Exclusion Statistics Research Articles

BCS model with asymmetric pair scattering: a non-Hermitian, exactly solvable Hamiltonian exhibiting generalized exclusion statistics

We demonstrate the occurrence of free quasi-particle excitations obeying generalized exclusion statistics in a BCS model with asymmetric pair scattering. The results are derived from an exact solution of the Hamiltonian, which was obtained via the algebraic Bethe ansatz utilizing the representation theory of an underlying Yangian algebra. The free quasi-particle excitations are associated with highest weight states of the Yangian algebra, corresponding to a class of analytic solutions of the Bethe ansatz equations.

Fractional exclusion statistics: the concept and some applications

I present briefly the concept of fractional exclusion statistics (FES) and I analyze two models of interacting particle systems. After I show that the models are equivalent by transforming one into the other by redefining some terms in the Hamiltonians, I calculate the heat capacity of the system and compare it with the heat capacity of an ideal Bose or Fermi gas. I show that under certain conditions the heat capacities may not be equal. I show that the interacting particle system may always be described as an ideal gas obeying FES and I show the method to construct such a gas.

Fractional exclusion statistics in systems with localized states

We develop a model based on the fractional exclusion statistics to describe systems with localized states. The local distribution of the energy levels is captured in the formalism by including the positions in the definition of the species. The particle distributions on the energy axis, as well as in the real space are determined for test-case systems with a peak/dip profile in the local density of states.

Through the lens of home-educated children: engagement in education

Engagement and participation are important for successful outcomes in education, yet disaffection in the UK, according to some exclusion and absence statistics, shows a growing trend. The purpose of this research was to develop a starting point for a theory of children’s engagement in education using grounded theory method. Evidence from home-educating families suggests that disaffection is a problem that is unique to school education. The research investigated home educated children's perceptions and experiences of home education. A flexible, semi-structured, qualitative design, encouraging children to determine the structure of their narratives was employed, using photovoice. Analysis followed grounded theory method and the results highlight themes for future development, refinement and further investigation. Important findings are that perceptions of self, and flexible and supportive learning contexts that provide children with a sense of active and autonomous involvement are linked to learning engagement and enthusiasm for learning. The conclusion outlines reasons why educational psychologists should become involved in discussion and research relating to home education. Limitations of the research are discussed and issues for future research are considered.

Quantum Maximum Entropy Principle for Fractional Exclusion Statistics

Using the Wigner representation, compatibly with the uncertainty principle, we formulate a quantum maximum entropy principle for the fractional exclusion statistics. By considering anyonic systems satisfying fractional exclusion statistic, all the results available in the literature are generalized in terms of both the kind of statistics and a nonlocal description for excluson gases. Gradient quantum corrections are explicitly given at different levels of degeneracy and classical results are recovered when ℏ→0.

Finite temperature analytical results for a harmonically confined gas obeying exclusion statistics in d-dimensions

Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a d-dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile distribution continues to provide a good description of the system, with deviations only arising well outside of the degenerate regime. Our results illustrate that the exceedingly simple functional form of the Gentile distribution is an excellent alternative to the generally only implicit form of the Haldane distribution at low temperatures.

Classical and quantum correlations in an ideal gas of particles obeying fractional exclusion statistics

We study the correlations in a two component gas of non-interacting particles obeying fractional exclusion statistics characterized by a parameter α ∈ [0, 1] that interpolates between bosonic (α = 0) and fermionic (α = 1) statistics. We discuss the behaviour of the total correlations, entanglement, classical correlations and quantum discord in the two spin density matrix (‘spin’ here refers to the two internal degrees of freedom) as a function of the statistical parameter α and the relative distance between the two spins. We show that the spins are entangled only for statistics α > α ∗ = 2/3 (independent of the number of spatial dimensions) and provided that the relative distance between the spins is less than a certain critical value r e which depends on the Fermi wave number and the statistics parameter. However, the quantum discord is non-vanishing even in the absence of entanglement, vanishing (for zero temperature) only in the limit of very large relative separations, i.e., at distances where quantum statistics play no role.

Fractional exclusion statistics: the method for describing interacting particle systems as ideal gases

I show that if the total energy of a system of interacting particles may be written as a sum of quasiparticle energies, then the system of quasiparticles can be viewed, in general, as an ideal gas with fractional exclusion statistics (FES). The general method for calculating the FES parameters is also provided. The interacting particle system cannot be described as an ideal gas of Bose and Fermi quasiparticles except in trivial situations.

Deconfined quantum criticality and generalized exclusion statistics in a non-Hermitian BCS model

We present a pairing Hamiltonian of the Bardeen–Cooper–Schrieffer form which exhibits two quantum critical lines of deconfined excitations. This conclusion is drawn using the exact Bethe ansatz equations of the model which admit a class of analytic solutions. The deconfined excitations obey generalized exclusion statistics. A notable property of the Hamiltonian is that it is non-Hermitian. Although it does not have a real spectrum for all choices of coupling parameters, we provide a rigorous argument to establish that real spectra occur on the critical lines. The critical lines are found to be invariant under a renormalization group map.

Pauli Paramagnetic Susceptibility of an Ideal Anyon Gas within Haldane Fractional Exclusion Statistics

The finite-temperature Pauli paramagnetic susceptibility of a three-dimensional ideal anyon gas obeying Haldane fractional exclusion statistics is studied analytically. Different from the result of an ideal Fermi gas, the susceptibility of an ideal anyon gas depends on a statistical factor g in Haldane statistics model. The low-temperature and high-temperature behaviors of the susceptibility are investigated in detail. The Pauli paramagnetic susceptibility of the two-dimensional ideal anyons is also derived. It is found that the reciprocal of the susceptibility has the similar factorizable property which is exhibited in some thermodynamic quantities in two dimensions.

Adiabatic sound velocity and compressibility of a trapped d-dimensional ideal anyon gas

The adiabatic sound velocity and compressibility for harmonically trapped ideal anyons in arbitrary dimensions are calculated within Haldane fractional exclusion statistics. The corresponding low-temperature and high-temperature behaviors are studied in detail. To compare with the experimental result of unitary fermions, the sound velocity for anyons in the cigar-shaped trap is derived. The sound velocity for anyons in the disk-shaped trap is also calculated. With the parameter g=0.287, the sound velocity of cigar-shaped unitary fermions modeled by anyons is in good agreement with the experimental result, while that of disk-shaped unitary fermions is v0/vF=0.406 with Fermi velocity vF.

Generalized Pauli principle for particles with distinguishable traits

The s=3/2 Ising spin chain with uniform nearest-neighbor coupling, quadratic single-site potential, and magnetic field is shown to be equivalent to a system of 17 species of particles with internal structure. The same set of particles (with different energies) is shown to generate the spectrum of the s=1/2 Ising chain with dimerized nearest-neighbor coupling. The particles are free of interaction energies even at high densities. The mutual exclusion statistics of particles from all species is determined by their internal structure and encoded in a generalized Pauli principle. The exact statistical mechanical analysis can be performed for thermodynamically open or closed systems and with arbitrary energies assigned to all particle species. Special circumstances make it possible to merge two or more species into a single species. All traits that distinguish the original species become ignorable. The particles from the merged species are effectively indistinguishable and obey modified exclusion statistics. Different mergers may yield the same end product, implying that the inverse process (splitting any species into subspecies) is not unique. In a macroscopic system of two merged species at thermal equilibrium, the concentrations of the original species satisfy a functional relation governed by their mutual statistical interaction. That relation is derivable from an extremum principle. In the Ising context the system is open and the particle energies depend on the Hamiltonian parameters. Simple models of polymerization and solitonic paramagnetism each represent a closed system of two species that can transform into each other. Here they represent distinguishable traits with different energies of the same physical particle.

Economic Downturn and the Improvement of Monetary Poverty and Social Exclusion Statistics System

The economic crisis has also challenged statistics. This article deals with the consequences of the economiccrisis for social statistics, and particularly monetary poverty indicators, which in this situation have shown certain signs ofweakness. The authors will discuss possible alternatives for the calculation of these indicators, as well as present theexperience of Statistics Latvia when reacting to the challenges raised by the economic crisis in the field of social statistics.

Fractional exclusion statistics applied to relativistic nuclear matter

The effect of statistics of the quasiparticles in the nuclear matter at extreme conditions of density and temperature is evaluated in the relativistic mean-field model generalized to the framework of the fractional exclusion statistics (FES). In the model, the nucleons are described as quasiparticles obeying FES and the model parameters were chosen to reproduce the ground state properties of the isospin-symmetric nuclear matter. In this case, the statistics of the quasiparticles is related to the strengths of the nucleon–nucleon interaction mediated by the neutral scalar and vector meson fields. The relevant thermodynamic quantities were calculated as functions of the nucleons density, temperature and fractional exclusion statistics parameter α. It has been shown that at high temperatures and densities the thermodynamics of the system has a strong dependence on the statistics of the particles. The scenario in which the nucleon–nucleon interaction strength is independent of the statistics of particles was also calculated, but it leads in general to unstable thermodynamics.

Joule-Thomson coefficient of ideal anyons within fractional exclusion statistics

The analytical expressions of the Joule-Thomson coefficient for homogeneous and harmonically trapped three-dimensional ideal anyons which obey Haldane fractional exclusion statistics are derived. For an ideal Fermi gas, the Joule-Thomson coefficient is negative, which means that there is no maximum Joule-Thomson inversion temperature. With careful study, it is found that there exists a Joule-Thomson inversion temperature in the fractional exclusion statistics model. Furthermore, the relations between the Joule-Thomson inversion temperature and the statistical parameter g are investigated.

Exclusion statistics for quantum Hall states in the Tao–Thouless limit

We consider spin-polarized Abelian quantum Hall states in theTao–Thouless limit, i.e. on a thin torus. For any filling factorν = p/q a well-defined sector of low energy states is identified and the exclusionstatistics of the excitations is determined. We study numerically, at and nearν = 1/3 and2/5, how the low energy states develop as one moves away from the Tao–Thouless limittowards the physical regime. We find that the lowest energy states in the physical regimedevelop from states in the low energy sector but that the exclusion statistics ismodified.

Thermodynamic geometry of fractional statistics

We extend our earlier study about the fractional exclusion statistics to higher dimensions in full physical range and in the nonrelativistic and ultrarelativistic limits. Also, two other fractional statistics, namely, Gentile and Polychronakos fractional statistics, will be considered and similarities and differences between these statistics will be explored. Thermodynamic geometry suggests that a two dimensional Haldane fractional exclusion gas is more stable than higher dimensional gases. Also, a complete picture of attractive and repulsive statistical interaction of fractional statistics is given. For a special kind of fractional statistics, by considering the singular points of thermodynamic curvature, we find a condensation for a nonpure bosonic system which is similar to the Bose-Einstein condensation and the phase transition temperature will be worked out.

Stochastic simulations for the time evolution of systems which obey generalized statistics:fractional exclusion statistics and Gentile’s statistics

We present a stochastic method for the simulation of the time evolution in systems whichobey generalized statistics, namely fractional exclusion statistics and Gentile’sstatistics. The transition rates are derived in the framework of canonical ensembles.This approach introduces a tool for describing interacting fermionic and bosonicsystems in non-equilibrium as ideal FES systems, in a computationally efficientmanner. The two types of statistics are analyzed comparatively, indicating theirintrinsic thermodynamic differences and revealing key aspects related to the speciessize.

A Quantum Boltzmann Equation for Haldane Statistics and Hard Forces; the Space-Homogeneous Initial Value Problem

The paper considers equations of Boltzmann type for Haldane exclusion statistics. Existence and some basic properties of the solutions are studied for the space homogeneous initial value problem with hard forces and angular cut-off. The approach uses strong L1 compactness. Some of the technical estimates are based on L∞ decay properties, and the control of the filling factor on range estimates for the solutions.

Realization of effective super Tonks-Girardeau gases via strongly attractive one-dimensional Fermi gases

A significant feature of the one-dimensional super Tonks-Girardeau gas is its metastable gas-like state with a stronger Fermi-like pressure than for free fermions which prevents a collapse of atoms. This naturally suggests a way to search for such strongly correlated behavior in systems of interacting fermions in one dimension. We thus show that the strongly attractive Fermi gas without polarization can be effectively described by a super Tonks-Girardeau gas composed of bosonic Fermi pairs with attractive pair-pair interaction. A natural description of such super Tonks-Girardeau gases is provided by Haldane generalized exclusion statistics. In particular, they are equivalent to ideal particles obeying more exclusive statistics than Fermi-Dirac statistics.