Classical and Quantum Coherent States

p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously (V. V. Kisil, p-Mechanics as a physical theory. An Introduction, E-print:arXiv:quant-ph/0212101, 2002; International Journal of Theoretical Physics 41(1), 63–77, 2002). We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allows us to evaluate classical observables at any point of phase space and simultaneously to evaluate quantum probability amplitudes. The example of the forced harmonic oscillator is used to demonstrate these concepts.     

Bimolecular annihilation reactions: Immobile reactants and multipolar interactions

We study the A + B 0 annihilation reaction via multipolar interactions wr^–s (r distance) in one and two dimensions. For equal numbers of immobile A and B particles we present computer simulations and numerical calculations of the decay. We find at large times that ind dimensions the particle concentration follows nt^–d/(2s–d)This work is dedicated to Prof. George H. Weiss.     

Quantum Hall Effect on the Hyperbolic Plane in the Presence of Disorder

We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in Comm. Math. Phys. 190 (1998), 629–673, to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.     

A Law of Large Numbers for the Zeroes of Heine–Stieltjes Polynomials

We determine the limiting density of the zeroes of Heine–Stieltjes polynomials (or of any set of points satisfying the conclusion of Heine–Stieltjes Theorem) in the thermodynamic limit and use this to prove a strong law of large numbers for the zeroes.     

Fluctuations in coagulating systems

Time-dependent fluctuations in a system of coagulating particles are studied, using the master equation for the probability distributionsP(m,t) for the occupation numbersm={m _k} (k=1,2,...) of thek-cluster states. Van Kampen's-expansion is used to determine the deterministic (order ⁰) and fluctuating part (order ^–1/2) of the solution. We calculate the time-dependent behavior of the fluctuations in the cluster size distribution. The model under consideration is of special interest since it exhibits a phase transition (gelation). For monodisperse initial states we give explicit expressions for the probability distribution of the fluctuations and for the equal-time and two-time correlation functions also near the phase transition. For general initial conditions we study the fluctuations (1) for large cluster sizes, (2) in the scaling limit (near the critical point), and (3) for large times. Our results show that the deterministic approach to coagulation processes (Smoluchowski theory) is invalid very close to the gelpointt _c and at large times (tt _M), where the distance from the gelpoint and the timet _M depend upon the size of the system.     

A note on q-weak law of large numbers

q-limit theorems for random variables are arising from non-extensive statistical mechanics. In this note we will prove q-weak law of large numbers using the notions of q-Fourier transform, q-independence, q-weak convergence.     

Many worlds and the appearance of probability in quantum mechanics

The interpretation of the squared norm as probability and the apparent stochastic nature of observation in the quantum theory are derived from the law of large numbers and the algebraic properties of infinite sequences of simultaneous quantum observables. It is argued that this result validates the many-worlds view of quantum reality.     

Causal Dissipative Cosmology With Variable G and Λ

A cosmological model with variable G and is considered in the framework of Israel–Stewart–Hiscock (ISH) causal theory. Power law as well as inflationary solutions are obtained. The gravitational constant is found to increase with time.     

Fundamentals of fuzzy probability theory

The canonical classical extension of quantum mechanics studied recently by E. G. Beltrametti and S. Bugajski opens a new way toward generalizing the standard probability theory. The emerging fuzzy probability theory is able to give a full account of both classical and quantal probabilities, and—like the standard probability theory—could be of universal use, far outside the borders of physics. A specific feature of this hypothetical theory of probability is its mixed, classical-quanta character: classical as well as quantal random variables are described on an equal footing in a unified framework. Some new features of the fuzzy probability theory are shown on simple examples.     

Surface tension in Ising systems with Kac potentials

We consider an Ising spin system with Kac potentials in a torus of ^d,d>-2, and fix the temperature below its Lebowitz-Penrose critical value. We prove that when the Kac scaling parameter vanishes, the log of the probability of an interface becomes proportional to its area and the surface tension, related to the proportionality constant, converges to the van der Waals surface tension. The results are based on the analysis of the rate functionals for Gibbsian large deviations and on the proof that they -converge to the perimeter functional of geometric measure theory (which extends the notion of area). Our considerations include nonsmooth interfaces, proving that the Gibbsian probability of an interface depends only on its area and not on its regularity.     

Information and Entropy Flow in the Kalman–Bucy Filter

We investigate the information theoretic properties of Kalman–Bucy filters in continuous time, developing notions of information supply, storage and dissipation. Introducing a concept of energy, we develop a physical analogy in which the unobserved signal describes a statistical mechanical system interacting with a heat bath. The abstract universe comprising the signal and the heat bath obeys a non-increase law of entropy; however, with the introduction of partial observations, this law can be violated. The Kalman–Bucy filter behaves like a Maxwellian demon in this analogy, returning signal energy to the heat bath without causing entropy increase. This is made possible by the steady supply of new information. In a second analogy the signal and filter interact, setting up a stationary non-equilibrium state, in which energy flows between the heat bath, the signal and the filter without causing any overall entropy increase. We introduce a rate of interactive entropy flow that isolates the statistical mechanics of this flow from marginal effects. Both analogies provide quantitative examples of Landauers Principle.     

Electron Energy Loss Spectra of the Organic Complexes of Europium

In the electron energy loss spectra (EELS) of the organic europium complexes Eu³⁺ (BTFA)₃TPPO and Eu³⁺(BrBTFA)₃TPPO in a gas phase obtained on excitation by monokinetic beams of electrons of different energies in the range 12–50 eV, we have identified the bands associated with the electron transitions S ₀–S ₁, S ₀–S ₂, and S ₀–S ₃. The connection of these transitions with the structural groups of the complexes is established. The addition of the bromine atom to the phenyl ring of diketonate leads to the rise in the relative intensity of the S ₀–S ₂ band. The singlettriplet transitions manifest themselves in the region 2.5–3.2 eV and contribute to the S ₀–S ₂ band of the electron energy loss spectra.     

Type-II intermittency characteristics in the presence of noise

We consider a type of intermittent behavior that occurs as the result of the interplay between dynamical mechanisms giving rise to type-II intermittency and random dynamics. We analytically deduce the law for the distribution of the laminar phases, which has never been obtained hitherto. The already known dependence of the mean length of the laminar phases on the criticality parameter [Phys. Rev. E 68, 036203 (2003)] follows as a corollary of the carried out research. We also prove that this dependence obtained earlier under the assumption of the fixed form of the reinjection probability does not depend on the relaminarization properties, and, correspondingly, the obtained expression of the mean length of the laminar phases on the criticality parameter remains correct for different types of the reinjection probability.     

Response to Nagle’s Criticism of My Proposed Definition of the Entropy

In a recent paper, Nagle criticized the new definition of entropy that I had proposed in an earlier work. In the examples for which Nagle claims my definition fails, he took a formula that I had derived for one set of experiments and used it to represent my definition for other experiments. However, the formulas obtained from my definition depend on the specific experimental observables. If my definition is correctly applied to Nagles experiments, no contradictions remain.     

PAC Measurements on New Ferromagnetic Compound Pd2TiSn

Magnetic hyperfine field (mhf) acting on ¹⁸¹Ta at the Ti site has been investigated in the alloy Pd₂TiSn by Perturbed Angular Correlation (PAC) measurements using the (133–482) keV – cascade in ¹⁸¹Ta following the ^– decay of ¹⁸¹Hf. The magnetic hyperfine field was measured as a function of temperature in the range of 45–1000K. The magnetization measurements were carried out with a vibrating sample magnetometer. The interpretation of the experimental results is based on the assumption that the probe nuclei occupy both the regular Ti sites as well as the Ti atom sites randomly distributed on the Pd sublattice.     

Fluctuation Relation beyond Linear Response Theory

The Fluctuation Relation (FR) is an asymptotic result onthe distribution of certain observables averaged over timeintervals τ as τ → ∞ and it is a generalization of thefluctuation–dissipation theorem to far from equilibrium systemsin a steady state, which reduces to the usual Green–Kubo (GK)relation in the limit of small external non-conservative forces.FR is a theorem for smooth uniformly hyperbolic systems, and it isassumed to be true in all dissipative ‘chaotic enough’ systemsin a steady state. In this paper, we develop a theory of finitetime corrections to FR, needed to compare the asymptoticprediction of FR with numerical observations, which necessarilyinvolve fluctuations of observables averaged over finite timeintervals τ. We perform a numerical test of FR in two cases inwhich non-Gaussian fluctuations are observable, while GK does notapply and we get a non-trivial verification of FR that is independent of and different from linear response theory.Our results are compatible with the theory of finite timecorrections to FR, while FR would be observably violated,well within the precision of our experiments, if such correctionswere neglected.     

A strong law of large numbers for iterated functions of independent random variables

We study sequences of random variables obtained by iterative procedures, which can be thought of as nonlinear generalizations of the arithmetic mean. We prove a strong law of large numbers for a class of such iterations. This gives rise to the concept of generalized expected value of a random variable, for which we prove an analog of the classical Jensen inequality. We give several applications to models arising in mathematical physics and other areas.     

Measurement in quantum mechanics as a stochastic process on spaces of fuzzy events

The measurement of one or more observables can be considered to yield sample points which are in general fuzzy sets. Operationally these fuzzy sample points are the outcomes of calibration procedures undertaken to ensure the internal consistency of a scheme of measurement. By introducing generalized probability measures on -semifields of fuzzy events, one can view a quantum mechanical state as an ensemble of probability measures which specify the likelihood of occurrence of any specific fuzzy sample point at some instant. These sample points are the possible outcomes of any infinitely rapid succession of measurements at that instant of any sequence of observables of the system.     

Loop Observables for BF Theories in Any Dimension and the Cohomology of Knots

A generalization of Wilson loop observables for BF theories in any dimension is introduced within the Batalin–Vilkovisky framework. The expectation values of these observables are cohomology classes of the space of imbeddings of a circle. One of the resulting theories discussed in the Letter has only trivalent interactions and, irrespective of the actual dimension, looks like a three-dimensional Chern–Simons theory.     

Shearing apparatus for neutron scattering studies on fluids: Preliminary results for colloidal suspensions

In this paper we study the transient diffusion-limited A+B0, A₀ = B₀ annihilation via deterministic reaction-diffusion equations and via simulation of the stochastic many-particle problem. We show that the two approaches are not equivalent and that the deterministic expressions capture only part of the picture. A lower bound is derived for the density decay which indicates that the overall density follows the power law t^–x with = mm(d/4, 1). Hierarchical oscillations superimposed on the power-law decay are observed for reactions on Sierpinski gaskets.This work is dedicated to Prof. George H. Weiss.