Narrow Results

It is important to consider the charged current weak interaction in the (semi-)inclusive deeply inelastic scattering to determine (transverse momentum dependent) parton distribution functions with flavor identifications. To this end, we calculate the charged current semi-inclusive deeply inelastic neutrino and anti-neutrino nucleus scattering and propose a measurable quantity in this paper to determine the transverse momentum dependent parton distribution functions. Semi-inclusive means that an unpolarized or a spin-0 hadron is detected in the current fragmentation region in addition to the scattered charged lepton. The target nucleus is assumed to be spin-1 for the complete calculation. We find that only the chiral-even terms survive and the chiral-odd terms vanish in this charged current deeply inelastic (anti-)neutrino nucleus scattering process. The measurable quantity is the yield difference or asymmetry of the positive pion and the negative pion. We notice that it is only determined by (anti-)strange quark distribution functions under isospin symmetry and would be reduced to a function of kinematic variable y if (anti-)strange quark distribution functions are neglected. The kaon meson production can be discussed in a similar way. Our calculations are limited at the leading twist level.

The thermodynamic limit in statistical thermodynamics of many-particle systems is an important but often overlooked issue in the various applied studies of condensed matter physics. To settle this issue, we review tersely the past and present disposition of thermodynamic limiting procedure in the structure of the contemporary statistical mechanics and our current understanding of this problem. We pick out the ingenious approach by Bogoliubov, who developed a general formalism for establishing the limiting distribution functions in the form of formal series in powers of the density. In that study, he outlined the method of justification of the thermodynamic limit when he derived the generalized Boltzmann equations. To enrich and to weave our discussion, we take this opportunity to give a brief survey of the closely related problems, such as the equipartition of energy and the equivalence and nonequivalence of statistical ensembles. The validity of the equipartition of energy permits one to decide what are the boundaries of applicability of statistical mechanics. The major aim of this work is to provide a better qualitative understanding of the physical significance of the thermodynamic limit in modern statistical physics of the infinite and "small" many-particle systems.

The dominance of Boltzmann–Gibbs distribution (BG) in statistical physics appears to endow an impression that this would be the only statistical approach available. In fact, a large class of statistical approaches already exists. This is generalizing BG statistics referring to various types of nonextensivity, nonadditivity, nonequilibrium, nonlinearity, etc. For instance, $log$ and $exp$ functions play crucial roles in both extensive and nonextensive domains of statistical mechanics. Emerging in a physical system, BG statistics defines extensive entropy which relates the number of microstates to thermodynamic quantities or macroscopic states. In this regard, the Boltzmann distribution, extensive statistics, refers to a well-defined probability distribution, while the various types of nonextensive statistics categorically violate the fourth Shannon–Khinchen additivity axiom. The $log$ and $exp$ distribution functions in BG, Tsallis, and generic statistics are systematically compared. We focus on the mathematical properties of both distribution functions and conclude that their compatibility exclusively depends on the nonextensive parameters, i.e. the mathematical properties of both distribution functions depend on the nonextensive parameters either that of Tsallis- or that of the generic-type of nonextensivity. We also conclude that the statistical nature of the underlying ensemble should be taken into consideration when applying the statistical approach.

Depending on the outcome of the triphoton experiment now underway, it is possible that the new local realistic Markov Random Field (MRF) models will be the only models now available to correctly predict both that experiment and Bell's theorem experiments. The MRF models represent the experiments as graphs of discrete events over space-time. This paper extends the MRF approach to continuous time, by defining a new class of realistic model, the stochastic path model, and showing how it can be applied to ideal polaroid type polarizers in such experiments. The final section discusses possibilities for future research, ranging from uses in other experiments or novel quantum communication systems, to extensions involving stochastic paths in the space of functions over continuous space. As part of this, it derives a new Boltzmann-like density operator over Fock space, which predicts the emergent statistical equilibria of nonlinear Hamiltonian field theories, based on our previous work of extending the Glauber–Sudarshan P mapping from the case of classical systems described by a complex state variable α to the case of classical continuous fields. This extension may explain the stochastic aspects of quantum theory as the emergent outcome of nonlinear PDE in a time-symmetric universe.

Single-bead walls were fabricated using stainless steel (SS) 304 wire and arc additive manufacturing (WAAM). The primary objective is to statistically characterize critical material properties of the build to make comparisons with wrought material. The quality of the build is demonstrated by the size distribution and spatial location of porosity in the build. Deviations in the thermal history as a function of the location in the build, especially as the vertical height of the build increases, cause differences in material properties which are statistically investigated. The material behavior and statistical variances along the horizontal direction of the build compared to perpendicular differences in location is significant. Optical microscopy and electron back-scatter diffraction are used to estimate statistically key geometric properties of grains. A preliminary investigation for fatigue using the strain life approach has been conducted to evaluate the effect of the build process and the location dependence on life. A generalized Weibull distribution is proposed to model the statistical variability of fatigue life data.

Let φ denote Euler's totient function, and G be the distribution function of φ(n)/n. Using functional equations, it is shown that φ(n)/n is statistically close to 1 essentially when prime factors of n are large. A function defined by a difference-differential equation gives a quantitative measure of the statistical influence of the size of prime factors of n on the closeness of φ(n)/n to 1.

As a corollary, an asymptotic expansion at any order of G(1)-G(1-ε) is obtained according to negative powers of log(1/ε), when ε tends to 0⁺. This improves a result of Erdős (1946) in which he gives the first term of it. By optimally choosing the order of this expansion, an estimation of G(1)-G(1-ε) is deduced, involving an error term of the same size as the best known error term involved in prime number theorem.

Soit φ l'indicatrice d'Euler. Nous étudions le comportement au voisinage de 1 de la fonction G de répartition de φ(n)/n, via la mise en évidence d'équations fonctionnelles. Nous obtenons un résultat mesurant l'influence statistique de la taille du plus petit facteur premier d'un entier générique n quant à la proximité de φ(n)/n par rapport à 1. Ce résultat met en jeu une fonction définie par une équation différentielle aux différences.

Nous en déduisons un développement limité à tout ordre de G(1)-G(1-ε) selon les puissances de 1/(log 1/ε), améliorant ainsi un résultat d'Erdős (1946) dans lequel il obtient le premier terme de ce développement. Une troncature convenable de ce développement fournit un terme d'erreur comparable à celui actuellement connu pour le théorème des nombres premiers.

A valuable approach to the analysis of hadron physics observables is provided by QCD’s equations-of-motion; namely, the Dyson-Schwinger equations. Drawing from a diverse collection of predictions, we revisit: γγ^* → neutral pseudoscalar transition form factors, their corresponding valence-quark distribution amplitudes and a recent result on the pion distribution functions.