Fevzi Büyükkılıç

A fractal approach to entropy and distribution functions

Abstract The spectrum of fractal dimensions D q of non-interacting classical and quantum gases in non-equilibrium, whose states are described by the set of occupation numbers { n 1 , n 2 , … n K }, has been obtained. With this point of departure, the fractal inspired entropies of the related gases have beem deduced. The generalized non-exponential forms of the distribution functions have been found by the application of Boltzmanns H -theorem.

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.

Generalization of the Maier–Saupe theory of the nematics within Tsallis thermostatistics

In this study, one of the mean-field theories in nematics, the Maier–Saupe theory (MST), is generalized within Tsallis thermostatistics (TT). The variation of the order parameter versus temperature has been investigated and compared with experimental data for p-azoxyanisole (PAA). So far, we believe that this is the first attempt at the application of TT in liquid crystals. It is well known that MST fails to explain the experimental data for some of the nematics, one of which is PAA. However, generalized MST is able to account for the experimental data for a long range of temperatures. Also, in this study, the effect of nonextensivity is shown for various values of the entropic 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.

Extension of Nikiforov-Uvarov method for the solution of Heun equation

We report an alternative method to solve second order differential equations which have at most four singular points. This method is developed by changing the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU) method. This is called extended NU method for this paper. The eigenvalue solutions of Heun equation and confluent Heun equation are obtained via extended NU method. Some quantum mechanical problems such as Coulomb problem on a 3-sphere, two Coulombically repelling electrons on a sphere, and hyperbolic double-well potential are investigated by this method.

Nonextensive thermostatistical investigation of the blackbody radiation

Abstract Thermodynamical quantities of the blackbody radiation such as free energy, entropy, total radiation energy, specific heat are calculated within the Tsallis thermostatistics where factorization method is incorporated. It is shown that basic thermodynamical relation of the blackbody radiation is form invariant with respect to the nonextensivity entropic index q. Furthermore, the nonextensive thermodynamical quantities related to the blackbody radiation are separately obtained in terms of q and the standard thermodynamical quantities of the blackbody radiation. It is indicated that the formulation may give a way to determine q which determines the degree of the nonextensivity that is one of the aims of the present study.