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We review and further develop a mathematical framework for non-equilibrium 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 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.

. 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-Wakimoto 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 Connes-Stoermer 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 logar...

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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.

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.

The spin-fermion model describes a two level quantum system S (spin 1/2) coupled to finitely many free Fermi gas reservoirs Rj which are in thermal equilibrium at inverse temperatures βj. We consider non-equilibrium initial conditions where not all βj are the same. It is known that, at small coupling, the combined system S + P j Rj has a unique non-equilibrium steady state (NESS) characterized by strictly positive 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. The Green-Kubo formula for the spin-fermion system 2 1

Given a W -algebra M with a W -dynamics , we prove the existence of the perturbed W -dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If has a -KMS state, and the perturbation satises some mild assumptions related to the Golden-Thompson inequality, we prove the existence of a -KMS state for the perturbed W -dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations. 1 2 1

. 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. 1. Introduction In this paper we consider the dissipative dynamics of a quantum mechanical 2-level system --- the spin --- characterized by its two eigenstates of energy e \Sigma = \Sigma1. More specifically we investigate the long time behavior of the dynamics of a spin 1 2 allowed to interact with a large reservoir. The reservoir is an infinitely extend...