Doğan Demirhan

Some bounds upon the nonextensivity parameter using the approximate generalized distribution functions

Abstract In this study, the approximate generalized quantal distribution functions and their applications which appeared in the literature so far, have been summarized. Making use of the generalized Planck radiation law, which has been obtained by us [Physica A 240 (1997) 657], some alternative bounds for the nonextensivity parameter q have been estimated. It has been shown that these results are similar to those obtained by Tsallis et al. [Phys. Rev. E 52 (1995) 1447] and by Plastino et al. [Phys. Lett. A 207 (1995) 42].

Generalized distribution functions and an alternative approach to generalized Planck radiation law

In this study, recently introduced generalized distribution functions are summarized and by using one of these distribution functions, namely generalized Planck distribution, an alternative approach to the generalized Planck law for the blackbody radiation has been tackled. The expression obtained is compared with the expression given by C. Tsallis et al. [Phys. Rev. E 52 (1995) 1447], and it is found that this approximate scheme provides bounds to the exact result, depending on the values of q-index.

A unified grand canonical description of the nonextensive thermostatistics of the quantum gases: Fractal and fractional approach

In this paper, the particles of quantum gases, that is, bosons and fermions are regarded as g-ons which obey fractional exclusion statistics. With this point of departure the thermostatistical relations concerning the Bose and Fermi systems are unified under the g-on formulation where a fractal approach is adopted. The fractal inspired entropy, the partition function, distribution function, the thermodynamics potential and the total number of g-ons have been found for a grand canonical g-on system. It is shown that from the g-on formulation; by a suitable choice of the parameters of the nonextensivity q, the parameter of the fractional exclusion statistics g, nonextensive Tsallis as well as extensive standard thermostatistical relations of the Bose and Fermi systems are recovered.

Time fractional development of quantum systems

In this study, the effect of time fractionalization on the development of quantum systems is taken under consideration by making use of fractional calculus. In this context, a Mittag–Leffler function is introduced as an important mathematical tool in the generalization of the evolution operator. In order to investigate the time fractional evolution of the quantum (nano) systems, time fractional forms of motion are obtained for a Schrodinger equation and a Heisenberg equation. As an application of the concomitant formalism, the wave functions, energy eigenvalues, and probability densities of the potential well and harmonic oscillator are time fractionally obtained via the fractional derivative order α, which is a measure of the fractality of time. In the case α=1, where time becomes homogenous and continuous, traditional physical conclusions are recovered. Since energy and time are conjugate to each other, the fractional derivative order α is relevant to time. It is understood that the fractionalization of...

Generalization of the mean-field Ising model within Tsallis thermostatistics

In this study, the mean-field Ising model, using the Bogolyubov inequality which has been obtained in the framework of the generalized statistical thermodynamics (GST), suitable for non-extensive systems, has been investigated. Generalized expressions for the mean-field magnetization and free energy have been established. These new results have been verified by the fact that they transform to the well-known Boltzmann-Gibbs results in the q → 1 limiting case. For the index q which characterizes the fractal structure of the magnetic system, an interval has been established where the generalized mean-field free energy has a minimum and mean-field magnetization has a corresponding finite value. The interval of q is consistent with paramagnetic free spin systems.

Effect of YBa2Cu3O7-x thin films on the textures and orientational properties of liquid crystals

Abstract.In the present work, the effect of thin films of YBa2Cu3O7-x complex compound on the mesomorphic and orientational properties of thermotropic nematic liquid crystals has been studied. Homogeneous, stable and reproducible homeotropic and tilted oriented textures of nematic liquid crystals were obtained. The effect of YBa2Cu3O7-x thin films on the morphologic, orientational and optical properties of thermotropic nematic liquid crystals are discussed.

Investigation of the pseudo-harmonic oscillator by su(1, 1) spectrum generating algebra

Abstract The discrete and continuous energy spectra of the pseudo-harmonic oscillator (PHO) have been investigated by the spectrum generating algebra. The PHO potential has been obtained depending on the dimension and the angular momentum. The zero-point energy of the two-dimensional PHO is found to be a minimum. The results are consistent with those obtained by other methods.

A statistical mechanical investigation of the pseudo-harmonic oscillator

Abstract The pseudo-harmonic oscillator whose potential is V = Ax −2 + Bx 2 is considered to be in contact with a heat source of temperature T . The vibrational partition function, mean energy, heat capacity, Helmholtz free energy and entropy are calculated. The effect of the potential shape parameter α=(8μ/ℏ 2 ) A on the physical quantities is investigated. The change of thermodynamic quantities with temperature is similar to that of the Schrodinger harmonic oscillator and does not depend on α.

Investigation of cumulative growth process via Fibonacci method and fractional calculus

In this study, cumulative growth of a physical quantity with Fibonacci method and fractional calculus is handled. The development of the growth process is described in terms of Fibonacci numbers, Mittag-Leffler and exponential functions. A compound growth process with the contribution of a constant quantity is also discussed. For the accumulation of residual quantity, equilibrium and lessening cases are discussed. To the best of our knowledge; compound growth process is solved for the first time in the framework of fractional calculus. In this sense, differintegral order of fractional calculus α has been achieved a physical content. It is emphasized that, in the basis of qualification of the fractional calculus for describing genuine complex physical systems with respect to ordinary descriptions is the cumulative growth mechanism with Fibonacci method. It is concluded that compound diminution and growth process mechanisms can be taken as a basis for the comprehension of derivative and integral operations in fractional calculus.

Investigation of the Bose?Einstein condensation based on fractality using fractional mathematics

Although atomic Bose gases are investigated in the dilute gas regime, the physical properties of the Bose–Einstein condensates are affected by interparticle interactions and the fractal nature of the space where the Bose systems are evolving. Theoretical predictions of the traditional Bose–Einstein thermostatistics do not account for the deviations from the experimental results, which are related to internal energy, specific heat, transition temperature, etc. On the other hand, in this study, fractional calculus is introduced where effects of the fractality of space are taken into account. Meanwhile, the order of the fractional derivative α is handled as a measure of the fractality of space. In this content, some improvements which take into account the effects of the fractal nature of the system are made in the standard physical results of the Bose–Einstein condensation phenomena. As an example, for the dilute atomic gas 7Li, the measured transition temperature of Bose–Einstein condensation could be obtained for the value of α ≈ 0.976, and one could predict that the interparticle interactions have a weak attractive nature consistent with experiment (Bradley et al 1995 Phys. Rev. Lett. 75 1687). Thus, a fractional mathematical theory is established in coherence with experimental results of Bose–Einstein condensation.