Irreducible Integrals Research Articles

Application of Mayer’s activity expansions to the Ising problem

Recent successes in approximate evaluations of high-order cluster integrals have greatly advanced the quantitative applicability of Mayer’s cluster expansion for various statistical models of matter and even real substances. In this paper, the thermodynamics of the Ising model is represented in a general form of the virial series in powers of activity, which is not restricted by any special geometry or the simplification of nearest-neighbor interactions. In order to make the presented theoretical expressions applicable in quantitative studies, a number of techniques are considered, which approximate the unlimited (almost infinite) set of reducible cluster integrals, bn, based on only a few number of irreducible integrals, βk. Namely, such short βk sets are defined here for a number of magnetic models of various geometry, dimensionality, as well as interaction ranges (not limited by the nearest coordination sphere), and the calculations performed for these models in subcritical regimes (below the Curie point) indicate the adequacy of the proposed approach in general and good agreement with the known exact solution in a particular case of the two-dimensional square model with nearest-neighbor interactions. In addition, the quantitative similarity of subcritical magnetization curves is observed for all the considered models at the same values of a certain reduced temperature and the nature of this similarity may deserve further attentive studying.

On the integrability of the equations of dynamics in a non-potential force field

A range of issues related to the integration of the equations of motion of mechanical systems in non-potential force fields (often called circulatory systems) are discussed. The approach to integration is based on the Euler-Jacobi-Lie theorem: for exact integration of a system with $n$ degrees of freedom it is necessary to have $2n-2$ additional first integrals and symmetry fields (taking the conservation of the phase volume into account) which are in certain natural relations to one another. The cases of motion in non-potential force fields that are integrable by separation of variables are specified. Geometric properties of systems with non-Noether symmetry fields are discussed. Examples of the existence of irreducible polynomial integrals of the third degree in the momentum are given. The problem of conditions for the existence of single-valued polynomial integrals of circulatory systems with two degrees of freedom and toric configuration spaces is considered. It is shown that in a typical case the equations of motion do not admit non-constant polynomial integrals. Bibliography: 32 titles.

Approximation of monotone clothoid segments by degree 7 Pythagorean–hodograph curves

The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution.

High-order cluster integrals for the lattice-gas model

In the paper, for the known statistical lattice-gas model, a new method is proposed to define the high-order reducible cluster integrals on the basis of the Cauchy--Hadamard theorem and recently established strict information about the convergence radius of the virial expansions for pressure and density in powers of activity. Compared with previous studies (where the whole set of reducible cluster integrals were evaluated on the basis of the extremely limited number of low-order irreducible integrals), the proposed complex method to approximate the reducible cluster integrals yields the theoretical values of pressure at the saturation and boiling points considerably close to each other as well as to the Lee--Yang solution for the two-dimensional lattice gas.

Asymptotics of activity series at the divergence point

For statistical models of imperfect gases, a new method is proposed to evaluate the reducible cluster integrals of very high (actually unlimited) orders on the basis of information on the irreducible integrals (virial coefficients) or information on the corresponding radius of convergence for the virial series in powers of activity. This method is used to transform conventional expansions for pressure and density in powers of activity to a functional form that allows the analytical study of those series at the vicinity of their divergence point. In particular, the results of this study confirm the adequacy of the cluster-based approach at the condensation region and agree with the results of the previous studies of partition function in terms of irreducible integrals.

Divergence of activity expansions: Is it actually a problem?

For realistic interaction models, which include both molecular attraction and repulsion (e.g., Lennard-Jones, modified Lennard-Jones, Morse, and square-well potentials), the asymptotic behavior of the virial expansions for pressure and density in powers of activity has been studied taking power terms of high orders into account on the basis of the known finite-order irreducible integrals as well as the recent approximations of infinite irreducible series. Even in the divergence region (at subcritical temperatures), this behavior stays thermodynamically adequate (in contrast to the behavior of the virial equation of state with the same set of irreducible integrals) and corresponds to the beginning of the first-order phase transition: the divergence yields the jump (discontinuity) in density at constant pressure and chemical potential. In general, it provides a statistical explanation of the condensation phenomenon, but for liquid or solid states, the physically proper description (which can turn the infinite discontinuity into a finite jump of density) still needs further study of high-order cluster integrals and, especially, their real dependence on the system volume (density).

Counting the number of master integrals for sunrise diagrams via the Mellin-Barnes representation

A number of irreducible master integrals for L-loop sunrise and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via the Mellin-Barnes representation.

Kondensaciya gratkovoho hazu ta yiyi zv’yazok z rozbizhnistyu virial"nyx rozkladiv za stepenyamy aktyvnosti

An efficient algorithm for the calculation of high-order reducible cluster integrals on the basis of irreducible integrals (virial coefficients) has been proposed. The algorithm is applied to study the behavior of the well-known virial expansions of the pressure and density in power series of activity up to very high-order terms, as well as recently derived symmetric power expansions in the reciprocal activity, in the framework of a specific lattice gas model. Our results are consistent with those obtained in other modern studies of the partition function in terms of the density. They disclose the physical meaning of the divergence that the mentioned expansions demonstrate in the condensation region.

High-density equation of state for a lattice gas.

For the lattice gas models of arbitrary geometry and dimensions with absolute repulsion between particles at zero distance (a hard core identical to a single lattice site) and arbitrary repulsion or attraction at other distances, the "hole-particle" symmetry of the system potential energy has been stated and an equation of state has been derived on the basis of the classical Gibbs statistics. The equation is completely analogous to the well-known virial equation of state, except that it is more accurate at high-density states, while the virial equation has the low-density limitation. Both equations contain the common set of the so-called irreducible integrals, related to the corresponding virial coefficients, and can be used together to describe the behavior of a lattice gas in a wide range of densities.

Virial expansion and condensation with a new generating function

Mayer’s convergence method for virial expansion and condensation is studied using a new generating function for canonical partition function, which directly depends on irreducible cluster integral, βk, unlike Mayer’s work where it depends on reducible cluster integral, bl. The virial expansion, criteria for its validity and criteria for condensation, etc. are derived from our generating function. All earlier Mayer’s results are obtained from this new generating function.

Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space

We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.

Equation of State Beyond the Radius of Convergence of the Virial Expansion

The well-known problem of the virial expansion low-density limitation is considered within Mayer's cluster expansion method. The expression for the configuration integral and the corresponding equation of state are presented based on this approach but not limited by the convergence radius of the series for density and activity. When taking into account any number of irreducible integrals at the thermodynamic limit, this equation of state demonstrates the exact coincidence with the virial one inside the domain of its convergence but specifies the condensation process directly outside that domain. Thus, the assumption of some researchers that the condensation should appear in the domain where the proof of the virial expansion is limited may now be regarded as confirmed, exclusively using the classical Gibbs statistics.

Nonlinear Viscoelastic Fracture Mechanics Using Boundary Elements

The boundary element methodology is applied to the fracture mechanics of non-linear viscoelastic solids. The adopted non-linear model is based on the ‘free volume’ concept, which is introduced into the relaxation moduli entering the linear viscoelastic relations through a time shift depending on the volumetric strain. Nonlinearity generates an irreducible domain integral into the original boundary integral equation governing the behaviour of linear viscoelastic solids. This necessitates the evaluation of domain strains, which relies on a non-standard differentiation of an integral with a strong kernel singularity. A time domain formulation based on constant shape functions over boundary elements and domain cells is computer-implemented through a numerical integration algorithm. The effectiveness of the developed numerical tool is demonstrated through the analysis of a plate with a central crack. The predicted stress field around the crack tip is compared with respective results obtained by the finite element method.

The Number of Master Integrals is Finite

For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.

The effective electroweak mixing angle [formula omitted] with two-loop fermionic contributions

We present the results from a calculation of the full electroweak two-loop fermionic contributions to the effective leptonic mixing angle of the Z boson, sin 2 θ eff , in the Standard Model. On-shell renormalization and analytic calculations are performed for the three-point vertex functions at zero external momenta, whereas irreducible three-point integrals for non-vanishing external momenta are evaluated semi-analytically applying two different methods. Comparisons with a previous calculation show complete agreement.

Statistical properties of limit of detection test statistics

There are a number of possible ways to define the instrumental limit of detection (LOD) figure of merit and in the present work we define four such sample test statistics and derive their probability density functions (PDFs). Although the derived PDFs were found to be irreducible integrals, they are easily evaluated via numerical integration and can be used to obtain expectation values, population precisions and confidence intervals. Monte Carlo simulation methods were used to prepare normalized histograms of one million LOD variates each, for homoscedastic linear calibration curve systems, and these were found to be in excellent agreement with the numerically obtained PDFs and associated statistics. The software used in all aspects of the present work is available for free, with full, commented source code, and is designed to facilitate exploration of calibration curve systems of immediate interest to the reader.

Billiard systems with polynomial integrals of third and fourth degree

The problem of the existence of polynomial-in-momenta first integrals for dynamical billiard systems is considered. Examples of billiards with irreducible integrals of third and fourth degree are constructed with the help of the integrable problems of Goryachev-Chaplygin and Kovalevsky from rigid body dynamics.

Progress towards 2 → 2 scattering at two loops

We discuss the two-loop integrals necessary for evaluating massless 2 → 2 scattering amplitudes. As a test process, we consider the leading colour two-loop contribution to q q → q′q q′ . We show that for physical scattering processes the two Smirnov-Veretin planar box graphs I 1 and I 2 are accompanied by factors of 1 (D−4) thereby necessitating a knowledge of both I 1 and I 2 to O(ϵ). Using an alternative basis I 1 and the irreducible numerator integral I 3, the factors of 1 (D−4) disappear.

Irreducible Hamiltonian BRST Approach to Topologically Coupled Abelian Forms

An irreducible Hamiltonian BRST approach to topologically coupled p- and (p + 1)-forms is developed. The irreducible setting is enforced by means of constructing an irreducible Hamiltonian first-class model that is equivalent from the BRST point of view to the original redundant theory. The irreducible path integral can be brought to a manifestly Lorentz covariant form.

A counterexample to an integer analogue of Carathéodory's theorem

is called an integral polyhedral cone generated by {z1, . . . , zk}. It is called pointed if the origin is a vertex of C and it is called unimodular if the set of generators {z1, . . . , zk} of C forms part of a basis of the lattice Z. By Gordan’s lemma the semigroup C ∩ Z is finitely generated for any integral polyhedral cone C, i.e., there exist finitely many vectors h, . . . , h such that every z ∈ C ∩ Z has a representation of the form z = ∑m i=1mih , mi ∈ Z≥0. It was pointed out by van der Corput [Cor31] that for a pointed integral polyhedral cone C there exists a uniquely determined minimal (w.r.t. inclusions) finite generating system H(C) of C ∩Zn which may be characterized as the set of all irreducible integral vectors contained in C. This set H(C) is usually called the Hilbert basis of C. Although Hilbert bases play a role in various fields of mathematics, their structure is not very well understood yet. A first systematic study was given by Sebo [Seb90]. In particular, the following three conjectures about the “nice” geometrical structure of Hilbert bases of an integral pointed polyhedral cone C ⊂ R are due to him: (Unimodular Hilbert Partitioning) There exist unimodular cones Ci, i ∈ I, generated by elements of H(C) such that i) C = ∪i∈ICi and ii) Ci∩Cj is a face both of Ci and Cj , i, j ∈ I. (Unimodular Hilbert Cover) There exist unimodular cones Ci, i ∈ I, generated by elements of H(C) such that C = ∪i∈ICi. (Integral Caratheodory Property) Each integral vector z ∈ C can be written as a nonnegative integral combination of at most n elements of H(C).