Paul Vanheuverzwijn

Completely positive maps on the CCR-algebra

Given any operator on the testfunction space, the general form of the induced completely positive map of the C*-algebra of the canonical commutation relations is characterized.

Completely positive quasi-free maps of the CCR-algebra

Abstract The class of completely positive (CP) quasi-free maps on the CCR-algebra is studied. We characterize the pure maps, study invariant states under semigroups, construct a particular dilation and consider the problem of implementation.

Energy-Entropy Inequalities for Classical Lattice Systems

A new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities. Their physical meaning is found to express thermodynamic stability. We demonstrate the applicability of the inequalities in specific models.

Quantum energy‐entropy inequalities: A new method for proving the absence of symmetry breaking

For quantum systems we develop a new method, based on a general energy‐entropy inequality, to rule out spontaneous breaking of symmetries. The main advantage of our scheme consists in its clear‐cut physical significance and its new areas of applicability; in particular we can handle discrete symmetry groups as well as continuous ones. Finally a few illustrations are discussed.

Energetically stable systems

For quantum systems as well as for classical continuous systems energetic stability is defined. It is proved that stability, supplemented with a cluster property, characterizes equilibrium states.

Implementable positive maps on standard forms

Abstract We determine those unital positive maps on a von Neumann algebra in standard form that can be isometrically implemented, thereby generalizing the result that in this situation any automorphism can be unitarily implemented and that any normal state is a vector state.

Dispersive properties and observables at infinity for classical kms systems

For infinite classical dynamical systems, satisfying the KMS condition, relations between asymptotic dispersive and cluster properties are proved. The local structure of the algebra of observables is explicitly characterized by the Poisson bracket commutant, and it is proved that the algebra of observables at infinity are constants of the motion.

Discrete lattice systems and the equivalence of microcanonical, canonical and grand canonical Gibbs states

It is proven that a microcanonical Gibbs measure on a classical discrete lattice system is a mixture of canonical Gibbs measures, provided the potential is “approximately periodic,” has finite range and possesses a commensurability property. No periodicity is imposed on the measure. When the potential is not approximately periodic or does not have the commensurability property, the inclusion does not hold.As a by-product, a new proof is given of the fact that for a large class of potentials, a canonical Gibbs measure is a mixture of grand canonical measures. Thus the equivalence of ensembles is obtained in the sense of identical correlation functions.

On the canonical Gibbs states associated with certain Markov chains

SummaryWe study the Gibbs states and canonical Gibbs states of a onedimensional inhomogeneous model with nearest neighbour interactions depending on a parameter β>0. For β≦1/2 all extreme canonical Gibbs states are also Gibbs states. For β>1 each extreme canonical Gibbs state is concentrated on a set of configurations having a fixed finite number of +1 (or −1) sites. For 1/2 <β≦1 extreme canonical Gibbs states of both the above types occur, a phenomenon not observed in noninteracting models.

Inhomogeneous mean field models

The infinite set of coupled mean field equations for a classical inhomogeneous Ising ferromagnet is studied with respect to existence and uniqueness of its solutions.