Al-Azhar University - Gaza Deanship of Postgraduate Studies Faculty of Science Department of Physics Applications of Fractional Dimensional Space in Statistical Mechanics By Kareema Fathee Abo Mossa Supervised By Prof.Dr. Samee Ismail Muslih A Thesis is Submitted in Partial Fulfillment of Requirements for the Degree of Master of Physical Sciences, Faculty of Graduate Studies, Al- Azhar University- Gaza. 2018 Dedication It the vastness of space and immensity of time, it is my joy to spend a planet and an epoch with my parents, my supervisor, physics department, Al-Azhar university and my friends. ii شكر و عرفان باتساع الفضاء و ضيق الوقت إنه من دواعي سروري التي تتسع الكون و العصر أن أهدي هذه الرسالة لوالدي العزيزين ، مشرفي الفاضل أ. د. سامي مصلح ، قسم الفيزياء ، جامعة اﻷزهر و أصدقائي . iii Acknowledgements I wish to express my sincere gratitude to Prof. Samee Muslih, for providing me an opportunity to do my letter with him. I wish also to express my gratitude to the officials and other staff members of Al-Azhar university- especially physics department – who rendered their help during the period of my letter work. Last but not Least I wish to avail myself this opportunity to express a sense of gratitude and love to beloved parents and my friends for their support, strength, help and everything else. iv إهداء أعبر عن امتناني لﻷستاذ الدكتور سامي مصلح ﻹعطائي الفرصة أن أنجز رسالتي تحت إشارفه . كما و أتمنى أن أعبر أيضا عن امتناني للمسئولين و هيئة الموظفين في جامعة اﻷزهر و خاصة قسم الفيزياء الذين قدموا مساعدتهم دون توان خﻻل عملي في الرسالة. و أخيار و ليس باخر أتمنى أن أنفع نفسي بهذه الفرصة ﻷعبر بأجمل إحساس و امتنان لوالدي الحبيبين و أصدقائي لدعمهم ليز و شكار للجميع على كل شيء . v Abstract The concept of fractional exclusion statistics (FES) has been of much interest in recent years. Many investigations of FES have been done on low- dimensional systems such as the fractional quantum Hall system, and the Calogero-Sutherland model (CSM), and the concept of FES played a very important role on these systems. However, in higher-dimensional systems, much work has not yet been made, except the work of Nayak and Wilczek. The general formulation of quantum statistical mechanics (QSM) that enables one to calculate thermal properties such as the equation of state for an ideal gas with FES is still laking, although the concept of FES was given for arbitrary dimensional systems. In this Letter, we present a generalization of the method to an ideal gas and it's common types (e.g., non-relativistic, relativistic and ultra-relativistic) and Dirac Hamiltonian with FES in D-dimensional space. vi ملخص بالعربية إن مفهوم اﻹحصاء الكسري أصبح ذو أهمية في السنوات اﻷخيرة. الكثير من اﻹنجاازت حققت في اﻷنظمة ذات البعد الصغير مثل نظام اﻹحصاء الكمي الكلي و نظام كالوجيرو سثرﻻند. مفهوم اﻹحصاء الكسري لعب دوار هاما في حل و تحديد تلك اﻷنظمة. من جهة أخرى اﻷنظمة ذات اﻷبعاد الكبيرة لم يعمل بها بعد ما عدا عمل نياك و فلكزك. الصيغة العامة للميكانيكا اﻹحصائية الكمية تمكننا من حساب الخصائص الحاررية مثل معادلة الحالة للغاز المثالي في اﻹحصاء الكسري بالرغم من أن مفهوم اﻹحصاء الكسري أعطي في تلك الفترة لﻷنظمة ذات البعد العشوائي. في هذه الرسالة سنقدم الطريقة العامة ﻹيجاد الكميات الحاررية للغاز المثالي و أنواعه و كذلك ديارك هاملتونيان في الفضاء الكسري. vii CONTENTS PAGE 1 FRACTAL 1 1.1 History 2 1.2 Introduction 3 1.3 Role of scaling 6 1.4 D is not a unique descriptor 8 1.5 Examples 9 CONCLUSION 15 2 SCALING METHOD AND ITS APPLICATIONS TO PROBLEMS IN FRACTIONAL DIMENSIONAL SPACE 16 2.1 Introduction 17 2.2 The volume and the surface area of a D-hypersphere 20 2.3 The scaling method and the dimension of the space 25 2.4 Dimensional regularization techniques and its applications 29 2.5 Fox function 36 CONLUSION 40 3 PARTITION FUNCTION 41 3.1 Definition 42 3.2 Canonical Partition Function 43 viii 3.2.1 Definition 43 3.2.2 Examples 48 3.2.2.1 Single particle in box 48 3.2.2.2 Harmonic oscillator in one dimension 49 3.2.2.3 Harmonic oscillator in three dimension 51 CONCLUSION 47 4 APPLICATIONS OF FRACTIONAL DIMENSIONAL SPACE IN STATISTICAL MECHANICS 54 4.1 Classical statistical mechanics 55 4.2 Fractional oscillators system 58 4.3 Quantum statistical mechanics 60 4.4 Non-relativistic ideal gas 66 4.5 Relativistic ideal gas 71 4.6 Ultra- relativistic ideal gas 75 4.7 Dirac Hamiltonian 78 CONCLUSION 83 REFERENCES 84 ix FIGURES PAGE [1] Traditional notions of geometry for defining scaling and dimension 7 [2] The first iterations of the Koch snowflake 7 [3] Patent drawing for a Sierpinski fractal dipole antenna 11 [4] Cell phone 11 [5] Feigenbaum attactor 12 [6] Pascal triangle 12 [7] Curve filling the Koch snowflake 13 [8] Microwave antenna 14 x CHAPTER ONE FRACTAL 1 Chapter 1 FRACTAL 1.1 History Fractals starts from theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way[1][2]. In 1883, Georg Cantor, who attended lectures by Weiestrass[2], published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals[1].Also, in the last period of the century, Felix Klein and Henri Poincare introduced a category of fractal that has come to be called "self-inverse" fractals[3]. In 1904, Helge von Koch, extending ideas of Poincare and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition called the Koch snowflacke[1],[2]. A decade later in 1915, when Waclaw Sierpinski constructed his famous triangle and then, his carpet. By 1918, two French mathematicians, Pierre Fatou and Gaston Julia, through working independently, arrived essentially simultaneously at results describing what are now seen as fractal behavior associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e, points that attract or repel other points), which have become very important in the study of fractals[1],[2],[4]. Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension''[2], significantly for the evolution of the definition of fractals, to allow for sets to have non-integer 2 dimensions. The idea of self-similar curves was taken further by Paul Levy, who, in his 1938 paper[4] plane or space curves and surfaces consisting of Parts similar to the whole described a new fractal curve, the Levy C curve. In the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How long is the coast of Britain? Statistical Self-Similarity and Fractional Dimension[5],[6], which built on ealier work by Lewis Fry Richardson. In 1975[7], Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set, captured of popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal"[8]. Currently, fractal studies are essentially exclusively computer-based. In 1980, Loren Carpenter gave a presentation at the special Interest group on Computer graphics and Interactive Techniques where he introduced his software for generating and rendering fractally generated landscapes[9]. 1.2 Introduction Fractals provide a simple description of complex structures in nature and are currently used in almost every branch of condensed matter physics. Mandelbrot gave a simple definition of the term fractal: a fractal is a shape made of parts similar to the whole in some way. The idea of the fractal is based on the self- similarity of complex structures. An example in condensed matter physics is the 3 band theory of electron in solids. Energy spectra of electrons[10] can be obtained by icoporating group theory based on the translational and rotational symmetry of the systems. The use of this mathematical tool greatly simplifies the treatment of systems composed of 1022 atoms. If the energy spectrum of a unit cell molecule is solved, the whole energy spectrum of the solid can be computed by applying group theory. In this context, the problem of an ordered solid is reduced to that of a unit cell. Weakly disordered systems can be handled by regarding impurities as a small perturbation to the corresponding ordered systems[11]. However, a different approach is required for elucidating the physical properties of strongly disordered/complex systems with correlations, or of medium-scale objects, for which perturbative approaches are not practical. For these systems, the concept of fractals plays an important role. Fractal structures fall into two categories. One is the deterministic fractal[12]; when the system is rescaled by a dilation transformation, it is identical with a part of the original. The other is the random fractal[13] in which the dilatational symmetry has only a statistical meaning. Unlike the deterministic fractal, the original random fractal structure does not overlap at different magnifications. Fractal structures observed in nature belong mainly to the second category. Fractal dimensions are sets by quantifying their complexity as a ratio of the change in detail to the change in scale[9]. Several types of fractal dimension can be measured theoretically and empirically[14],[15]. Fractal dimensions are used to characterized a broad spectrum of objects ranging from the abstract[15],[16] to practical phenomena, including turbulence[9], river networks, urban growth, human physiology, 4 medicine, and market trends. Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture.