Mathematical aspects of refined cluster functions. II

The refined cluster model theory of bound states is extended to describe reactions of nuclei. The intend of the method given is to use sets of linear matrix equations only which can be derived in close analogy to the procedure given in the first paper of this series.     

6Li formfactors in the refined cluster model

A soft core central potential is used to calculate independently⁶Li ground state and excited state wave functions in the refined cluster model. Charge density, elastic formfactor and transition formfactor to the 3⁺ state are derived and agree reasonably well with experiment.     

Molecular cluster perturbation theory. I. Formalism

We present second-order molecular cluster perturbation theory (MCPT(2)), a linear scaling methodology to calculate arbitrarily large systems with explicit calculation of individual wave functions in a coupled-cluster framework. This new MCPT(2) framework uses coupled-cluster perturbation theory and an expansion in terms of molecular dimer interactions to obtain molecular wave functions that are infinite order in both the electronic fluctuation operator and all possible dimer (and products of dimers) interactions. The MCPT(2) framework has been implemented in the new SIA/Aces4 parallel architecture, making use of the advanced dynamic memory control and fine-grained parallelism to perform very large explicit molecular cluster calculations. To illustrate the power of this method, we have computed energy shifts, lattice site dipole moments, and harmonic vibrational frequencies via explicit calculation of the bulk system for the polar and non-polar polymorphs of solid hydrogen fluoride. The explicit lattice size (without using any periodic boundary conditions) was expanded up to 1000 HF molecules, with 32,000 basis functions and 10,000 electrons. Our obtained HF lattice site dipole moments and harmonic vibrational frequencies agree well with the existing literature.     

He5 scattering states in the refined cluster function representation

As a first application of the refined cluster model reaction theory we calculated the scattering states of He⁵ using a specific ansatz for the nucleon-nucleon forces. He⁴-n and H³-d channels are considered only. Our results agree qualitatively with the experimental values if available; in particular we were able to explain the narrow resonance in theD _3/2 He⁴-n channel as a consequence of the strong coupling to theS _3/2 T-d channel.     

Mathematical aspects of relativistic kinetic theory

The linearized relativistic Boltzmann equation is studied with respect to its position in the theory of integral equations. A sufficient condition is given for the convergence of the series which are usually employed in the calculation of the transport coefficients pertaining to systems described by this equation. As an illustration, the theory is applied to the neutrino gas.     

Mathematical and physical aspects of gauge theories

Conclusion It is now a broadly-admitted fact that gauge theories provide intense research stimulation for both mathematicians and physicists. We tried to show that, as far as bundles and connections are the adequate minimal concepts to formalize the very act of beings which produce their own space in a consistent way, or, which is the same, the very act of informing a basis world B through a parametrized space of operations E, gauge theories realize a true translation between geometrical and physical information.In Section I.A, we tried to show how the concept of communication via experimentation finds this equivalence and makes a necessary collision between mathematics and physics.We gave some striking theoretical and practical facts: the obtaining of interaction Lagrangians, Higgs phenomenon and good energy behaviour of graphs arising from broken symmetries, the miraculous encounter between connection and path integral formalism. We could have added the relation with twistor theory, the theory of strong interactions, super gravity theories where the basis world B is not even a manifold. Some authors even speak of geometrodynamics and pregeometry [21] and would like to understand physics as a cosmogony.In a way, this makes us understand Einstein's unsatisfied mind when he wrote down his famous equation. The equivalence between a curvature tensor and an impulsion-energy tensor requires a careful analysis of the act of setting consistent spaces (via experimentation) when one doesn't reduce material beings (monads) to material points ruled by the universal Newton's observer.So, as far as gauge theories pretend to found and explain the process of translation between Geometry and Physics, they provide an extremely solid epistemological basement for the so-called mathematical Physics.     

Stochastic aspects in the theory of spectral-line broadening. II. Noncommutative cluster expansions

Generalizing ideas of von Waldenfels we develop a systematic procedure to define truncatedn-point operators which are reminiscent of Ursell functions of statistical mechanics. The truncation procedure is adapted to factorization relations obeyed by the operators in question. The results are applied to spectral-line broadening in plasmas. We derive cluster expansions for the line-shape function in terms of these truncated operators, where the ions are treated quasistatistically. The first order approximation for the line-shape function is discussed. The results are carried over to several moving perturber species, in particular to nonquasistatic ions.     

New method of computing correlation functions. I

A new method is proposed for computing the temperature and time correlation functions which permits taking correctly into account the desired order in the interaction.     

Fibrillar crystals of isotactic polystyrene. I. Morphological aspects

Fibrillar crystals have been prepared by the crystallization of isotactic polystyrene from stirred solutions in 1,3,5-trimethylben-zene and cyclohexanol. Similar microfibrils have been prepared in the nascent state by the polymerization of styrene in 1,3,5-trimethyl-benzene with a heterogeneous Ziegler-Natta catalyst. The microfibrils varied in width between 140 and 250 Å and were characterized by periodic lamellar overgrowths which were believed to give rise to a discrete X-ray reflection having a d-spacing of 90 A. Thermal analysis suggested that high growth temperatures favored greater crystal perfection. During crystallization from stirred cyclohexanol solutions, a fractionation of high molecular weight chains occurred which was more efficient at higher crystallization temperatures. In comparison to crystallizations under quiescent conditions, a marked facility of the crystallization process was observed in the formation of stirrer-crystallized and Ziegler-Natta polystyrene. The ease of crystallization of the shear-regenerated and nascent microfibrils has been related to a reduction in the free energy of nucleation. Mechanisms have been proposed to account for a favored nucleation process.     

Graphical representations and cluster algorithms I. Discrete spin systems

Graphical representations similar to the FK representation are developed for a variety of spin-systems. In several cases, it is established that these representations have (FKG) monotonicity properties which enables characterization theorems for the uniqueness phase and the low-temperature phase of the spin system. Certain systems with intermediate phases and/or first-order transitions are also described in terms of the percolation properties of the representations. In all cases, these representations lead, in a natural fashion, to Swendsen-Wang-type algorithms. Hence, at least in the above-mentioned instances, these algorithms realize the program described by Kandel and Domany, Phys. Rev. B 43 (1991) 8539–8548. All of the algorithms are shown to satisfy a Li-Sokal bound which (at least for systems with a divergent specific heat) implies critical slowing down. However, the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points.     

Mathematical Methods of Subjective Modeling in Scientific Research: I. The Mathematical and Empirical Basis

mathematical formalism for subjective modeling, based on modelling of uncertainty, reflecting unreliability of subjective information and fuzziness that is common for its content. The model of subjective judgments on values of an unknown parameter x ∈ X of the model M(x) of a research object is defined by the researcher–modeler as a space¹ (X, p(X), \(P{I^{\bar x}}\), \(Be{l^{\bar x}}\)) with plausibility\(P{I^{\bar x}}\) and believability \(Be{l^{\bar x}}\) measures, where x is an uncertain element taking values in X that models researcher—modeler’s uncertain propositions about an unknown x ∈ X, measures \(P{I^{\bar x}}\), \(Be{l^{\bar x}}\) model modalities of a researcher–modeler’s subjective judgments on the validity of each x ∈ X: the value of \(P{I^{\bar x}}(\tilde x = x)\) determines how relatively plausible, in his opinion, the equality \((\tilde x = x)\) is, while the value of \(Be{l^{\bar x}}(\tilde x = x)\) determines how the inequality \((\tilde x = x)\) should be relatively believed in. Versions of plausibility Pl and believability Bel measures and pl- and bel-integrals that inherit some traits of probabilities, psychophysics and take into account interests of researcher–modeler groups are considered. It is shown that the mathematical formalism of subjective modeling, unlike “standard” mathematical modeling, ?enables a researcher–modeler to model both precise formalized knowledge and non-formalized unreliable knowledge, from complete ignorance to precise knowledge of the model of a research object, to calculate relative plausibilities and believabilities of any features of a research object that are specified by its subjective model \(M(\tilde x)\), and if the data on observations of a research object is available, then it: ?enables him to estimate the adequacy of subjective model to the research objective, to correct it by combining subjective ideas and the observation data after testing their consistency, and, finally, to empirically recover the model of a research object.     

Coexistence of cluster and shell-model aspects in nuclear systems

We discuss cluster phenomena in light nuclei. As examples of typical cluster structures, we first review cluster structures of ¹²C, ¹⁶O, and ²⁰Ne, and then introduce some topics of cluster phenomena in light neutron-rich nuclei such as Be and C isotopes. A particular attention is paid on coexistence of cluster and shell-model aspects.     

Mathematical modeling of randomness and fuzziness phenomena in scientific studies: I. Mathematical and empirical foundations

The possibility theory as a mathematical model of randomness and fuzziness phenomena is considered in a variant that enables the modeling of both probabilistic randomness, including that inherent in unpredictably evolving stochastic objects whose probabilistic models cannot be empirically reconstructed and nonprobabilistic randomness (fuzziness) inherent in real physical, technical, and economical objects, human–machine and expert systems, etc. Some principal distinctions between the considered variant and the known possibility theory variants, in particular, in mathematical formalism and its relationship with probability theory, substantive interpretation, and applications exemplified by solving the problems of identification and estimation optimization, empirical reconstruction of a fuzzy model for a studied object, measurement data analysis and interpretation, etc. (in the paper “Mathematical Modeling of Randomness and Fuzziness Phenomena in Scientific Studies. II. Applications”) are shown.     

Nonequilibrium statistical mechanics. I. Mathematical formalism. Theorem of N. S. Krylov

Based on an algebraic formulation of quantum mechanics we introduce concepts playing a fundamental role in the constructionof the statistical mechanics of systems having direct classical analogs. We give the definition of a macrostate, mixing in the quantum version, and also demonstrate the existence of an upper bound to the relaxation time for an isolated system. It is shown that the theory constructed here contains both quantum and classical mechanics as limiting cases, but these two cases are not reducible to each other. The existence of a transition range not describable by the Schrodinger equation is noted.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 11, pp. 42–45, November, 1981.     

New aspects on the thermodynamic functions of iron

It is shown that the phase transformations of iron can be simply explained if all phases are attributed equal magnetic entropies at high temperatures corresponding to directional disorder of local moments. It is also pointed out that the b.c.c. or f.c.c. stabilization in iron alloys is consistent with a rigid band model for the conduction electrons.     

A general approach to association using cluster partition functions

A systematic and fundamental approach to associating mixtures is presented. It is shown how the thermodynamic functions may be computed starting from a partition function based on the cluster concept such as occurs in chemical theory. The theory provides a basis for and an extension of the existing chemical theory of (continuous) association. It is applicable to arbitrary association schemes. Analysis of separate cases is not necessary. The assumptions that were made to allow the development were chosen such as to make the principle of reactivity valid. It is this same principle that links various theories: the chemical theory of continuous association, the lattice fluid hydrogen bonding model, and first-order perturbation theory. The equivalence between these theories in appropriate limits is shown in a general and rigorous way. The theory is believed to provide a practical framework for engineering modeling work. Binary interaction parameters can be incorporated. The association scheme is accounted for by a set of generic equations, which should facilitate robust implementation in computer programs.     

Intersecting disks (and spheres) and statistical mechanics. I. Mathematical basis

The calculations of a previous paper on intersecting disks are completed. Further quantities of interest in connection with intersecting disks and spheres are defined. The above considerations are extended to spherical boundary conditions. Then two applications are stated: The penetrable-sphere model of Widom and Rowlinson, and the hard-sphere system. Finally, the generalization toD-dimensional spheres is outlined.