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Physical properties of some Gaussian distributions in quantum optics are considered. A useful definition for a temperature for a general Gaussian distribution is presented and used for analyzing the quantum optics version of Cramer’s theorem.

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Processes driven by randomly interrupted Gaussian white noise are considered. An evolution equation for single-event probability distributions is presented. Stationary states are considered as a solution of a second-order ordinary differential equation with two imposed conditions. A linear model is analyzed and its stationary distributions are explicitly given.

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The growth process of an initially ideal sphere in a convective fluctuating velocity field is considered. The influence of fluctuations of the velocity field on the growth process is examined.

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Langevin Equations of Ginzburg–Landau form, with multiplicative noise, are proposed to study the effects of fluctuations in domain growth. These equations are derived from a coarse-grained methodology. The Cahn–Hiliard–Cook linear stability analysis predicts some effects in the transitory regime. We also derive numerical algorithms for the computer simulation of these equations. The numerical results corroborate the analytical predictions of the linear analysis. We also present simulation results for spinodal decomposition at large times.

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An elementary account of Fefferman’s analysis of the quantum electron–proton system is presented. Assuming an optimal form of the stability of matter lower bound, it is shown that in an appropriate low density and low temperature limit, the electron–proton system behaves as a free gas of hydrogen atoms.

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The symmetry decomposition of the Frobenius–Perron operator and the associated Zeta functions is worked out for the case of reflection-symmetric 1-d maps.

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Nonlinear dynamical states of a spatially extended micromagnetic system — the Bloch wall — were analyzed by means of spatio-temporal diagrams and power spectral analysis in the spatial frequency domain. The system studied exhibits chaotic dynamics with propagating coherent spatial structures — Bloch lines — which have soliton properties. Although it is spatially extended only temporal chaos occurs. The symptoms of this type of chaos (spatially complex patterns changing violently with the time) in such a spatially extended system should not be confused with chaos in the space and the time simultaneously. The system is not spatially chaotic due to the existence in it of coherent spatial structures with a fixed length scale (kink solitons). In particular, the spatio-temporal diagrams certainly look complicated enough but spatial power spectra show only a low number of modes at a given time.

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The probabilistic approach provides a useful tool for understanding the nature of dynamics of cellular automata. It allows not only clarification of different results of the evolution but also gives explanation to the physical meaning of the rules.

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Real and complex perturbations of commutations relations are discussed. Using this approach we present some new ideas in the quantum probability inspired by \(C^*\)-algebraic description of quantum statistical mechanics as well as from the ideas of Voiculescu.

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Using the example of high temperature superconductors, we show that in the strongly correlated multiband Hubbard model the low energy excitations may be described by an effective spin-orbital model. The Hund’s rule exchange interaction leads to a competition between different possible magnetic order in the ground state. In a doped material this completion is additionally enhanced by the differences in the kinetic energies which leads to the formation of three-particle bound states. Superconducting ground state may be then stabilized by the exchange of \(d\)-\(d\) excitons.

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The quantum dynamics of a quartic double well, subjected to a harmonically oscillating field, is studied in the framework of the Floquet formalism. The modifications of the familiar tunneling process due to driving and dissipation are investigated numerically and explained in terms of the quasienergy spectrum. In absence of dissipation, there is a one-dimensional manifold in the parameter space spanned by amplitude and frequency of the driving force, where tunneling is almost completely suppressed by the coherent driving. The influence of dissipation is described on basis of the reduced density matrix in the Floquet representation. In particular, we consider the effect of weak Ohmic damping. In the classical limit, this system corresponds to a damped bistable Dulling oscillator. The interplay of coherent and incoherent transport processes is studied in terms of the transient time evolution of a temporal autocorrelation function. We find that the coherent suppression of tunneling is stabilized by reservoir-induced noise for a suitably chosen temperature. By computing stroboscopic Husimi distributions, we also compare the quantal stationary states with the corresponding classical deterministic attractors.

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The classic Granato–Lücke model for dislocation damping is revisited by accounting for possible refinements to the basic vibrating string mechanism. We argue that the perturbation approach, which consists in separating the (linear) dislocation dynamics from the equilibrium lattice environment, is not suitable, no matter how accurate the description of the coupling, to explain the finite decrement function observed experimentally in a variety of samples at vanishingly small frequencies. A few ideas for a more general theory are discussed in some detail.

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We present several models of time fluctuating media with finite memory, consisting in one and two-dimensional lattices, the nodes of which fluctuate between two internal states according to a Poisson process. A particle moves on the lattice, the diffusion by the nodes depending on their internal state. Such models can be used for the microscopic theory of reaction constants in a dense phase, or for the study of diffusion or reactivity in a complex medium. In a number of cases, the transmission probability of the medium is computed exactly; it is shown that stochastic resonances can occur, an optimal transmission being obtained for a convenient choice of parameters. In more general situations, approximate solutions are given in the case of short and moderate memory of the obstacles. The diffusion in an infinite two-dimensional lattice is studied, and the memory is shown to affect the distribution of the particles rather than the diffusion law.