Based on the framework established in a recent work (Ubriaco in Physica A 414:128, 2014), wherein a generalized bosonic model was formulated using the Dunkl operator formalism (DOF), we extend this approach to its different applications with introducing a statistical distribution—termed the Dunkl-deformed Bose–Einstein (DDBE) distribution—constructed by generalizing the usual Bose–Einstein distribution through the DOF. In this framework, we study further general thermodynamical and statistical properties of a gas model of the Dunkl-deformed bosons obeying the DDBE statistics. Emphasis is given to a careful analysis on the conditions under which the Bose–Einstein-like condensation would occur in the present Dunkl-deformed boson gas. It is shown that the critical temperature for such a system is higher than that of the usual (undeformed) Bose gas for values of the model deformation parameter θ in the range θ > 0 . As another application of the DDBE statistics, we investigate the thermostatistics of the Dunkl-deformed Debye solid and the effects of the deformation parameter θ on the high- and low-temperature behavior of the model specific heat are discussed. Our results obtained in this study could provide much physical insight for approaching not only in the treatment of interacting bosons theories including collective excitations, but also in the condensation characteristics of quantum systems beyond the standard formalism.

A q-deformed field theory at finite temperature is presented and its related thermo field dynamics is constructed. This enables us to introduce the quon field, whose Lagrangian leads to a q-deformed partition function. Accordingly, the general thermostatistical properties of a gas model of quon fields, such as the q-deformed statistical distribution function describing an intermediate-statistics behavior and the equation of state in two and three dimensions, are investigated. For low temperatures, the conditions under which Bose-Einstein-like condensation would occur in the quon gas model are discussed. It is shown that the critical temperature of such a gas is higher than that of the usual Bose gas for values of the model parameter q in the range q<1. For high temperatures, possible anyonic behavior of the present quon gas model in two spatial dimensions is studied through an analysis of the effect of deformation on the second and third virial coefficients in the equation of state. The results obtained in this work reveal the ability of the model for analyzing a parastatistical behavior of systems with quasiparticles and put forward it as a possible candidate for effectively modeling the properties of some exotic quantum states, such as in the case of dark matter constituents.

After introducing the algebraic and representative properties of the extended Weyl–Heisenberg algebra along with its special case containing the Wigner algebra, a new deformed boson algebra involving the reflection operator is proposed. Such an algebra differs from the other well-known deformed particle algebras in the literature so that its number operator spectrum constitutes a novel deformed number called ν -number, which is changed for both even n and odd n in its energy levels. As physical applications of the quantum algebra developed here, the thermostatistical properties of a gas model of the ν -deformed bosons and the blackbody radiation phenomena covering the ν -deformed photon gas are studied in detail. Another application is carried out onto lattice oscillations via the Debye crystal model containing the ν -deformed phonons. The effects of ν -deformation onto the low-temperature behavior of the model specific heat of the ν -deformed Debye solid are explored and are compared with the results of both the standard phonon gas and the ones with the non-extensive Tsallis statistics. Finally, possible implications of our results on other application areas of research such as strongly correlated quantum matter and star formation in cosmological systems are concisely discussed.

We introduce a specific deformed Bose gas model, whose underlying quasiparticle algebra is related to the κ-deformed bosonic oscillator algebra. We then develop the statistical distribution function of a gas model of the κ-deformed bosons containing finite and infinite dimensional cases. We investigate interpolating statistics behavior of this deformed model and apply it to lattice oscillations via the Debye crystal model. The effect of the deformation parameter κ onto the low-temperature behavior of the model specific heat is discussed and is compared with the results of both the standard phonon gas and the ones with the Tsallis non-extensive statistics. Another application is carried out onto the Bose-like condensation of this deformed model and the conditions under which the κ-deformed boson condensation would occur in such a system are discussed. It is shown that the critical temperature of the κ-deformed boson gas with the infinite dimensional case is higher than that of the ideal Bose gas, while it has lower values than those of the ideal Bose gas for the finite dimensional case. We consider that the results obtained in this work may provide much physical insight into further studies on strongly correlated quantum materials as well as interacting theories of bosons including collective excitations, where unconventional quantum statistics might have an important role.

In this paper, we consider a system of the q-deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q-calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q-derivative, q-integral and q-exponential function are introduced. Using these properties, we construct a new formalism for the q-deformed quantum mechanics, which accordingly involve the q-adjoint operator and the q-Hermitian operator properties. We then derive the q-deformed Heisenberg relation, and develop the q-Hermitian momentum operator. The q-deformed Schrodinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q-deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.

Starting on the basis of Fibonacci calculus and Fibonacci oscillator algebra, we introduce the main properties to develop a new formalism for the two-parameter (q_1,q_2) -deformed quantum mechanics, where q_1 and q_2 are real positive independent deformation parameters. As applications of such a two-parameter deformed formalism, we investigate the behavior of a quantum particle in some different physical phenomena covering the free particle and the inverse-harmonic potential case. The effect of two deformation parameters on the wave functions for these applications is studied. Another application is carried out onto the quantum statistics of lattice oscillations through a model of the (q_1,q_2) -deformed phonon gas, and it is shown that the high- and low-temperature behavior of the model specific heat differs notably from the classical theories for the interval 0<(q_1,q_2)<∞ . We also construct a two-parameter deformed non-extensive entropy based on some elements of the Fibonacci calculus and discuss its possible connection with the Tsallis entropy in non-extensive statistical mechanics. Finally, other possible application areas of the present two-parameter (q_1,q_2) -deformed construction on quantum mechanics are discussed.

In this work, after introducing the main algebraic and representative properties of the quantum mechanics with Dunkl derivative, two-particle system with exchange symmetry is studied. In this framework, a new deformed derivative involving the exchange operator is proposed. With the help of such an operator, the new deformed quantum theory for systems with two quantum particles in one dimension is developed. As physical applications of the present construction, systems with two quantum particles interacting via the harmonic potential and the delta-function potential well are considered in detail. The wave functions and the energy spectra for these potentials are derived. Finally, possible implications of our results on other application areas of research such as in interacting theories of either bosons or fermions are concisely discussed.

In this study, we consider a deformed gas of the bosonic Fibonacci oscillators, whose properties enable us to develop both an intermediate-statistics behavior and an energy spectrum given by a generalized Fibonacci sequence in terms of the real independent deformation parameters q_1 and q_2 . By viewing photons and phonons as the particles obeying the commuting Fibonacci oscillator algebra, we extensively discuss several thermodynamical and statistical properties of the blackbody radiation and Debye crystal model. We then investigate possible roles of the deformation parameters q_1 and q_2 on the thermostatistics of such deformed photon and deformed phonon gases. Finally, we compare our results with the results of both the non-extensive Tsallis statistics and the usual (undeformed) Bose gas case. The approach presented here could give new insights for further studies of deformed quantum thermodynamics and its potential applications such as in understanding the properties of early universe as well as modeling a possible candidate of dark matter.

In this work, the general procedure for obtaining the superstatistical density of states, where the probability density function has the nonzero variance, is first developed. We then discuss the microcanonical ensemble based on the superstatistics with the free Hamiltonian as a stochastic variable. Finally, some applications of the formalism presented here are analyzed in detail within the framework of different probability distributions such as three states distribution, the Gamma distribution, the q-deformed Dirac delta distribution and the Poisson distribution.

The multi-dimensional q-deformed bosonic Newton oscillator algebra with SU(d)-symmetry is considered. In this framework, we first introduce some new properties concerning the q-deformed calculus related to the algebra, and we then discuss possible consequences of applying these deformed oscillators in some quantum optical issues such as in the construction of coherent states and their effects on the photon statistics. Second, we investigate the role of q-deformation on both the energy levels and the wave functions of the bosonic Newton oscillators by constructing the q-deformed Hermite polynomials. The results obtained in this work might have some implications for studies on quantum information based technologies such as in photonic quantum computing.

In this work, we present an approach to describe imprecise probability through an effective probability theory, called the f-probability. We develop a bijective and monotonous map from the precise probability in order to construct the f-probability theory based on the f-addition, f-subtraction, f-multiplication and f-division. We apply the f-probability to the Bernoulli trial and derive the f-binomial distribution. Finally, we obtain the non-extensive entropy through the f-probability theory, and give its statistical physical implications on several areas of potential applications. PACS number(s): 02.50.Cw, 05.20.-y; 05.90.+m

In this work, we present a new algebraic modelby constructing the modified multi-dimensional q-deformed bosonic and fermionic Newton oscillator algebras. This construction leads effectively to describe interpolating statistics, which can be used to approach the properties of quasi-particle excitations occurring in many-body interacting quantum systems. It is shown that the model algebras are endowed with the SUφ(d)- and SUθ(d)-symmetries, where φ and θ are real parameters describing the special phase shifts between two different modes of the models. Particular emphasis is given to the two-dimensional case, which reveals a suitable framework for analyzing anyonic behavior of the models. Furthermore, we discuss the effects of deformation on the general thermodynamical and statistical properties of gas models of these interpolating statistics particles such as the q-deformed statistical distribution function and the equation of state in two and three dimensions. Finally, we concisely point out other possible physical applications of the present deformed (quasi)particle models.

In this study, we first introduce a new quasi-particle algebra, which enables us to effectively describe a unified framework for both bosons and fermions. We then study general thermodynamical and statistical properties of a hybrid-type gas model of these quasi-particles. In this context, we specifically focus on the conditions under which the hybrid-type quasi-particle gas condensation would occur in the present model. The results obtained in this work reveal that the present gas model of quasi-particles can be used to approximate non-linear behavior observed in composite particle systems such as in studies on the phenomenon of high-Tc superconductivity in a given material.

We study the low and high temperature thermostatistical properties of a deformed boson gas constructed by the bosonic intermediate-statistics particles confined in low spatial dimensions. Many of the deformed thermodynamical functions of the system such as internal energy and entropy are investigated by means of some elements of the Fibonacci calculus. Particular emphasis is given to a careful analysis on low dimensional systems of such deformed bosons, and the conditions under which the Bose–Einstein condensation would occur in such systems are discussed. We show that low dimensional systems with deformed bosons exhibit the Bose–Einstein condensation for values of the model deformation parameters (p,q) greater than one. We also study possible anyonic behavior of the model for high temperatures. The results obtained in this work reveal that the present deformed boson gas model can be used for modeling nonlinear behavior of systems with quasiparticles encountered in several areas of research particularly in quantum science.

In this work, we propose a new model for describing an intermediate-statistics particles system. Starting with a deformed grand partition function, we investigate several thermodynamical and statistical properties of a gas model of two-parameter deformed particles. We specifically focus on the low-temperature behavior of the model and the conditions under which either boson condensation or fermion condensation would occur in such a model are discussed. Our results obtained in this study reveal that the present deformed gas model exhibits duality of boson and fermion, and can be useful for approaching the thermostatistics of condensation characteristics in quantum systems.

In this work, we first introduce some new properties concerning the Fibonacci calculus. We then discuss the thermostatistics of gas models of two-parameter deformed oscillators, called bosonic and fermionic Fibonacci oscillators, in the thermodynamical limit. In this framework, we analyze the behavior of two-parameter deformed mean occupation numbers describing the Fibonacci-type bosonic and fermionic intermediate-statistics particles. A virial expansion of the equation of state for the bosonic Fibonacci oscillators’ gas model is obtained in both two and three dimensions, and the first five virial coefficients are derived in terms of the real independent deformation parameters [Formula: see text] and [Formula: see text]. The effect of bosonic and fermionic [Formula: see text], [Formula: see text]-deformation on the thermostatistical properties of Fibonacci-type [Formula: see text], [Formula: see text]-boson and [Formula: see text], [Formula: see text]-fermion gas models are also discussed. The results obtained in this work can be useful for investigating some exotic quasiparticle states encountered in condensed matter systems.

A deformed fermion gas model aimed at taking into account thermal and electronic properties of quasiparticle systems is devised. The model is constructed by the fermionic Fibonacci oscillators whose spectrum is given by a generalized Fibonacci sequence. We first introduce some new properties concerning the Fibonacci calculus. We then investigate the low-temperature thermostatistical properties of the model, and derive many of the deformed thermostatistical functions such as the chemical potential and the entropy in terms of the model deformation parameters p and q. We specifically focus on the p,q-deformed Sommerfeld parameter for the heat capacity of the model, and its behavior is compared with those of both the free-electron Fermi theory and the experimental data for some materials. The results obtained in this study reveal that the present deformed fermion model leads to an effective approach accounting for interaction and compositeness of quasiparticles, which have remarkable implications in many technological applications such as in nanomaterials.

Starting with a deformed fermionic grand partition function, we study the high and low temperature thermostatistical properties of a special q-deformed fermion gas in two spatial dimensions. Many of the deformed thermostatistical functions such as the specific heat and the entropy are derived in terms of the real deformation parameter q for the range q < 1. For high temperatures, we specifically focus on the behavior of both the entropy function and the deformed virial coefficients in the equation of state for the q-fermion gas in two dimensions. Possible physical applications of the present q-fermion gas are briefly discussed.

This special volume of Journal of Physics: Conference Series is dedicated to the proceedings of "International Conference on Quantum Science and Applications (ICQSA-2016)". The conference was organized by the Centre for Quantum Research and Applications at Eskisehir Osmangazi University, Eskisehir, Turkey. It was held in Eskisehir Osmangazi University Congress and Culture Centre during May 25-27, 2016 http://icqsa2016.ogu.edu.tr. It gathered actively 143 participants from different disciplines in natural and applied sciences coming from 16 different countries from all over the word. It was the first international conference in its content on the scientific research fields of quantum science and applications in Turkey. It also consisted of 12 plenary lectures and 119 contributed oral presentations covering interdisciplinary fields of research.

A fermionic deformation scheme is applied to a study on the low-temperature quantum statistical behavior of a quasifermion gas model with intermediate statistics. Such a model does not satisfy the Pauli exclusion principle, and its quantum statistical properties are based on a formalism of the fermionic q-calculus. For low temperatures, several thermostatistical functions of the model such as the chemical potential, the heat capacity, and the entropy are derived by means of a function of the model deformation parameter q. The effect of fermionic q-deformation on the low-temperature thermostatistical properties of the model are discussed in detail. Our results show that the present deformed (quasi)fermion model provides remarkable connections of the model deformation parameter q, first, with the thermal effective mass of a quasiparticle, and second, with the temperature parameter. Hence, it turns out that the model deformation parameter q has also a role controlling the strength of effective quasiparticle interactions in the model. Finally, we conclude that this work can be useful for understanding the details of interaction mechanism of fermions such as quasiparticle states emergent in the fractional quantum Hall effect.