Va Zagrebnov

A QUANTUM CRYSTAL MODEL IN THE LIGHT-MASS LIMIT: GIBBS STATES

Ground and temperature quantum Gibbs states are constructed for a ferroelectric anharmonic quantum oscillator model with small masses. It is shown that they possess mixing properties. The construction relies on the Feynman–Kac–Nelson representation of the conditional reduced density matrices and on the cluster expansions for the corresponding Gibbs field of trajectories.

Phase-transitions and algebra of fluctuation operators in an exactly soluble model of a quantum anharmonic crystal

A complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions. In the one-phase region, the fluctuations are normal and its algebra is non-Abelian. In the two-phase region and on the critical line (Tc>0) the momentum fluctuation is normal, the displacement is critical, and the algebra is Abelian; atTc=0 (quantum phase transition) this algebra is non-Abelian with abnormal displacement and supernormal (squeezed) momentum fluctuation operators, both being dimension dependent.

No-go theorem for quantum structural phase-transitions

We prove that quantum fluctuations can suppress structural phase transitions. We give a rigorous proof for a one-component (R1) quantum crystal with local double-well anharmonism under the condition that the masses of the atoms in the lattice sites of Zd (d>or=3) are light enough.

On Bogoliubov's model of superfluidity

The authors point out that the conventional Bogoliubov model contains an attractive effective interaction, putting into question its stability. For positive chemical potentials they show instability, making the model unsuitable for explaining superfluidity from first principles. They present an extended model, yielding rigorously the relevant spectrum for superfluidity.

On Condensations in the Bogoliubov Weakly Imperfect Bose Gas

We show that condensation in the Bogoliubov weakly imperfect Bose gas (WIBG) may appear in two stages. If interaction is such that the pressure of the WIBG does not coincide with the pressure of the perfect Bose gas (PBG), then the WIBG may manifest two kinds of condensations: nonconventional Bose condensation in zero mode, due to the interaction (the first stage), and conventional (generalized) Bose–Einstein condensation in modes next to the zero mode due to the particle density saturation (the second stage). Otherwise the WIBG manifests only the latter kind of condensation.

On the parallel dynamics of the Q-state potts and Q-ising neural networks

Using a probabilistic approach, the parallel dynamics of theQ-state Potts andQ-Ising neural networks are studied at zero and at nonzero temperatures. Evolution equations are derived for the first time step and arbitraryQ. These formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks. An explicit analysis, including dynamical capacity-temperature diagrams and the temperature dependence of the overlap, is carried out forQ=3. Both types of models are compared.

Bose-Einstein condensation for homogeneous interacting systems with a one-particle spectral gap

We prove rigorously the occurrence of zero-mode Bose–Einstein condensation for a class of continuous homogeneous systems of boson particles with superstable interactions. This is the first example of a translation invariant continuous Bose-system, where the existence of the Bose–Einstein condensation is proved rigorously for the case of non-trivial two-body particle interactions, provided there is a large enough one-particle excitations spectral gap. The idea of proof consists of comparing the system with specially tuned soluble models.

Retrieval and chaos in extremely dilutedQ-Ising neural networks

Using a probabilistic approach, the deterministic and the stochastic parallel dynamics of aQ-Ising neural network are studied at finiteQ and in the limitQ→∞. Exact evolution equations are presented for the first time-step. These formulas constitute recursion relations for the parallel dynamics of the extremely diluted asymmetric versions of these networks. An explicit analysis of the retrieval properties is carried out in terms of the gain parameter, the loading capacity, and the temperature. The results for theQ→∞ network are compared with those for theQ=3 andQ=4 models. Possible chaotic microscopic behavior is studied using the time evolution of the distance between two network configurations. For arbitrary finiteQ the retrieval regime is always chaotic. In the limitQ→∞ the network exhibits a dynamical transition toward chaos.

Dynamics of quantum fluctuations in an anharmonic crystal model

For a model given previously by the authors describing a structural phase transition we compute theq-mode critical fluctuations of momentum and displacement as a function of the critical temperatures, the wave vectorq, and a fading-out external field. An explicit dependence on the rates of fading out is obtained. In order to define the critical fluctuation operators we prove a reconstruction theorem, which is of model-independent value. Finally we study the critical spectrum and get rigorous results on the soft modes and the central peak.

Gaussian, non-Gaussian critical fluctuations in the Curie-Weiss model

It is known that at the critical temperature the Curie-Weiss mean-field model has non-Gaussian fluctuations and that “internal fluctuations” can be Gaussian. Here we compute the distribution of theq-mode magnetization fluctuations as a function of the temperature, the wave vectorq, and a fading out external field. We obtain new classes of probability distributions generated by this external field as well as new critical behavior in terms of its rate of fading out. We discuss also the susceptibility as the limitq tending to zero.