Generalized entropies and the investigation of thermal fluctuation

The application field of Boltzmann-Gibbs statistic is rather restrictive. It is applied successfully to those systems which contain particles at least at the extent of Avogadro number, which is known as the thermodynamic limit, have short-range interaction potential between the particles and whose particles' motions are quite chaotic. These systems are so-called extensive systems. However, there are many systems in Physics, Chemistry, Biology which these conditions don't exist such as the materials growing at nano scale under their gravitation, the velocity vectors of a fluid in the case of turbulence, the diffusion of particles through a porous medium. These systems, which Boltzmann-Gibbs statistic is not obtained, are known as nonextensive systems. For the recent of three decades, a few entropies have been proposed to extend the Boltzmann-Gibbs statistic in a way that can also be applied to the nonextensive systems. These entropies, known as generalized entropies, has been obtained by imposing one or more than one parameter to the Boltzman-Gibbs entropy. These entropies reduce to Boltzmann-Gibbs entropy under the specific limits of their parameter(s). In recent years, it was noticed a deep similarity between the structural properties of entropy function and the formal group law. This similarity paved the way for unifying the generalized entropies under the name of universal entropy. The main subject of this thesis is to analyze universal entropy on the ground of group theory and to search whether this entropy satisfies the third law of thermodynamics and some main physical validity conditions, such as Shannon-Khinchin axioms and third law of thermodynamics. In addition, one of the aims of the thesis is to investigate the thermal fluctuation and examine the application of the nonextensive statistical mechanics based on Tsallis entropy. The results obtained at the end of the thesis can be pointed as follow. The generalized parameter(s) of the entropies, including the universal entropy, is (are) still vague and it seems impossible to give a physical meaning to the parameter (s) by any general manner. Secondly, it has been emphasized that the universal entropy is problematic because of choosing the closed form of the formel group law, which is also the generalizing of the SK-4, inconsistently with the physical validity conditions. In addition, it has been mentioned some antinomies staying on the ground of the generalized entropies and has been drawn attention to the misunderstanding of the ergodic hypothesis. Despite these unfavorable results, it can be reasonably seen that the aforementioned similarity is still one of the main ways to construct the generalized entropy which also comprises the Boltzmann-Gibbs entropy. On the other hand, it is possible to generalize the Boltzmann-Gibbs entropy by regarding the temperature fluctuation and without imposing any (unphysical) generalized parameter(s). Key Words: Entropy, Third law of thermodynamic, Extensive systems, Nonextensive systems, Thermal fluctuations, Generalized parameters.