Landau Diamagnetism in Noncommutative Space and the Nonextensive Thermodynamics of Tsallis

We consider the behavior of electrons in an external uniform magnetic field B where the space coordinates perpendicular to B are taken as noncommuting. This results in a generalization of standard thermodynamics. Calculating the susceptibility, we find that the usual Landau diamagnetism is modified. We also compute the susceptibility according to the nonextensive statistics of Tsallis for (1-q)<<1, in terms of the factorization approach. Two methods agree under certain conditions.Comment: Clarifications and new references. Version to appear in Phys.Lett.

Quantal distribution functions in non-extensive statistics and an early universe test revisited

Within the context of non-extensive thermostatistics, we use the factorization approximation to study a recently proposed early universe test. A very restrictive bound upon the non-extensive parameter is presented: $|q-1| < 4.01 \times 10^{-3}$.Comment: 4 pages, prl revtex style, no figures. To appear in Physica A, 199

Statistical mechanics and the description of the early universe II. Principle of detailed balance and primordial 4He formation

If the universe is slightly non-extensive, and the distribution functions are not exactly given by those of Boltzmann-Gibbs, the primordial production of light elements will be non-trivially modified. In particular, the principle of detailed balance (PDB), of fundamental importance in the standard analytical analysis, is no longer valid, and a non-extensive correction appears. This correction is computed and its influence is studied and compared with previous works, where, even when the universe was considered as an slightly non-extensive system, the PDB was assumed valid. We analytically track the formation of Helium and Deuterium, and study the kind of deviation one could expect from the standard regime. The correction to the capture time, the moment in which Deuterium can no longer be substantially photo-disintegrated, is also presented. This allows us to take into account the process of the free decay of neutrons, which was absent in all previous treatments of the topic. We show that even when considering a first (linear) order correction in the quantum distribution functions, the final output on the primordial nucleosynthesis yields cannot be reduced to a linear correction in the abundances. We finally obtain new bounds upon the non-extensive parameter, both comparing the range of physical viability of the theory, and using the latest observational data.Comment: 24 pages, to appear in Physica A (2001

Nonextensive Statistical Mechanics Application to Vibrational Dynamics of Protein Folding

The vibrational dynamics of protein folding is analyzed in the framework of Tsallis thermostatistics. The generalized partition functions, internal energies, free energies and temperature factor (or Debye-Waller factor) are calculated. It has also been observed that the temperature factor is dependent on the non-extensive parameter q which behaves like a scale parameter in the harmonic oscillator model. As $q\to 1$, we also show that these approximations agree with the result of Gaussian network model.Comment: 8 pages, 2 figure

Faz geçişleri, mıknatıslanma özellikleri ve ısıng sistemi

Bu tezin, veri tabanı üzerinden yayınlanma izni bulunmamaktadır. Yayınlanma izni olmayan tezlerin basılı kopyalarına Üniversite kütüphaneniz aracılığıyla (TÜBESS üzerinden) erişebilirsiniz.SUMMARY The theory of phase transitions and critical phenomena is generally the subject of this thesis. Therefore, firstly the necessary parts of statistical mechanics for the study of phase transitions are given briefly, then the systems which exhibit phase transition and the models which are put forward for the solutions of these are introduced. Since all physical systems which can be described by a two-valued variable are characterized by Ising model, İt is fairly an important model and frequently used in current researches. Hence this model is especially considered and applications to magnetic systems are examined. In this study, particularly one-dimensional Ising model is handled. By the help of renormalization group theory, on finite lattice it is indicated that the model does not exhibit phase transition in one dimension. In addition, by using generalized statistical mechanics, some thermodynamic quantities have been obtained as a function of reduced temperature (kBT/J ) and the obtained results are compared to the exact results. The two-dimensional Ising model has been examined for the spin systems with few number of sites. Firstly, by the help of direct calculation of thermal averages method, some thermodynamic quantities have been found for square and triangular lattices. Lastly, by using the renormalization group theory, with two types of boundary conditions for square lattice, the values of critical point (J*), exponent related to scaling factor (yT), specific heat critical exponent (a) and corelation length critical exponent (v) have been calculated. 65ÖZET Bu tezin konusunu genel olarak, faz geçişleri ve kritik olaylar teorisi oluşturmaktadır. Bu nedenle önce, faz geçişlerini incelemek için gerekli olan istatistiksel mekanik kısaca verilmekte, daha sonra ise faz geçişi gösteren sistemler ile bu sistemlerin çözümü için ortaya konan modeller tanıtılmaktadır. Ising modeli, iki-değerli bir değişkenle tanımlanabilecek tüm fiziksel sistemleri modelleyebileceğinden dolayı, oldukça önemli bir modeldir, ve halen süre gelen araştırmalarda sıkça kullanılmaktadır. Bu yüzden özellikle, bu model üzerinde durulmakta, manyetik sistemlere uygulanması incelenmektedir. Bu çalışmada özel olarak, bir-boyutlu Ising modeli ele alınmaktadır. Renormalizasyon grup teorisi yardımıyla, modelin bir-boyutta faz geçişine uğramadığı, sonlu örgü üzerinde gösterilmektedir. Ayrıca, genelleştirilmiş istatistiksel mekanik kullanılarak bazı termodinamik nicelikler, indirgenmiş sıcaklığın (kBT/J) fonksiyonu olarak elde edilmekte ve tam sonuç değerleriyle karşılaştırılmaktadır. Az sayıda örgü noktası içeren spin sistemleri için iki-boyutlu Ising modeli incelenmektedir. İlk olarak, ısısal ortalamaların doğrudan hesabı yöntemiyle, kare örgü ve eşkenar üçgen örgü için bazı termodinamik nicelikler bulunmakta ve örgü noktası sayısının değişimi ile neler olduğu tartışılmakladır. Son olarak, renormalizasyon teorisi yardımıyla, kare örgü için iki tip sınır koşulu kullanılarak kritik nokta (F), ölçekleme çarpanına ait üs (yT), Özgül ısı kritik üssü (a) ve korelasyon uzunluğu kritik üssü (v) hesaplanmaktadır. 6

Nonekstensif fiziksel sistemler için yeni bir formalizm: Genelleştirilmiş istatistiksel termodinamik ve uygulamaları

In this study, in the framework of the generalized statistical thermodynamics (GST), the applications of the previously obtained generalized distribution functions within the factorization approach to some physical systems have been considered and some bounds upon the entropic index q, which is known as the nonextensivity parameter, have been deduced. In addition to this, the connection between GST and quantum groups, a subject of recent interest, has been investigated by means of studying Landau diamagnetism within both formalisms. It is seen that the results are in good agreement with those obtained before. Finally, considering the time evolution of quantum systems within GST, the generalized evolution operator has been obtained. Applying these results to a simple quantum mechanical system, the effect of nonextensivity to the evolution of quantum systems has been examined. Keywords: Nonextensive systems, Tsallis statistics, boson and fermion systems, distribution functions, quantum groups.Bu çalışmada, genelleştirilmiş istatistiksel termodinamik (GİT) çerçevesinde faktorizasyon yaklaşımı kullanılarak daha önce elde edilmiş olan genelleştirilmiş dağılım fonksiyonlarının bazı fiziksel sistemlere uygulamaları üzerinde durulmuş ve nonekstensiflik parametresi olarak bilinen q entropi indisi için bazı sınır değerler belirlenmiştir. Bunun yanı sıra, son zamanlarda oldukça ilgi çeken bir konu olan GİT ile kuantum grupları arasındaki ilişki, Landau diamanyetizmasını her iki formalizm altında incelemek suretiyle araştırılmıştır. Elde edilen sonuçların, daha önce bulunmuş sonuçlarla uyumlu olduğu görülmüştür. Son olarak, kuantum sistemlerinin zaman içinde evrimi GİT çerçevesinde ele alınarak genelleştirilmiş evrim operatörü elde edilmiştir. Bulunan sonuçlar, basit bir kuantum mekaniksel sisteme uygulanarak nonekstensifliğin bu tip sistemlerin evrimine etkisi araştırılmıştır

Necessary Condition of Self-Organisation in Nonextensive Open Systems

In this paper, we focus on evolution from an equilibrium state in a power law form by means of q-exponentials to an arbitrary one. Introducing new q-Gibbsian equalities as the necessary condition of self-organization in nonextensive open systems, we theoretically show how to derive the connections between q-renormalized entropies (Delta(S) over tilde (q)) and q-relative entropies (KLq) in both Bregman and Csiszar forms after we clearly explain the connection between renormalized entropy by Klimantovich and relative entropy by Kullback-Leibler without using any predefined effective Hamiltonian. This function, in our treatment, spontaneously comes directly from the calculations. We also explain the difference between using ordinary and normalized q-expectations in mean energy calculations of the states. To verify the results numerically, we use a toy model of complexity, namely the logistic map defined as Xt +1 = 1 - aX(t)(2), where a is an element of [0, 2] is the map parameter. We measure the level of self-organization using two distinct forms of the q-renormalized entropy through period doublings and chaotic band mergings of the map as the number of periods/chaotic-bands increase/decrease. We associate the behaviour of the q-renormalized entropies with the emergence/disappearance of complex structures in the phase space as the control parameter of the map changes. Similar to Shiner-Davison-Landsberg (SDL) complexity, we categorize the tendencies of the q-renormalized entropies for the evaluation of the map for the whole control parameter space. Moreover, we show that any evolution between two states possesses a unique q = q* value (not a range for q values) for which the q-Gibbsian equalities hold and the values are the same for the Bregmann and Csiszar forms. Interestingly, if the evolution is from a = 0 to a = a(c) similar or equal to 1.4011, this unique q* value is found to be q* similar or equal to 0.2445, which is the same value of qsensitivity given in the literature.U.T. is a member of the Science Academy, Bilim Akademisi, Turkey and acknowledges partial support from TUBITAK (Turkish Agency) under the Research Project number 121F269.TUBITAK (Turkish Agency) [121F269