Results 1  10
of
33
Mathematical Theory of NonEquilibrium Quantum Statistical Mechanics
, 2002
"... We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium steady states, entropy production and ..."
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Cited by 33 (3 self)
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We review and further develop a mathematical framework for nonequilibrium quantum statistical mechanics recently proposed in [JP4, JP5, JP6, Ru3, Ru4, Ru5, Ru6]. In the algebraic formalism of quantum statistical mechanics we introduce notions of nonequilibrium 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.
H.: Equilibrium statistical mechanics of Fermion lattice systems
"... We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle wit ..."
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Cited by 26 (7 self)
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We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the noncommutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C ∗dynamical systems for the Fermion (CAR) algebra A with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics αt commutes with the evenoddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δα of αt contains the set A ◦ of all strictly local elements of A. (III) The set A ◦ is the core of δα.
An analogue of the KacWakimoto formula and black hole conditional entropy
 Commun. Math. Phys
"... Abstract. A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac ..."
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Cited by 21 (10 self)
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Abstract. A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by KacWakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the ConnesStœrmer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.
Introduction to Representations of the Canonical Commutation and Anticommutation Relations
, 2005
"... ..."
Positive Commutators in NonEquilibrium Quantum Statistical Mechanics
 Commun. Math. Phys
, 2000
"... The method of positive commutators, developed for zero temperature problems over the last twenty years, has been powering progress in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to nonequilibrium quantum statistical mechanics. We ..."
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Cited by 17 (7 self)
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The method of positive commutators, developed for zero temperature problems over the last twenty years, has been powering progress in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to nonequilibrium quantum statistical mechanics. We use the positive commutator technique to give an alternative proof of a fundamental property of large quantum systems, called Return to Equilibrium. This property says that equilibrium states are (asymptotically) stable: if a system is slightly perturbed from its equilibrium state, then it converges back to that equilibrium state as time goes to infinity. Keywords: positive commutator, Mourre estimate, return to equilibrium, virial theorem, Fermi golden rule Mathematics Subject Classification (2000): 82C10, 81Q10 1 Introduction In this paper, we study a class of open quantum systems consisting of two interacting subsystems: a finite system, called the particle system coupled to a reservo...
C.A.: The GreenKubo formula and Onsager reciprocity relations in quantum statistical mechanics
"... Dedicated to David Ruelle on the occasion of his 70th birthday We study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the GreenKubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Our deriva ..."
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Cited by 14 (5 self)
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Dedicated to David Ruelle on the occasion of his 70th birthday We study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the GreenKubo 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 nonequilibrium steady states. The GreenKubo formula and the Onsager reciprocity relations in quantum statistical mechanics 2 1
Noncommutative spectral invariants and black hole entropy
 Commun. Math. Phys
, 405
"... We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the “heat kernel ” semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system w ..."
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Cited by 10 (7 self)
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We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the “heat kernel ” semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler characteristic) is proportional to log µA, where µA is the global index of A, and the second spectral invariant is again proportional to c. We give a further general method to define a mean entropy by considering conformal symmetries that preserve a discretization of S1 and we get the same value proportional to c. We then make the corresponding analysis with the proper Hamiltonian associated to an interval. We find here, in complete generality, a proper mean entropy proportional to log µA with a first order correction defined by means of the relative entropy associated with canonical states.
Level shift operators for open quantum systems
 J. Math. Anal. Appl
, 2007
"... Level shift operators describe the second order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the s ..."
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Cited by 6 (4 self)
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Level shift operators describe the second order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems. 1 Introduction and main results Level shift operators emerge naturally in the context of perturbation theory of (embedded) eigenvalues, where they govern the shifts of levels (resonances) at second order in perturbation.