1}(\rho_{\omega_0}))_{n\in\mathbb{N}}$ driven by a Markov chain $(\omega_n)_{n\in\mathbb{N}}$. We show that the almost sure large time behavior of the system can be extracted from the large $n$ asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator $\mathbb{L}$. As a physical application, we consider the case where the $\mathcal{L}_\omega$'s are the reduced dynamical maps describing the repeated interactions of a system $\mathcal{S}$ with thermal probes $\mathcal{C}_\omega$. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.
Published: 2022
Full Text: View/download PDF

2. Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics.

Author: Vojkan Jaksic, Yoshiko Ogata, Claude-Alain Pillet, and Robert Seiringer
Published: 2011

3. 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
Full Text: View/download PDF

4. On entropy production of repeated quantum measurements II. Examples

Author: Claude-Alain Pillet, Vojkan Jaksic, Tristan Benoist, Noé Cuneo, 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), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), 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-20-CE40-0024,QTraj,Trajectoires quantiques(2020), ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), McGill University = Université McGill [Montréal, Canada], ANR-20-CE40-0024-01,QTraj,Quantum Trajectories, 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), Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Dynamical Systems (math.DS), 01 natural sciences, 010305 fluids & plasmas, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Arrow of time, 0103 physical sciences, FOS: Mathematics, Statistical physics, Mathematics - Dynamical Systems, 010306 general physics, Mathematical Physics, Central limit theorem, Mathematics, Quantum Physics, Markov chain, Entropy production, Quantum instrument, Probability (math.PR), Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Mathematical theory, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Large deviations theory, Quantum Physics (quant-ph), Rate function, Mathematics - Probability
Abstract: International audience; We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation-dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory.
Published: 2020
Full Text: View/download PDF

5. Entropic Fluctuations in Thermally Driven Harmonic Networks

Author: Claude-Alain Pillet, Armen Shirikyan, 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), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), CNRS PICS RESSPDE, PICS RESSPDE, Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Boundary (topology), Physics - Classical Physics, Dynamical Systems (math.DS), Interval (mathematics), Langevin dynamics, 01 natural sciences, Matrix (mathematics), 0103 physical sciences, FOS: Mathematics, Uniqueness, Mathematics - Dynamical Systems, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 0101 mathematics, 010306 general physics, Fluctuation relations, Mathematical Physics, nonequilibrium statistical mechanics, Mathematical physics, Physics, Hamiltonian matrix, Fluctuation theorem, Probability (math.PR), 010102 general mathematics, Classical Physics (physics.class-ph), Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Coupling (probability), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Stochastic dfferential equations, Rate function, Mathematics - Probability
Abstract: We consider a general network of harmonic oscillators driven out of thermal equilibrium by coupling to several heat reservoirs at different temperatures. The action of the reservoirs is implemented by Langevin forces. Assuming the existence and uniqueness of the steady state of the resulting process, we construct a canonical entropy production functional $$S^t$$ which satisfies the Gallavotti–Cohen fluctuation theorem. More precisely, we prove that there exists $$\kappa _c>\frac{1}{2}$$ such that the cumulant generating function of $$S^t$$ has a large-time limit $$e(\alpha )$$ which is finite on a closed interval $$[\frac{1}{2}-\kappa _c,\frac{1}{2}+\kappa _c]$$ , infinite on its complement and satisfies the Gallavotti–Cohen symmetry $$e(1-\alpha )=e(\alpha )$$ for all $$\alpha \in {\mathbb {R}}$$ . Moreover, we show that $$e(\alpha )$$ is essentially smooth, i.e., that $$e'(\alpha )\rightarrow \mp \infty $$ as $$\alpha \rightarrow \tfrac{1}{2}\mp \kappa _c$$ . It follows from the Gartner–Ellis theorem that $$S^t$$ satisfies a global large deviation principle with a rate function I(s) obeying the Gallavotti–Cohen fluctuation relation $$I(-s)-I(s)=s$$ for all $$s\in {\mathbb {R}}$$ . We also consider perturbations of $$S^t$$ by quadratic boundary terms and prove that they satisfy extended fluctuation relations, i.e., a global large deviation principle with a rate function that typically differs from I(s) outside a finite interval. This applies to various physically relevant functionals and, in particular, to the heat dissipation rate of the network. Our approach relies on the properties of the maximal solution of a one-parameter family of algebraic matrix Riccati equations. It turns out that the limiting cumulant generating functions of $$S^t$$ and its perturbations can be computed in terms of spectral data of a Hamiltonian matrix depending on the harmonic potential of the network and the parameters of the Langevin reservoirs. This approach is well adapted to both analytical and numerical investigations.
Published: 2016
Full Text: View/download PDF

6. Entropic Fluctuations in Gaussian Dynamical Systems

Author: 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], 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-05-PADD-0009,EcoSerre,Viabilité des systèmes de cultures protégées dans un contexte d'agriculture durable(2005), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY)
Subjects: Dynamical systems theory, Gaussian, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Dynamical Systems (math.DS), Dynamical system, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, 0103 physical sciences, FOS: Mathematics, Gaussian dynamical systems, Statistical physics, Mathematics - Dynamical Systems, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], [MATH]Mathematics [math], 010306 general physics, Mathematical Physics, nonequilibrium statistical mechanics, Mathematics, Entropy production, Probability (math.PR), Statistical and Nonlinear Physics, Observable, Mathematical Physics (math-ph), Function (mathematics), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Phase space, symbols, Large deviations theory, Mathematics - Probability
Abstract: International audience; We study non-equilibrium statistical mechanics of a Gaussian dynamical system and compute in closed form the large deviation functionals describing the fluctuations of the entropy production observable with respect to the reference state and the non-equilibrium steady state. The entropy production observable of this model is an unbounded function on the phase space, and its large deviation functionals have a surprisingly rich structure. We explore this structure in some detail.
Published: 2016
Full Text: View/download PDF

7. Large deviations and entropy production in viscous fluid flows

Author: Armen Shirikyan, Vahagn Nersesyan, Vojkan Jaksic, Claude-Alain Pillet, McGill University = Université McGill [Montréal, Canada], Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-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), Université de Cergy Pontoise (UCP), Université Paris-Seine, CNRS PICS Fluctuation theorems in stochastic systems, Initiative d'excellence Paris-Seine, NSERC, ANR-17-CE40-0006,NONSTOPS,Systèmes stochastiques et ouverts hors équilibre(2017), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), and ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011)
Subjects: regular densities, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Markov process, FOS: Physical sciences, 01 natural sciences, entropy production, Navier-Stokes system, symbols.namesake, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics - Optimization and Control, Mathematical Physics, Probability measure, Mathematics, Lebesgue measure, Entropy production, Mechanical Engineering, Lagrangian trajectories, 010102 general mathematics, Mathematical analysis, Probability (math.PR), Mathematical Physics (math-ph), 16. Peace & justice, 010101 applied mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Large deviations, 35Q30, 35R60, 60B12, 60F10, 76D05, 93B05, Optimization and Control (math.OC), Bounded function, symbols, Large deviations theory, Vector field, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Rate function, Mathematics - Probability, Analysis, Analysis of PDEs (math.AP)
Abstract: We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier-Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component possesses a positive smooth density with respect to the Lebesgue measure in any finite time. This allows one to define a natural concept of the entropy production, and to show that its time average is a bounded function of the trajectory. The proofs are based on a new criterion for the validity of the level-3 LDP for Markov processes and an application of a general result on the image of probability measures under smooth maps to the laws associated with the motion of the particle., Comment: 52 pages
Published: 2019
Full Text: View/download PDF

8. Large Deviations from a Stationary Measure for a Class of Dissipative PDEs with Random Kicks

Author: Armen Shirikyan, Claude-Alain Pillet, Vahagn Nersesyan, Vojkan Jaksic, 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), 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), 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), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), ANR-11-BS01-0017,EMAQS,Estimation et manipulation à l'échelle Quantique(2011), ANR-11-BS01-0015,STOSYMAP,Systèmes stochastiques en mathématiques et physique mathématique(2011), 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)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
Subjects: General Mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Markov process, Measure (mathematics), large deviations, symbols.namesake, Mathematics - Analysis of PDEs, 35Q30, 76D05, 60B12, 60F10, Integer, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Navier–Stokes system, Mathematical Physics, Ginzburg–Landau equation, Mathematics, Markov chain, Applied Mathematics, Probability (math.PR), Mathematical analysis, Mathematical Physics (math-ph), Coupling (probability), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Bounded function, dissipative PDE’s, occupation measures, symbols, Dissipative system, Large deviations theory, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract: We study a class of dissipative PDE's perturbed by a bounded random kick force. It is assumed that the random force is non-degenerate, so that the Markov process obtained by the restriction of solutions to integer times has a unique stationary measure. The main result of the paper is a large deviation principle for occupation measures of the Markov process in question. The proof is based on Kifer's large deviation criterion, a Lyapunov-Schmidt type reduction, and an abstract result on large-time asymptotic for generalised Markov semigroups., Comment: 42 pages
Published: 2015
Full Text: View/download PDF

9. Landauer–Büttiker and Thouless Conductance

Author: Laurent Bruneau, Claude-Alain Pillet, 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), 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: Physics, Quantum Physics, scattering theory, Spectral theory, Heat current, Scattering, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Conductance, Statistical and Nonlinear Physics, spectral theory, Condensed Matter::Mesoscopic Systems and Quantum Hall Effect, Condensed Matter::Disordered Systems and Neural Networks, Mathematics - Spectral Theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, Independent electron approximation, Scattering theory, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Quantum statistical mechanics, Scaling, quantum transport, Mathematical Physics, nonequilibrium statistical mechanics
Abstract: In the independent electron approximation, the average (energy/charge/entropy) current flowing through a finite sample ${\mathcal{S}}$ connected to two electronic reservoirs can be computed by scattering theoretic arguments which lead to the famous Landauer–Buttiker formula. Another well known formula has been proposed by Thouless on the basis of a scaling argument. The Thouless formula relates the conductance of the sample to the width of the spectral bands of the infinite crystal obtained by periodic juxtaposition of ${\mathcal{S}}$. In this spirit, we define Landauer–Buttiker crystalline currents by extending the Landauer–Buttiker formula to a setup where the sample ${\mathcal{S}}$ is replaced by a periodic structure whose unit cell is ${\mathcal{S}}$. We argue that these crystalline currents are closely related to the Thouless currents. For example, the crystalline heat current is bounded above by the Thouless heat current, and this bound saturates iff the coupling between the reservoirs and the sample is reflectionless. Our analysis leads to a rigorous derivation of the Thouless formula from the first principles of quantum statistical mechanics.
Published: 2015
Full Text: View/download PDF

10. 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
Full Text: View/download PDF

11. 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
Full Text: View/download PDF

12. 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
Full Text: View/download PDF

13. 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
Full Text: View/download PDF

14. 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

15. 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
Full Text: View/download PDF

16. The Green–Kubo Formula for Locally Interacting Fermionic Open Systems

Author: Claude-Alain Pillet, Vojkan Jaksic, and Yoshiko Ogata
Subjects: Physics, Nonequilibrium statistical mechanics, Nuclear and High Energy Physics, Entropy production, 010102 general mathematics, Statistical and Nonlinear Physics, 01 natural sciences, Green–Kubo relations, Open quantum system, Heat flux, Reciprocity (electromagnetism), Quantum mechanics, Kubo formula, 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, Fermi gas, Mathematical Physics, Mathematical physics
Abstract: We consider a model describing finitely many free Fermi gas reservoirs coupled by local interactions and prove the Green-Kubo formulas and the Onsager reciprocity relations for heat and charge fluxes generated by temperature and chemical potential differentials.
Published: 2007
Full Text: View/download PDF

17. 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
Full Text: View/download PDF

18. Full statistics of energy conservation in two-time measurement protocols

Author: Annalisa Panati, Yan Pautrat, Vojkan Jaksic, Tristan Benoist, 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), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], 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), 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, 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), NSERC, ANR-14-CE25-0003,StoQ,Méthodes Stochastiques en Mécanique Quantique(2014), ANR-12-JS01-0008,SQFT,Théories spectrales et de la diffusion pour des modèles de théorie quantique des champs(2012), 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), 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: Physics, Quantum Physics, Current (mathematics), Statistical Mechanics (cond-mat.stat-mech), FOS: Physical sciences, Mathematical Physics (math-ph), Time limit, Energy conservation, Distribution (mathematics), Probability theory, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Statistics, Statistical physics, Quantum Physics (quant-ph), Quantum statistical mechanics, Anderson impurity model, Mathematical Physics, Condensed Matter - Statistical Mechanics, First law of thermodynamics
Abstract: The first law of thermodynamics states that the average total energy current between different reservoirs vanishes at large times. In this note we examine this fact at the level of the full statistics of two times measurement protocols also known as the Full Counting Statistics. Under very general conditions, we establish a tight form of the first law asserting that the fluctuations of the total energy current computed from the energy variation distribution are exponentially suppressed in the large time limit. We illustrate this general result using two examples: the Anderson impurity model and a 2D spin lattice model., 5 pages, 1 figure. Accepted for publication in Phys. Rev. E
Published: 2015
Full Text: View/download PDF

19. [Untitled]

Author: Claude-Alain Pillet and Walter H. Aschbacher
Subjects: Jordan–Wigner transformation, Scattering, Inverse, Statistical and Nonlinear Physics, Statistical mechanics, Decoupling (cosmology), 01 natural sciences, 010305 fluids & plasmas, Positive entropy, Quantum mechanics, Lattice (order), 0103 physical sciences, Scattering theory, 010306 general physics, Mathematical Physics, Mathematical physics, Mathematics
Abstract: We study the non-equilibrium statistical mechanics of the two-sided XY chain. We start from an initial state in which the left (x M) part of the lattice are at inverse temperatures betaL and betaR. Using a simple scattering theoretic analysis, we construct the unique non-equilibrium steady state (NESS). This state depends on betaL and betaR, but not on the choice of the decoupling parameter M. We prove that in the non-equilibrium case, betaL>betaR, this state has strictly positive entropy production.
Published: 2003
Full Text: View/download PDF

20. 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
Full Text: View/download PDF

21. [Untitled]

Author: Claude-Alain Pillet and Vojkan Jaksic
Subjects: Physics, Quantum discord, Entropy (statistical thermodynamics), 010102 general mathematics, Statistical and Nonlinear Physics, 01 natural sciences, Quantum relative entropy, Open quantum system, Quantum probability, Quantum process, 0103 physical sciences, Statistical physics, 0101 mathematics, 010306 general physics, Quantum statistical mechanics, Quantum dissipation, Mathematical Physics
Abstract: We review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics recently proposed in refs. 1–7. In the algebraic formalism of quantum statistical mechanics we introduce notions of non-equilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.
Published: 2002
Full Text: View/download PDF

22. 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
Full Text: View/download PDF

23. [Untitled]

Author: Vojkan Jaksic and Claude-Alain Pillet
Subjects: Pure mathematics, Spectral theory, 010102 general mathematics, Ergodicity, Zero (complex analysis), Statistical and Nonlinear Physics, State (functional analysis), 16. Peace & justice, Dynamical system, 01 natural sciences, Combinatorics, Mixing (mathematics), 0103 physical sciences, Ergodic theory, 010307 mathematical physics, 0101 mathematics, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics
Abstract: Let L be the Liouvillean of an ergodic quantum dynamical system ($\mathfrak{M}$, τ, ω). We give a new proof of the theorem of Jadczyk that eigenvalues of L are simple and form a subgroup of $\mathbb{R}$. If ω is a (τ, β)-KMS state for some β≠0 we show that this subgroup is trivial, namely that zero is the only eigenvalue of L. Hence, for KMS states ergodicity is equivalent to weak mixing.
Published: 2001
Full Text: View/download PDF

24. Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures

Author: Claude-Alain Pillet, Luc Rey-Bellet, Jean-Pierre Eckmann, Département de Physique Théorique, University of Geneva [Switzerland], Section de mathématiques [Genève], Université de Genève (UNIGE), 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), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de Genève = University of Geneva (UNIGE)
Subjects: [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Dynamical Systems (math.DS), ddc:500.2, 01 natural sciences, Hamiltonian system, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, Mathematics - Dynamical Systems, ddc:510, 0101 mathematics, 010306 general physics, Mathematical Physics, Physics, Stochastic process, 010102 general mathematics, Mathematical analysis, Anharmonicity, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Statistical mechanics, Invariant (physics), Nonlinear Sciences - Chaotic Dynamics, 3. Good health, Phase space, Hypoelliptic operator, [NLIN.NLIN-CD]Nonlinear Sciences [physics]/Chaotic Dynamics [nlin.CD], Invariant measure, Chaotic Dynamics (nlin.CD)
Abstract: We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of H\"ormander used in the study of hypoelliptic differential operators., Comment: 43 pages
Published: 1999
Full Text: View/download PDF

25. [Untitled]

Author: Claude-Alain Pillet, Luc Rey-Bellet, and Jean-Pierre Eckmann
Subjects: Physics, Entropy production, 010102 general mathematics, Energy flux, Statistical and Nonlinear Physics, 01 natural sciences, Hamiltonian system, Nonlinear system, 0103 physical sciences, Statistical physics, 0101 mathematics, 010306 general physics, Entropy (arrow of time), Mathematical Physics, Stationary state
Abstract: We consider a finite chain of non-linear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system.
Published: 1999
Full Text: View/download PDF

26. Invariant Manifolds for a Class of Dispersive, Hamiltonian, Partial Differential Equations

Author: C. Eugene Wayne, 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, Pennsylvania State University (Penn State), Penn State System-Penn State System, 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: Partial differential equation, Applied Mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 010102 general mathematics, Mathematical analysis, Invariant manifold, Invariant (physics), 01 natural sciences, Schrödinger equation, 010101 applied mathematics, symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Periodic orbits, Scattering theory, 0101 mathematics, Hamiltonian (quantum mechanics), Mathematics::Symplectic Geometry, Analysis, Center manifold, Mathematics, Mathematical physics
Abstract: International audience; We construct an invariant manifold of periodic orbits for a class of non-linear Schrödinger equations. Using standard ideas of the theory of center manifolds, we rederive the results of Soffer and Weinstein (Comm. Math. Phys.133, 119–146 (1997);J. Differential Equations98, 376–390 (1992)) on the large time asymptotics of small solutions (scattering theory).
Published: 1997
Full Text: View/download PDF

27. [Untitled]

Author: Claude-Alain Pillet and Vojkan Jakišć
Subjects: Thermal reservoir, 010102 general mathematics, Statistical and Nonlinear Physics, Stationary ergodic process, 01 natural sciences, 010305 fluids & plasmas, Langevin equation, Nonlinear system, Colors of noise, 0103 physical sciences, Brownian dynamics, Ergodic theory, Statistical physics, 0101 mathematics, Mathematical Physics, Brownian motion, Mathematics
Abstract: We discuss the dissipative dynamics of a classical particle coupled to an infinitely extended heat reservoir. We announce a number of results concerning the ergodic properties of this model. The novelty of our approach is that it extends beyond Markovian dynamics to the case where the Langevin equation is driven by colored noise. Our method works in arbitrary space dimension, and for fully nonlinear systems.
Published: 1997
Full Text: View/download PDF

28. On a model for quantum friction III. Ergodic properties of the spin-boson system

Author: Claude-Alain Pillet and Vojkan Jaksic
Subjects: Physics, Open quantum system, Ladder operator, Quantum dynamics, Quantum process, Quantum mechanics, Quantum system, Quantum operation, Statistical and Nonlinear Physics, Quantum number, Quantum statistical mechanics, Mathematical Physics
Abstract: We investigate the dynamics of a 2-level atom (or spin 1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared. From a more philosophical point of view our results show that, contrary to the conventional wisdom, quantum dynamics can be simpler at positive than at zero temperature.
Published: 1996
Full Text: View/download PDF

29. Large deviations and Gallavotti-Cohen principle for dissipative PDE's with rough noise

Author: Claude-Alain Pillet, Armen Shirikyan, Vahagn Nersesyan, Vojkan Jaksic, 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), 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 - 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), Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-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 Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY)
Subjects: Markov process, FOS: Physical sciences, 35Q30, 35Q56, 37L55, 60B12, 60F10, large deviation principle, reaction-diffusion system, Space (mathematics), entropy production, Navier-Stokes system, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Condensed Matter - Statistical Mechanics, Mathematical Physics, Mathematics, Statistical Mechanics (cond-mat.stat-mech), dissipative PDE's, Ginzburg-Landau equation, Entropy production, Mathematical analysis, Probability (math.PR), Statistical and Nonlinear Physics, Mathematical Physics (math-ph), 16. Peace & justice, Gallavotti-Cohen fluctuation relation, Symmetry (physics), Burgers equation, Burgers' equation, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], occupation measures, Dissipative system, symbols, Large deviations theory, Rate function, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract: We study a class of dissipative PDE's perturbed by an unbounded kick force. Under some natural assumptions, the restrictions of solutions to integer times form a homogeneous Markov process. Assuming that the noise is rough with respect to the space variables and has a non-degenerate law, we prove that the system in question satisfies a large deviation principle in tau-topology. Under some additional hypotheses, we establish a Gallavotti-Cohen type symmetry for the rate function of an entropy production functional and the strict positivity and finiteness of the mean entropy production in the stationary regime. The latter result is applicable to PDE's with strong nonlinear dissipation., Comment: 47 pages; to appear in Communications in Mathematical Physics
Published: 2013
Full Text: View/download PDF

30. Inside-outside duality for planar billiards: A numerical study

Author: Jean-Pierre Eckmann, Barbara Dietz, Claude-Alain Pillet, Uzy Smilansky, Iddo Ussishkin, 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), Département de Physique Théorique, University of Geneva [Switzerland], Section de mathématiques [Genève], Université de Genève (UNIGE), Laboratoire de Spectrométrie Physique (LSP), Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS), Department of Physics of Complex Systems, Weizmann Institute of Science [Rehovot, Israël], Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), and Université de Genève = University of Geneva (UNIGE)
Subjects: quantum billiards, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Duality (optimization), ddc:500.2, 01 natural sciences, 010305 fluids & plasmas, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ddc:510, Connection (algebraic framework), 010306 general physics, Eigenvalues and eigenvectors, Physics, scattering theory, Scattering, Mathematics::Spectral Theory, Eigenfunction, Nonlinear Sciences - Chaotic Dynamics, Classical mechanics, Scattering theory, Chaotic Dynamics (nlin.CD), Dynamical billiards, inside-outside duality, Laplace operator, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiard Laplacian and the scattering phases, basically that every energy at which a scattering phase is $2\pi$ corresponds to an eigenenergy of the Laplacian. Interesting phenomena appear when the shape of the domain does not allow an extension of the eigenfunction to the exterior. In this paper these phenomena are studied and illustrated from several points of view., Comment: uuencoded tar-compressed (using uufiles) postscript file, 15 pages
Published: 1995
Full Text: View/download PDF

31. 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
Full Text: View/download PDF

32. 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
Full Text: View/download PDF

33. 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
Full Text: View/download PDF

34. 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

35. 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
Full Text: View/download PDF

36. Non-equilibrium steady-states for interacting open systems: exact results

Author: Claude-Alain Pillet, Horia D. Cornean, Valeriu Moldoveanu, 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), Department of Mathematical Sciences [Aalborg], Aalborg University [Denmark] (AAU), National Institute of Materials Physics (NIMP), Romanian Academy of Sciences, 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: Current (mathematics), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Non-equilibrium thermodynamics, FOS: Physical sciences, 01 natural sciences, 010305 fluids & plasmas, Condensed Matter - Strongly Correlated Electrons, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, 0103 physical sciences, Mesoscale and Nanoscale Physics (cond-mat.mes-hall), Statistical physics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], 010306 general physics, Mathematical Physics, quantum transport, Independence (probability theory), nonequilibrium statistical mechanics, Physics, Mesoscopic physics, Condensed Matter - Mesoscale and Nanoscale Physics, Strongly Correlated Electrons (cond-mat.str-el), Scattering, nonequilibrium steady states, Coulomb blockade, State (functional analysis), Mathematical Physics (math-ph), Condensed Matter Physics, Electronic, Optical and Magnetic Materials, Exact results, [PHYS.COND.CM-MS]Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], PACS number(s): 73.23.Hk, 85.35.Ds
Abstract: Under certain conditions we prove the existence of a steady-state transport regime for interacting mesoscopic systems coupled to reservoirs (leads). The partitioning and partition-free scenarios are treated on an equal footing. Our time-dependent scattering approach is {\it exact} and proves, among other things the independence of the steady-state quantities from the initial state of the sample. Closed formulas for the steady-state current amenable for perturbative calculations w.r.t. the interaction strength are also derived. In the partitioning case we calculate the first order correction and recover the mean-field (Hartree-Fock) results., Comment: To appear in Phys. Rev. B
Published: 2011
Full Text: View/download PDF

37. 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
Full Text: View/download PDF

38. 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

39. 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
Full Text: View/download PDF

40. 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
Full Text: View/download PDF

41. 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
Full Text: View/download PDF

42. Topics in Non-Equilibrium Quantum Statistical Mechanics

Author: Yan Pautrat, Claude-Alain Pillet, Walter H. Aschbacher, and Vojkan Jaksic
Subjects: Open quantum system, Kullback–Leibler divergence, Entropy production, Statistical physics, Quantum statistical mechanics, Linear response theory, Mathematical physics, Mathematics
Abstract: 1 Zentrum Mathematik M5, Technische Universitat Munchen, D-85747 Garching, Germany e-mail: hbar@ma.tum.de 2 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada e-mail: jaksic@math.mcgill.co 3 Laboratoire de Mathematiques, Universite Paris-Sud, 91405 Orsay cedex, France e-mail: yan.pautrat@math.u-psud.fr 4 CPT-CNRS, UMR 6207, Universite du Sud, Toulon-Var, B.P. 20132, 83957 La Garde Cedex, France e-mail: pillet@univ-tln.fr
Published: 2006
Full Text: View/download PDF

43. Quantum Dynamical Systems

Author: Claude-Alain Pillet
Subjects: Quantum network, Mathematics::Dynamical Systems, Quantum dynamics, 010102 general mathematics, 01 natural sciences, Quantum technology, Theoretical physics, Open quantum system, Quantum process, 0103 physical sciences, Quantum operation, Quantum algorithm, 010307 mathematical physics, Statistical physics, 0101 mathematics, Mathematics, Quantum computer
Abstract: This notes are an expanded version of the lectures given by the author at the Grenoble "Open Quantum Systems" summer school in 2003. They provide a short introduction to quantum dynamical systems and their ergodic properties with particular emphasis on the quantum Koopman–von Neumann spectral theory.
Published: 2006
Full Text: View/download PDF

44. 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
Full Text: View/download PDF

45. Open Quantum Systems II

Author: Stéphane Attal, Claude-Alain Pillet, and Alain Joye
Subjects: Quantum geometry, Quantum dynamics, 05 social sciences, 01 natural sciences, Open quantum system, Quantum probability, Quantization (physics), Quantum process, 0502 economics and business, 0103 physical sciences, Quantum operation, Statistical physics, 010306 general physics, Quantum dissipation, 050203 business & management, Mathematics
Abstract: Understanding dissipative dynamics of open quantum systems remains a challenge in mathematical physics. This problem is relevant in various areas of fundamental and applied physics. From a mathematical point of view, it involves a large body of knowledge. Significant progress in the understanding of such systems has been made during the last decade. These books present in a self-contained way the mathematical theories involved in the modeling of such phenomena. They describe physically relevant models, develop their mathematical analysis and derive their physical implications. Volume II is dedicated to the Markovian formalism of classical and quantum open systems. A complete exposition of noise theory, Markov processes and stochastic differential equations, both in the classical and the quantum context, is provided. These mathematical tools are put into perspective with physical motivations and applications.
Published: 2006
Full Text: View/download PDF

46. Open Quantum Systems III

Author: Yan Pautrat, Claude-Alain Pillet, Vojkan Jaksic, and Walter H. Aschbacher
Subjects: Physics, Quantum discord, 010102 general mathematics, 01 natural sciences, Quantum relative entropy, Open quantum system, Quantum process, 0103 physical sciences, Quantum operation, 010307 mathematical physics, Statistical physics, Supersymmetric quantum mechanics, 0101 mathematics, Quantum statistical mechanics, Quantum dissipation
Abstract: These notes are an expanded and revised version of the lectures given by the second and fourth autor in the summer school "Open Quantum System" held in Grenoble, June 16-July 4, 2003. They provide an introduction to recent developments in non-equilibrium statistical mechanics of open quantum systems, including a completely worked out (simple) example. We discuss non-equilibrium steady states (NESS) and their structural properties, entropy production, linear response theory and weak coupling limit. The emphasis is on Ruelle's scattering approach to the construction of NESS.
Published: 2006
Full Text: View/download PDF

47. Open Quantum Systems I

Author: Alain Joye, Stéphane Attal, and Claude-Alain Pillet
Subjects: medicine.medical_specialty, Quantum network, 010308 nuclear & particles physics, Computer science, Quantum simulator, Markov process, 01 natural sciences, Quantum technology, Open quantum system, symbols.namesake, Stochastic differential equation, 0103 physical sciences, Calculus, symbols, Quantum nanoscience, medicine, 010306 general physics, Quantum
Abstract: Understanding dissipative dynamics of open quantum systems remains a challenge in mathematical physics. This problem is relevant in various areas of fundamental and applied physics. From a mathematical point of view, it involves a large body of knowledge. Significant progress in the understanding of such systems has been made during the last decade. These books present in a self-contained way the mathematical theories involved in the modeling of such phenomena. They describe physically relevant models, develop their mathematical analysis and derive their physical implications. Volume II is dedicated to the Markovian formalism of classical and quantum open systems. A complete exposition of noise theory, Markov processes and stochastic differential equations, both in the classical and the quantum context, is provided. These mathematical tools are put into perspective with physical motivations and applications.
Published: 2006
Full Text: View/download PDF

48. 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
Full Text: View/download PDF

49. 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

50. 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
Full Text: View/download PDF

Catalog

Books, media, physical & digital resources

See catalog results

Searchworks

Select search scope, currently: Articles Catalog books, media & more in Jio Institute collections Articles journal articles & other e-resources

Search

Search Constraints

Refine your results

Search Limiters

Topic

Publication Year Range

Language

Category

Publication Type

Journal

Database

Publisher

67 results on '"Claude-Alain Pillet"'

Search Results

Catalog

Select search scope, currently: Articles

Catalog

books, media & more in Jio Institute collections

Articles

journal articles & other e-resources