Nonextensive thermostatistics and deformed structures

We obtain a generalized Planck law within the framework of nonextensive statistics making use of a deformed oscillator system. Our results are used to fit the data from the COBE (Cosmic Background Explorer) satellite. Best fit values for the entropy parameter q, the deformation parameter r and for the temperature are found.     

q-Deformation of symplectic dynamical symmetries in algebraic models of nuclear structure

With a view toward further nuclear structure applications of approaches based on quantum-deformed (or q-deformed) algebras, introduced to the authors by Yu.F. Smirnov, we construct a q analog of a boson realization of the symplectic noncompact sp(4, R) algebra together with a q analog of a fermion realization of the symplectic compact sp(4) algebra. The first study, on the q-deformed Sp(4,R) symmetry, is applied to the development of a q analog of the two-dimensional Interacting Boson Model with q-deformed SU(3) the underpinning dynamical symmetry group. An explicit realization in terms of q-tensor operators with respect to the standard su _q (2) algebra is given. The group-subgroup structure of this framework yields the physical interpretation of the generators of the groups under consideration. The second symplectic algebra, the q-deformed sp(4), is applied to studying isovector pairing correlations in atomic nuclei. A specific q deformation of the sp(4) algebra is realized in terms of q deformed fermion creation and annihilation operators of the shell model. The generators of the algebra close on four distinct realizations of the u _q (2) subalgebra. These reductions, which correspond to different types of pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q deformation is based on a comparison of the results for energies of the lowest isovector-paired 0⁺ states in the deformed and nondeformed cases.     

Propagator for the Chebyshev q object

We propose an alternative role of the harmonic oscillator algebra. Observing that the q-deformed harmonic oscillator algebra defines the Chebyshev q object, we show that the q-free particle and the pulsed oscillator are special cases of the Chebyshev q object, characterized by a common deformation parameter q and reduced to a usual free particle as q tends to unity. For the deformed free particle, q is a real number, whereas for the pulsed oscillator it belongs to S ¹. Then, we derive the propagator for the Chebyshev q object, from which we obtain the propagators for the deformed free particle and the pulsed oscillator.     

Thermodynamic consistency of the q-deformed Fermi-Dirac distribution in nonextensive thermostatics

The q-deformed statistics for fermions arising within the nonextensive thermostatistical formalism has been applied to the study of various quantum many-body systems recently. The aim of the present note is to point out some subtle difficulties presented by this approach in connection with the problem of thermodynamic consistency. Different possible ways to apply the q-deformed quantum distributions in a thermodynamically consistent way are considered.     

Stability of incomplete entropy and incomplete expectation

Based on the principle of maximum entropy, the q-exponential distribution can be derived from several different nonextensive entropies including the incomplete entropy. It is widely used in nonextensive statistical mechanics. In the present paper, it is shown that the incomplete expectation value and incomplete entropy are stable under small deformation of the probability distribution function of the system.     

Tsallis, Rényi and nonextensive Gaussian entropy derived from the respective multinomial coefficients

We explore the generalization of the ordinary multinomial coefficient, based on the deformed q-multiplication and q-division. Aim of this study is to construct the appropriate multinomial coefficients, from which one can obtain the Tsallis, Rényi and nonextensive Gaussian entropy, respectively. We show that for all three above entropies there are two possible ways to define the generalized multinomial coefficient. Its consequence is discussed.     

A family of nonextensive entropies

A generalized nonextensive two-parameter entropy is developed, along lines which unify current nonextensive frameworks. It recovers, as particular cases, the Tsallis and symmetric entropies, as well as the Boltzmann-Gibbs entropy. The properties of the new (q, q′)-entropy are analysed.     

Nonextensive entropies derived from Gauss? principle

Gauss? principle in statistical mechanics is generalized for a q-exponential distribution in nonextensive statistical mechanics. It determines the associated stochastic and statistical nonextensive entropies which satisfy Greene-Callen principle concerning on the equivalence between microcanonical and canonical ensembles.     

Equilibrium statistical mechanics for incomplete nonextensive statistics

The incomplete nonextensive statistics in the canonical and microcanonical ensembles is explored in the general case and in a particular case for the ideal gas. By exact analytical results for the ideal gas it is shown that taking the thermodynamic limit, with z=q/(1−q) being an extensive variable of state, the incomplete nonextensive statistics satisfies the requirements of equilibrium thermodynamics. The thermodynamical potential of the statistical ensemble is a homogeneous function of the first degree of the extensive variables of state. In this case, the incomplete nonextensive statistics is equivalent to the usual Tsallis statistics. If z is an intensive variable of state, i.e. the entropic index q is a universal constant, the requirements of the equilibrium thermodynamics are violated.     

Bipartite entanglement within the framework of real and ideal lasers

We study the nonlocal correlations and quantum entanglement for two deformed bosonic fields of arbitrary deformation parameters, q ₁ and q ₂, prepared in an entanglement of deformed coherent states. As a measure of entanglement, we use the von Neumann entropy and investigate its behavior for different strength regimes of the optical fields. We find that the photon number can enhance the von Neumann entropy, and the deformation parameters can restrain the system entanglement.     

Deformed fermion realization of sp(4) and its reductions

A specific q-deformation of the compact symplectic sp(4) algebra, one that is suitable for nuclear physics applications, is realized in terms of q-deformed fermion creation and annihilation operators of the shell-model. The generators of the algebra close on four distinct realizations of the u _q (2) subalgebra. These reductions, which correspond to different pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q-deformation is based on a comparison of the results for energies of the lowest isovector-paired 0⁺ states in the deformed and non-deformed cases.     

A q-deformation of the Bogoliubov transformations

An approach for q-deformed Bogoliubov transformations is presented. Assuming a left-right module action together with an ?-operation and deformed commutation relations, we construct a q-deformation of the nonlinear Bogoliubov transformation. Finally, we introduce a Hopf structure when q is a root of unity.     

Spin chains and classical strings in two parameters q-deformed AdS3 × S3

In this paper, we study the spin chain and string excitation in the two-parameters q-deformed AdS₃ × S³ proposed by Hoare [6]. We obtain the deformed spin chain model at the fast spin limit for choices of deformed parameters. General ansatz for giant magnons are studied in great detail and complicated dispersion relation is treated perturbatively. We also study several types of hanging string solutions and their charges and spins are analyzed numerically. At last, we explore its pp-wave limit and find its solution only depends on the difference of deformed parameters.     

f-oscillators deformation for Moyal algebras

Using general construction of star-product the q-deformed Wigner-Weyl-Moyal quantization procedure is elaborated. The q-deformed Groenewold kernel determining the product of quantum observables is given in explicit form for small nonlinearities corresponding to nonlinear vibrations of classical and quantum q-oscillators. The deformation of Groenewold kernel related to general kinds of nonlinear vibrations described by f-oscillators are considered.     

Bohmian formalism and Fisher information from q-deformed Schrödinger equation

We discuss the Bohmian mechanics using a deformed Schrödinger equation for position-dependent mass systems, in the context of a q-algebra inspired by the nonextensive statistical mechanics. We obtain the Bohmian quantum formalism by means of a deformed version of the Fisher information functional, from which a deformed Cramér–Rao bound is derived. Lagrangian and Hamiltonian formulations, inherited by the q-algebra, are also developed. Then, we illustrate the results with a particle confined in an infinite square potential well. The preservation of the deformed Cramér–Rao bound for eigenstates shows the role played by the q-algebraic structure.     

Nonextensive formalism and continuous Hamiltonian systems

A recurring question in nonequilibrium statistical mechanics is what deviation from standard statistical mechanics gives rise to non-Boltzmann behavior and to nonlinear response, which amounts to identifying the emergence of “statistics from dynamics” in systems out of equilibrium. Among several possible analytical developments which have been proposed, the idea of nonextensive statistics introduced by Tsallis about 20 years ago was to develop a statistical mechanical theory for systems out of equilibrium where the Boltzmann distribution no longer holds, and to generalize the Boltzmann entropy by a more general function Sq while maintaining the formalism of thermodynamics. From a phenomenological viewpoint, nonextensive statistics appeared to be of interest because maximization of the generalized entropy Sq yields the q-exponential distribution which has been successfully used to describe distributions observed in a large class of phenomena, in particular power law distributions for q>1. Here we re-examine the validity of the nonextensive formalism for continuous Hamiltonian systems. In particular we consider the q-ideal gas, a model system of quasi-particles where the effect of the interactions are included in the particle properties. On the basis of exact results for the q-ideal gas, we find that the theory is restricted to the range q<1, which raises the question of its formal validity range for continuous Hamiltonian systems.     

Dynamical symmetries inq-deformed quantum mechanics

The dynamical algebra of theq-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a welldefined algebraSU _q(1,1) with consistent conjugation properties and comultiplication. We obtain non lowest weight representations of this algebra.     

A Two-Parameter Deformed N = 2 SUSY Algebra

A two-parameter deformed N = 2 SUSY algebra is constructed by using the q-deformed bosonic and fermionic Newton oscillator algebras. The Fock space representation of the (q ₁,q ₂)-deformed N = 2 SUSY algebra is analyzed. The comparison between the algebra constructed and earlier versions of deformed N = 2 SUSY algebras is also made.     

Non-Gaussian effects on quantum entropies

A deduction of generalized quantum entropies within the non-Gaussian frameworks, Tsallis and Kaniadakis, is derived using a generalized combinatorial method and the so-called q and κ calculus. In agreement with previous results, we also show that for the Tsallis formulation the q-quantum entropy is well-defined for values of the nonextensive parameter q lying in the interval [0,2].