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 األزهر 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 لألستاذ الدكتور سامي مصلح إلعطائي الفرصة أن أنجز رسالتي تحت إشرافه . كما
و أتمنى أن أعبر أيضا عن امتناني للمسئولين و هيئة الموظفين في جامعة األزهر و خاصة قسم
الفيزياء الذين قدموا مساعدتهم دون توان خالل عملي في الرسالة. و أخيرا و ليس باخر أتمنى أن أنفع
نفسي بهذه الفرصة ألعبر بأجمل إحساس و امتنان لوالدي الحبيبين و أصدقائي لدعمهم ليز و شكرا
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 اإلحصاء الكسري أصبح ذو أهمية في السنوات األخيرة. الكثير من اإلنجازات حققت في
األنظمة ذات البعد الصغير مثل نظام اإلحصاء الكمي الكلي و نظام كالوجيرو سثرالند. مفهوم
اإلحصاء الكسري لعب دورا هاما في حل و تحديد تلك األنظمة. من جهة أخرى األنظمة ذات األبعاد
الكبيرة لم يعمل بها بعد ما عدا عمل نياك و فلكزك.
الصيغة العامة للميكانيكا اإلحصائية الكمية تمكننا من حساب الخصائص الحرارية مثل معادلة الحالة
للغاز المثالي في اإلحصاء الكسري بالرغم من أن مفهوم اإلحصاء الكسري أعطي في تلك الفترة
لألنظمة ذات البعد العشوائي.
في هذه الرسالة سنقدم الطريقة العامة إليجاد الكميات الحرارية للغاز المثالي و أنواعه و كذلك ديراك
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. For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension[17]. Thus, it is zero for sets describing points (0- dimensioal sets); 1 for sets describing lines (1-dimensional sets having length only); 2 for sets describing surfaces (2-dimensional sets having length and width); and 3for sets describing volumes (3-dimensional sets having length, width, and height). Unlike topological dimensions, the fractal index can take non-integer values[18],indicating that a set fills its space qualitatively differently from how an ordinary geometrical set does[15]. For instance, a curve with fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Similarly, a surface with fractal dimension 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume[18]. Overall, fractals show several types and degrees of self-similarity and detail that may not be easily visualized. These include, as examples, strange attractors for which the detail has been described as in essence, smooth portions piling up[18], the Julia set, which can be seen to be complex swirls upon swirls, and heart rates, which are patterns of rough spikes repeated and scaled in time[19].
5
1.3 Role of scaling
Now we would like to introduce the definition of fractal dimension as proposed by Hausdorff which stated as rests in unconventional views of scaling and dimension[20]. As Fig. (1) illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable N stands for the number of sticks, for the scaling factor, and D for the fractal dimension:
D N (1.1)
The symbol α above denotes proportionality. This scaling rule typifies conventional rules about geometry and dimension – for lines, it quantifies that, because if we take N=3 and D=1, we get
1 → 3 → 31, then 1/3.
Similarly, if we take N=9, D=2, we get
2 2 → 9 → 9 1, then .
6
Figure 1. Traditional notions of geometry for defining scaling and dimension.
Figure 2. The first four iterations of the Koch snowflake which has an
approximate Hausdorff dimension of 1.2619.
The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick
7 scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case, D=4 when =1/3, and the value of D can be found by rearranging Equation 1:
log N log N D (1.2) log
That is, for a fractal described by N=4 when =1/3,we get
log4 0.60206 D 1.26 log1/ 3 0.477
A non-integer dimension that suggests the fractal has a dimension not equal to the space it residues in. The scaling used in this example is the same scaling of the Koch curve and snowflake[1],[2]. Of note, the images shown are not true fractals because the scaling described by the value D cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, the theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an infinite number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619[9],[20].
1.4 D is not a unique descriptor
As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns. The value of D for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe nor provide enough information to reconstruct it. Many fractal structures or patterns could be
8 constructed that have the same scaling relationship but are dramatically different from the Koch curve.
1.5 Examples
The concept of fractal dimension described in Eq.(1.2)[20] is a basic view of a complicated construct. The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. In practice, however, fractal dimensions can be determined using techniques that approximate scaling and detail from limits estimated from regression lines over log vs log plots of size vs scale. Several formal mathematical definitions of different types of fractal dimension are listed below. Although for some classic fractals all these dimensions coincide, in general they are not equivalent:
(i) Box counting dimension: D0 is estimated as the exponent of a power law.
logN( ) D0 lim 0 1 (1.3) log
(ii) Information dimension: D1 considers how the average information needed to identify an occupied box scales with box size; p is a probability.
log p D1 lim 0 1 (1.4) log
9
(iii) Correlation dimension: D2 is based on M as the number of points used to
generate a representation of a fractal and g , the number of pairs of points closer than ε to each other.
2 log(g / M ) D2 lim (1.5) 0,M log
(iv) Generalized or Rényi dimensions
The box-counting, information, and correlation dimensions can be seen as special cases of a continuous spectrum of generalized dimensions of order α, defined by:
1 log pi 1 i D lim (1.6) 0 log
(v) Higuchi dimension
d logLk D (1.7) d logk
(vi) Multifractal dimensions: a special case of Rényi dimensions where scaling behaviour varies in different parts of the pattern.
(vii) uncertainty exponent.
(viii) Packing dimension.
(ix) Assouad dimension.
10
(x) Local connected dimension.
(xi) Fluid-mixers[21]:
Figure (3) Patent drawing for Figure (4) Cellphone with a
a Sierpinski fractal dipole Sierpinski Gasket antenna.
antenna.
(xii) Hausdorff dimension[1],[2]:
Presented some fractals by Hausdorff dimension[2], with the purpose of visualizing what it means for a fractal to have a low or a high dimension.
1: Feigenbaum attractor with Hausdorff dimension 0.538[2]
11
Figure (5) The Feigenbaum attractor
2: Pascal triangle[22] modulo 5 with Hausdorff dimension 1.6826
Figure (6) Pascal triangle
3: Curve filling the Koch snowflake[1],[2] with Hausdorff dimension 2
12
Figure (7) Curve filling the Koch snowflake
4:Microwave antenna[23] for Energy Harvesting Applications[24]
13
Figure (8) Microwave antenna
14
CONCLUSION
In this chapter we introduce the procedures which the scientists make to arrive the general form of fractional dimensional space in mathematics and many sciences. And we attach some figures to show its importance and applications in many sciences in our life.
15
CHAPTER TWO Scaling Method and its Applications to Problems in
Fractional Dimensional Space
61
Chapter 2
Scaling Method and its Applications to Problems in
Fractional Dimensional Space
2.1 Introduction
In this chapter we will introduce the scaling method to obtain volume element in Fractional dimensional space as presented by Agrawal and Muslih[25]. The first fractional physical phenomenon, which was observed sometime ago and is still a subject of many investigations, is the Brownian motion[26]. Feynman and
Hibbs[27] were the first to formulate the nonrelativistic quantum mechanics as an integral over Brownian paths, which is now known as the Feynman path integral. The concept of fractals has been applied in many other areas of physics ranging from the dynamics of fluids in porous media to the resistivity networks in electronics[28],[29]. One of the key issues in fractal geometry is measuring the dimension of the fractional space. Several methods have been proposed related to this issue. For example, the fractional mass dimension obeys the scaling rule M (r) krDM [26] , where M is the mass of the fractal medium, r is
the radius of the sphere, DM is the mass fractal dimension, and k is a constant that depends on the scaling rule. This definition could be used to compute the dimension of the fractional space. An experimental measurement suggests that
6 6 the fractional mass dimension DM of our real world is (3 10 ) [30]. The
61 same technique can be used to find the fractional charge dimension via the
DQ relation Q(r) kr , where the terms Q and DQ are defined similarly. This scaling rule has one more implication; it suggests that a charge may be understood as a distribution instead of a monopole, a dipole, or a quadrupole, as it is traditionally done. Several applications of fractional dimensional space could be cited. In the 1970s, Stillinger[30] described a procedure for integration on a fractional space of dimension D (where D is a non-integer number) and
generalized the Laplace operator D(r) in this space as
2 D 1 1 sin D2 . (2.1.1) D r 2 r r r 2 sin D2
Many investigations into low-dimensional semiconductors[31],[32] have used this Laplacian to solve the Schrodinger equation for hydrogen like atoms for anisotropic solids to obtain energy bounds and the optical spectra as a function of functional spatial dimension D.
As a next application, consider the N-quarks statistical mechanics with a
potential U(r1,r2 ,..., rN ) qi q j | ri rj | , where is a parameter. From the 1i jN theory of statistical mechanics of many body problem[33], that the potential energy term defined above will be thermodynamically stable if 0 2.
In quantum field theory, the dimensional regularization technique[34], i.e,
2 (D) 2 f (x)d(V ) f (x)x D1dx (2.1.2) D D ( ) 0 2
61
Eq.(2.1.2) suggests that D (the parameter representing non-integer dimension of the space) could be of relevance as a regularization parameter in the Feynman integral. The techniques used as a method of removing the divergent term from the Feynman integral. Clearly, the fractional dimensional space could be
important in many applications.
Related to fractal geometry and fractional dimensional space is the area of fractional derivatives and integrals which have recently been applied in many applications including fractional Hamiltonian systems[35], chaotic dynamics[28], astrophysics[36], physics of fractals and complex media[37], and recent studies of scaling phenomena. However, as demonstrated[37],[38] the areas of fractals, fractional dimensional space, and fractional derivatives are not completely independent.
We examine applications of fractional dimensional space and its relationship with fractional derivatives and integrals. To accomplish this, we first present a generalization of a differential volume element in a fractional space of dimension
D˃0. We use this definition (1) to obtain the volume and the surface area of a hypersphere in the D-dimensional space and (2) to define the solid angle about a point in the same space. This hypersphere is also denoted as a D-hypersphere or simply a D-sphere (and an N-hypersphere or simply an N-sphere when D=N is an integer). The proposed extension allows us to define a scaling rule for a fractional line element. It is shown that this definition leads to the same expressions for the volume and the surface area of the hypersphere, and the solid angle about a point in the fractional space. Furthermore, the overall dimension
61 of the space could be obtained by summing the dimensions of the fractional line elements. Also we can present a regularization condition for two-parameter functions defined in a D-dimensional space. Tarasov[39] used the formula of fractional regularization only for fractional space of dimension D, 0˂D≤3. The regularization condition defined here is used (1) to find a closed form expression for the fractional Gaussian integral, (2) to establish a relationship between a fractional dimensional space and a fractional integral, (3) to develop the Bochner theorem, and (4) to obtain an expression for the fractional integral of a Mittag–
Leffler function. Some possible extensions of this work are also discussed.
2.2 The volume and the surface area of D-hypersphere
In this section we define the volume and the surface area of a generalized D- hypersphere for D˃0 and the total solid angle around a point for the same space.
Consider the volume and the surface area of an N-hypersphere in an N- dimensional space, where N is an integer. Note that the experts in geometry and topology define an N-hypersphere differently[40]. Here, we shall follow the definition used by the experts in geometry[40],[41]. We assume that the radius of this N-hypersphere is R, and its center coincides with the origin of the
coordinate system in the space.
Using the Cartesian co-ordinates x (x1,..., xN ) , The differential volume element and the volume of the N-sphere in the N-dimensional space are given by
02
N (dV) N dx1...dxN dxi )2.2.2( j1
and
N V dx , (2.2.2) N j j1
N 2 2 2 where, is the product symbol and {x R | x1 ... xN R }.
To simplify the derivation, let us define the transformation equations between
the Cartesian coordinates and a set of polar coordinates (r,1 ,..., N 1 ) as[42],
j1 x j r cos( j )sin( k ), j=1,…,N-1 k0
N 1 xN rsin( k ), (2.2.3) k1
where 0 2,0 j ( j 1,2,..., N 2),0 N1 2,and r is the radial
N 2 1 2 coordinate defined as r ( x j ) . Using Eq. (2.2.3), the volume element in the j1
new coordinates can be written as
N1 (dV) N r drdN , (2.2.4) where
N 1 N 1k d N (sink ) dk (2.2.5) K 1
06 is the hypersolid angle at the origin defined by the hypersurface element
N1 (dS) N r dN (2.2.6) substituting Eqs.(2.2.5) and (2.2.6) into Eq.(2.2.1), integrating over the complete
range of angles ,1 ,..., N 1 , and using the identity
j 1 ( ) sin j d 2 , (2.2.7) j 2 0 ( ) 2 we obtain,
N R 2 2 N 1 VN (dV) N r dr, (2.2.8) (N ) 0 2 where is the gamma function. Eq. (2.2.8) suggests the following transformation:
N R 2 2 (dV) r N 1dr, N (2.2.9) 0 (N 2) where “a→b” indicates that "a could be replaced with b". Note that the transformation indicated in Eq.(2.2.9) must take place simultaneously.
Furthermore, Eq. (2.2.9) assumes that it is possible to integrate over all angles separately as is the case here. If this condition fails, then the transformation given in (2.2.9) will not be valid. Therefore, transformation (2.2.9) must be applied carefully. Note that the value of R could extend to . Using Eq. (2.2.8), we
obtain the volume of the N-sphere as
00
2 N 2 R N V (2.2.10) N N(N 2) differentiating Eq. (2.2.10) with respect to R, the hypersurface area of the N- sphere is given as
2 N 2 S R N 1 (2.2.11) N (N 2)
For N˃3, this hypersurface can be of dimension of more than 2. Note that S N is
N 1 N1 proportional to R . Dividing SN by R , we obtain the expression for the hypersolid angle about a point in the N-dimensional space as
2 N 2 (2.2.12) N (N 2)
Note that for N= 2 and N= 3, the hypersolid angle about a point, the hypersurface area, and the hypervolume of a sphere of radius R are given as
2 2 4 3 2 2,S2 2R,V2 R ,3 4,S3 4R ,andV3 R . 3
For N= 2, the hypersphere is denoted as the circle, the solid angle, the surface area, and the volume are simply called the angle, the circumference, and the area of the circle. For N= 1, the hypersurface area and the hypervolume is reduced to
2 points and a line, and we obtain 1 2,S1 2 , and V1 2R , and thus for N= 1, the hypervolume is essentially the length of the line. For N= 1, it is not
common to call 1,S1,and V1 as the hypersolid angle, the hypersurface area, and the hypervolume.
02
Equation (2.2.8) could be thought of as a transformation from an N-dimensional space to one-dimensional space. Therefore, one could write the transformation
D R 2 2 D1 (dV) D r dr, (2.2.13) (D ) D 0 2
where D˃0 is the dimension of the space (which could be a non-integer), D is
the domain of the D-sphere , and (dV) D is the fractional volume element in the
D-dimensional space. Like before, this transformation is valid only when the angles could be integrated separately. Furthermore, both transformations in
(2.2.13) must applied at the same time. Otherwise, this transformation may not be valid. Our interpretation of the meaning of D will be discussed in the following section.
Using Eq. (2.2.13), the volume of the D-sphere of radius R is given as
R 2 D 2 2 D 2 R D V (dV) r D1dr (2.2.14) D D (D 2) D(D 2) D 0 differentiating Eq. (2.2.14) with respect to R, The hypersurface area of the D- sphere is given as
2 D 2 S R D1 (2.2.15) D (D 2)
D1 D1 Note that this area is proportional to R , and therefore, dividing S D by R , we
get the hypersolid angle about a point in the D-dimensional space as
2 D 2 (2.2.16) D (D 2)
02
It could be verified that these definitions of the volume, the surface area of a hypersphere and the solid angle at a point in the D-dimensional space are also applicable when D is an integer. As discussed later, fractionalization of Cartesian coordinates also leads to the same results. One of the major significance of Eq.
(2.2.13) is that it allows us to propose a scaling rule.
2.3 The scaling method and the dimension of the space
Let us say that N coordinates x1 ,..., xN are needed to locate a point in a space. In the case where a space is filled with regular geometric objects, the curves and the surfaces are smooth, it is common to call this number N as the dimension of the space. Thus, a straight line, a plane surface, and a cube are of dimension 1, 2 and 3. This is also true if these spaces have curvatures. For example, motions along the circumference of a circle and on the surface of a sphere can be considered as motions in one and two-dimensional spaces even though our true motions may be in a three-dimensional space. In such cases, infinitesimal line, area, and volume elements in the Cartesian coordinates are defined as
dx1,dx1dx2 ,and dx1dx2 dx3 , respectively, and even in the case of a space with curvature, the distance between two points sufficiently closed to each other is given by a quadratic expression. However, this is not the situation in the case of fractal lines, surfaces, volumes, and hypervolumes. Thus, in these cases, there is a clear distinction between the number of coordinates used to locate a point and the dimension of the space.
02
The dimensions of the fractals can be defined in various ways. Here, we shall use the scaling method d x f () | x | 1 dx to relate the differential fractional line element d x with the differential straight line element dx where α is a parameter called the fractional dimension of the line and f(α) is a function of α.
Here, we take the absolute value of x in | x | 1 to indicate that our scaling is symmetric about the origin. A space filled with such lines is called a fractional space. Theoritically, one can speculate α to have any positive value. However, we shall restrict the value of α such that 0˂α≤1. The case of α˃1 is left for future considerations. For α=1, f(α) also must be 1 so that the scaling is a unit scaling ( or an identity transformation). Many functions would satisfy this requirement.
However, based on Sec. 2.2, we take f () 2 / ( / 2) [see Eq. (2.2.13)]. The presence of a factor 2 in Eq. (2.2.13) is due to the fact that for D=1, r is integrated from –R to R, and when the limits are taken as 0 and R, one gets a factor of 2.
Thus, if a point in a space is located using N points x (x1,..., xN ) , then the
i scaling between d xi and dxi is defined as
i 2 | x |i 1 i i d xi dxi , i=1,…, N. (2.3.1) (i / 2) we call this space as a fractional space, and
D 1... N (2.3.2)
01
As the dimension of the space. When D 1... N =1, we call the scaling as the unit or the identity scaling, and in this case D=N, i.e., the fractional dimension agrees with normal dimension of the space.
If 1 ... N , where 0˂α≤1, we call the scaling as isotropic scaling, other- wise an anisotropic scaling. Regardless of the isotropic or the anisotropic scaling, the dimension of the space is always given by Eq. (2.3.2). Furthermore, we call
the space a fractional dimensional space as long as i for at least one I is not
equal to 1. When j is not equal to 1 for some j, then we also say that the
coordinate x j has been fractionalized.
Following the above discussion, the volume element in fractional dimensional space D is defined as
1 N (dV)D d x1...d xN (2.3.3) and the volume of the D-sphere is given as
V d(V ) d i x ...d N x (2.3.4) D D 1 N D D
Let us examine the consequences of this definition of a fractional volume element. For this purpose, we compute the volume of the D-sphere once again.
First note that for 1 ... N 1, the definition of the volume element given by
Eq. (2.3.3).
01
Using Eqs. (2.3.1) and (2.3.2), the differential fractional volume element [Eq.
(2.3.3)] is given as
N i 1 N D 2 | xi | (dV) D dxi (2.3.5) i1 ( i 2) i1 using Eqs. (2.2.1),(2.2.3),(2.2.5),(2.2.6),(2.3.2), and (2.3.5), the volume of the
D-sphere [Eq. (2.3.4)] is given as
1 R 2 2 N 1 N 1 i1 i V 2 N D 2 ... (cos )i 1 sin D i k i1 i1 ( i 2) k0 0 1 0 N 10 N 1 D1 N 1k r dr(sin k ) d k k 1 (2.3.6)
Finally, using the identity
2 ( 2)( 2) 2 cos 1 sin 1d , , 0 (2.3.7) 0 (( ) / 2)
In Eq. (2.3.6) we obtain the volume of the D-sphere as
2 D 2 R D V (2.3.8) D D(D 2)
Which is the same as that given by Eq. (2.2.14). Accordingly, the scaling method gives the same results for the surface area of the D-sphere and the hypersolid angle around a point in the D-dimensional space as those given by Eqs. (2.2.11) and (2.2.12).
Theorem 1: If the scaling is defined by
01
i 2 | x |i 1 i i d xi dxi , i=1,…,N (i / 2)
,..., Then for arbitrary values of 1 N , the volume VD and the surface area S D of a
D-sphere and the hypersolid angle D around a point in a D-dimensional space
D D 2 D 2 D 2 2 R 2 D1 2 is given by VD , S D R and D , respectively D(D 2) D (D / 2) 2
(where D is defined by, D 1... N ).
Furthermore, VD , S D ,and D depend on the total dimension of the space and not explicitly on the individual dimensions of the fractional line elements.
2.4 Dimensional regularization technique and its applications
The scaling rule presented in Sec. (2.3) can be used to develop a regularization technique that will allow us to compute the integral of a function over a fractional
domain. Thus, let us assume that we have a function F(x1 ,..., xN ) defined over a fractional domain , and we want to compute integral
F(x ,..., x )d 1 x ,..., d N dx . Using the scaling Eq. (2.16), this integral is given 1 N 1 N as
N | x |i 1 N F(x ,..., x )d 1 x ,..., d N x F(x ,..., x ) D 2 i dx 1 N 1 N 1 N i (2.4.1) i1 ( i / 2) i1
The domain D and used here may be different from those defined above.
This conversion of an integral of a function from a fractional dimensional space to a regular dimensional space is called the regularization technique. This
01 regularization method is used to remove the divergent term in the evaluation of
Feynman diagram term. For example, in the space-time dimension of four, i.e.
D=4, the Feynman diagram loop integral
d 4 p 1 , diverges. To avoid this divergent and to change this integral (2 )4 ( p 2 m2 )2 into convergent loop integral, we introduce integration on the fractional dimensional space D 4 , where is small parameter closed to zero. In this case the convergent loop integral is obtained as
d 4 1 dp 2 (4 ) / 2 p3 lim lim 0 (2)4 ( p 2 m2 )2 0 (2)4 (4 / 2) ( p 2 m2 )2
The regularization condition in Eq.(2.4.1) can be specialized for special functions.
For radial function, this specialization is expressed as follows.
Theorem 2:
Let F(r) be a radial function. The fractional dimensional regularization condition for this function is given as
2 D / 2 R F(r)(dV) F(r)r D1dr (2.4.2) D (D / 2) D 0
where D is the dimension of the space, D is the domain of the hypersphere,
2 2 2 which in regular space is defined as {x | x1 ... xN R }, and R is the radius of the sphere which can be extended to infinity.
22
We now consider some applications of this dimensional regularization thechnique:
(1) Gaussian function in D-dimensional space
The Gaussian integral[43],[44]
2 ex dx (2.4.3)
Plays a fundamental role in probability and statistics, one can prove the equality in Eq.(2.4.3)using the Poisson approach.
For N-dimensional equivalent is the following identity
2 e r dV N / 2 (2.4.4) R N for fractional dimensional space, this identity is defined as follows.
Theorem 3:
The fractional Gaussian integral in the D-dimensional space satisfies the following identity:
D 2 2 (D / ) er (dV) (2.4.5) D R N (D / 2)
And, in particular, for 2 , we have
2 e r (dV) D / 2 (2.4.6) D R N
Proof:
26
Note that e r is a radial function. Thus, the identity in Equation (2.4.5) follows by using theorem (2), the definition of Gamma function, and Equation (2.4.6) follows by taking 2 . Note that both Eqs. (2.4.5) and (2.4.6) agree with
(2.4.4).
(2) The relationship between an integral over a fractional dimensional space and a fractional integral
The right Riemann-Liouville fractional integral of order α is defined as[45]:
1 b I f (z) (x z) 1 f (x)dx (2.4.7) z b () z setting z=0 in Eq. (2.4.7) using the scaling defined in Eq. (2.3.1), it follows
that
D 2(D) f (x)d D x I f (0) (2.4.8) z b (D / 2) where [0,b] . Here b can be extended to . Thus, one interpretation of a fractional integral of a function computed at the initial point (or at the origin) would be that it is a measure of the integral of the function over a fractional dimension space.
Let us take b= . In this case, z could take any value. Using simple translation of the axis, Eq. (2.4.7) could be written as
1 I f (z) x 1 f (x, z)dx (2.4.9) z () 0
20
Thus, a fractional integral could be throught of as an integral of a translated function over a fractional dimensional space.
(3) Bochner theorem in a fractional dimensional space
A natural way for the investigation of Riesz integro-differentiation[45] is the computation of the Fourier transform of the kernel | x | N , where N is the dimension of the space, x R N , and α is a real positive number.[45] This transform can be computed using the Bochner theorem, which can be stated as follows.
Bochner theorem:
The Fourier transform of a radial function is a radial function, and the following relation is valid:
(2 ) N / 2 R eiz.x (r)(dV) (r)r N / 2 J (| z | r)dr (2.4.10) D (N 2) / 2 (N )1 2 | z | 0
N Provided (r) is summable in x, where {x R || x | r R}, z R N ,
and J is the Bessel function of the first kind of order .
Furthermore
(2 ) N / 2 eiz.x (r)(dV) (r)r N / 2 J (| z | r)dr (2.4.11) N (N 2) / 2 (N / 2)1 R N | z | 0 for any function (r) provided that
r N 1 (1 r)(1N ) / 2 | (r) | dr (2.4.12) 0
22
The integral on the L.H.S of (2.4.12) is interpreted as conventionally
convergent. It converges absolutely if r N 1 | (r) | dr . 0
(4) integration of the Mittg-Leffler function over a fractional dimensional space and its relationship with the Fox H-function
The Mittag-Leffler function[46],[47] is the natural generalization of the exponential function. Being a special case of the Fox function introduced below, it is defined through the inverse Laplace transform
1 E ((t / ) ) L1{ }, (2.4.13) 4 u1 from which the series expansion
n ((t / ) ) E ((t / ) ) (2.4.14) n0 (1 n) can be deduced. The asymptotic behavior is
1 E ((t /) ) ((t /) (1)) (2.4.15)
For t , 0˂α˂1. Special cases of the Mittag-Leffler function are the exponential function
t / E1 (t / ) e (2.4.16) and the product of the exponential and the complementary error function
1/ 2 t / 1/ 2 E1 2 ((t / ) e erfc((t / ) ) (2.4.17)
22 we not in passing that the Mittag-Leffler function is the solution of the fractional relaxation equation
d(t) t D1 (t) (2.4.18) dt 0 t as already mentioned in the main text, and by Glockle and Nounenmacher, the
Mittag-Leffler function interpolates between the initial stretched exponential form
(t / ) E ((t / ) ) exp( ) (2.4.19) (1) and the long time inverse power-law behavior (2.4.15).
22
2.5 The fox function
The Fox function, also referred to as Fox's H-function, H-function, generalized
Mellin-Barnes, or generalized Meijer's G-function, has been applied in statistics before it was introduced into physics by Schneider as analytic representations for Levy distributions in x-space and as solns. of fractional equations.
In 1961 Fox defined the H-function in his studies of symmetrical Fourier kernels as the Mellin-Barners type path integral
(a p , Ap ) (a1 ,A1 ),(a2 , A2 ),...,(a p , Ap ) 1 m,n (z) m,n z m,n z ds(s)z s p,q p,q (b , B ) p,q (b , B ),(b , B ),...,(b , B ) 2i q q 1 1 2 2 q q L
(2.4.20) with the integral density
m n (b B s) (1 a A s) 1 j j 1 j j (2.4.21) (s) q p (1 b B s) (a A s) m1 j j m1 j j note that the path integral in Eq. (2.4.20) represents just the inverse Mellin transform of (s) .
H-function can be expressed as a computable series in the form
m n (b B (b ) / B ) (1 a A (b ) / B j j h h j j h h (b ) / B m (1) z h h mn (z) j1, jh . j1 . pq q p n1 0 !Bh (1 b j B j (bh ) / Bh (a j Aj (bh ) / Bh jm1 jn1
(2.4.22)
63 which is an alternating series and thus shows slow convergence. For large argument |z|→ .
The Fox functions can be expanded as a series over the residues.
m,n s p,q (z) res((s)z ) (2.4.23) 0
to be taken at the point s (a j 1)/ Aj , for j=1,…,n.
Some special functions and their Fox function representation
b z 1,0 z e 0,1 z | (b,1) |;
r 1,1 (1 r,1) 1/(1 z) 1,1 z ; (2.4.24) (0,1)
/ 1,1 ( /,1) z /1 az a 1,1 az . ( /,1)
Maitland's generalized hypergeometric or Wright's function:
(a1 , A1 ),...,(a p , Ap ) (1 a, A1 ),...,(1 a p , Ap ) z 1, p z p q (b , B ),...,(b , B ) p,q1 (0,1),(1 b , B ),...,(1 b , B ) 1 1 q q 1 1 q q
generalized Mittag-Leffler function (E,1 (z) E (z)):
j 1,1 (0,1) z E , (z) 1,2 z (2.4.25) (0,1),(1 ,) j0 ( j ) using the dimensional regularization techniques, the integral of the Mittag-
Leffler function over the functional space of dimension s˃0 is given
by
63
(s1) / 2 E (x)d s x E (x)x s1ds (2.4.26) 0 (s / 2) 0 note that the Mittag-Leffler function is a special case of the Fox function, and the relationship between these two functions is given by
1,1 (0,1) E , (x) 1,2 x (2.4.27) (0,1),(1 ,)
The Mellin transform of a single H-function[48] is given as
m m (b B s) (1 a A s) j j j j (a p , AP ) x s1 m,n ax dx a s j1 j1 (2.4.28) p,q (b , B ) q p 0 q q (1 b j B j s) (a j Aj s) jm1 jn1 where,
min R(b j | B j ) R(s) 1/ Aj max R(a j / Aj ), 1 jm 1 jn
| arg a | (1/ 2),
m p m q Aj Aj B j B j 0 j1 jn1 j1 jm1
setting m=n=p=a, A1 B1 1, b1 b2 0,q 2,s D, and B2 , we obtain
(0,1) x s11,1 x dx () (2.4.29) 1,2 0 (0,1),(1 ,) which leads to
s / 2(s) E (x)d s x (2.4.30) 0 (s / 2)
63
Eq. (2.4.24) is useful in finding the Mellin transform of the Feynman propagator of a free particle system, which allows us to obtain the Feynman quantum mechanical kernel for the systems[48].
63
CONCLUSION
In this chapter we presented a generalization of the regularization condition for a fractional space of dimension D.We introduce Bochner theorem and Fox function in the fractional space.
04
CHAPTER THREE
Partition Function
41
Chapter 3
Partition Function
3.1 Introduction
Partition functions are functions of the thermodynamic state variables- which describe the statistical properties of a system in thermodynamic equilibrium, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions.
The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances.
42
3.2 Canonical partition function
3.2.1 Definition
Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature T, and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of systems comprises an ensemble called a canonical ensemble. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous.
A: Classical discrete system
Now we would like to give a brief review on the quantum statistical systems in integer dimensional space for a canonical ensemble[49],[50] that is classical and discrete. The canonical partition function is defined as:
E e i (3.1) i where, i is the index for the microstates of the system.
is the thermodynamic beta defined as 1 K BT , K B is the Boltzmann constant.
43
Ei is the total energy of the system in the respective microstate.
The exponential factor exp( Ei ) is known as the Boltzmann factor.
A1: Classical continuous system
In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable.
In classical statistical mechanics[51], it is rather inaccurate to express the partition function as a sum of discrete terms. In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function[51] is defined as:
1 e (q, p) d 3qd 3 p (3.2) h3 where,
3 d is a shorthand notation serving as a reminder that the pi and xi are vectors in three dimensional spaces and H is the classical Hamiltonian. h is the Planck constant.
is the thermodynamic beta defined as before.
H(q,p) is the Hamiltonian of the system. q is the canonical position.
44 p is the canonical momentum.
To make it into a dimensionless quantity, we must divide it by h, which is some quantity with units of action (usually taken to be Planck's constant).
A2: Classical continuous system (multiple identical particles)
For a gas of N identical classical particles, the partition function is
1 exp[H ( p ,.... p , x ,..., x )]d 3 p ...d 3 p d 3 x ...d 3 x (3.3) N!h3N 1 N 1 N 1 N 1 N where,
pi indicate particle momenta .
xi indicate particle positions.
Here,we use x instead of q to indicate to the time.
The reason for the factorial factor N! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h3N (where h is usually taken to be Planck's constant).
B: Quantum Mechanical Discrete System
For a canonical ensemble[49],[50] that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor
45
tr(e H ) (3.4) where,
is defined as before .
H is the Hamiltonian operator.
H The dimension of e is the number of energy eigenstates of the system.
B1: Quantum mechanical continuous system
For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as:
1 q, p | e H | q, pdqdp (3.5) h where,
is the thermodynamic beta defined as 1 K B T ,
is the Hamiltonian operator, q is the canonical position.
In systems with multiple quantum states[52] sharing the same Es, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write instead of Eq. (3.1) the partition function in terms of the contribution from energy levels (indexed by j) as follows:
46
E j gi .e (3.6) j where,
g j is the degeneracy factor, or number of quantum states s which have the same
energy level defined by E j Ei .
The above treatment applies to quantum statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above.
In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis) as in Eq.
(3.4), which have Ĥ as the quantum Hamiltonian operator. The exponential of an operator can be defined using the exponential power series[53]. The classical form of Z is recovered when the trace is expressed in terms of coherent states[54] and when quantum-mechanical uncertainties in the position and momentum of a particle are regarded as negligible. Formally, using Bra–ket notation, one inserts under the trace for each degree of freedom the identity:
dxdp 1 | x, px, p | (3.7) h where, |x, p⟩ is a normalized Gaussian wave-packet centered at position x and momentum p. Thus,
47
dxdp dxdp tr(e | x, px, p |) x, p | e | x, p (3.8) h h
A coherent state is an approximate eigenstate of both operators x and p , hence also of the Hamiltonian Ĥ, with errors of the size of the uncertainties. If Δx and
Δp can be regarded as zero, the action of Ĥ reduces to multiplication by the classical Hamiltonian, and Z reduces to the classical configuration integral.
3.2.2 Examples
3.2.2.1 Single Particle in Box
The canonical partition function for a single high-temperature non-relativistic point-like particle in box
V (3.9) ppb A3
Where V is the volume of the container.
The subscript "ppb" stands for "point particle in a box"
The R.H.S is temperature dependent because A scales like . Here is the thermal de Broglie length
2 2 : (3.10) mK BT
In general, the partition function is defined in-terms of an infinite series, but in many cases it is possible to sum the series. In this case the result is a compact, closed-form expression.
48
Ideal gas of point particles
To generalize the single-particle partition function of a gas of multiple particles,
.We continue to assume all the particles are point-like. At not-too high temperatures, this is a good model for a monatomic gas[55].
3.2.2.2 Harmonic Oscillator[56] in one dimension
Before we can obtain the partition for the one—dimensional harmonic oscillator, we need to find the quantum energy levels. Because the system is known to exhibit periodic motion, we can again use Bohr-Sommerfeld quantization and avoid having to solve schrӧdinger's equation. The total energy is
P 2 kx2 P 2 mw2 x 2 E , (3.11) 2m 2 2m 2 where w k m is the classical oscillation frequency. Inverting this gives
P 2mE m2 w2 x 2 . (3.12)
We insert this into the Bohr-Sommerfeld quantization condition[57]:
Pdx 2mE m2 w2 x2 dx nh, (3.13) where the integral is over one full period of oscillation.
Let
x 2E sin() , (3.14) mw2
49
So that
m2 w2 x 2 2mE sin2 () . (3.15)
Then
2E 2 2E 1 2E Pdx 2mE cos2 ( )d 2 nh, (3.16) 2 mw 0 w 2 w
So, again making the switch E En ,
we obtain
hw E n nw . (3.17) n 2
The full solution to Schrodinger's equation (a lengthy process involving Hermite
1 Polynomials) gives E w(n ) . Except for the constant factor, Bohr- n 2
Sommerfeld quantization has done a fine job of determining the energy states of the harmonic oscillator.
Armed with the energy states, we can now obtain the partition function:
E Z exp( n ) exp(E ) 1 exp(w) exp(2w) .... K T n n B n
(3.18)
but this is just a geometric series; if we make a substitution y exp(w) , then 1 y y 2 ... . But I also know that y y y 2 y 3 ... since both Z and y have an infinite number of terms, I can substract them and all terms cancel except the first: y 1, which immediately yields
50
1 1 , or (3.19) 1 y 1 exp(w)
Now we can calculate the mean energy:
2 2 ln() Nk T 2 1 exp(w) w U Nk T Nk T 2 (1) 2 (1)exp(w) T T (1 exp(w)) k T
exp(w) Nw Nw Nwn(T), 1 exp(w) exp(w) 1
(3.20)
1 where, n(T) is the occupation factor. exp(w K BT) 1
w At very high temperatures T ,exp(w K BT) 1 (w K BT) . K B
So,
n(T) K BT w,U(T0) NK BT andCv (T0) NK B .
(3.21)
Notice that these high-temperature values are exactly twice those found for the one dimensional particle in a box, even though the energy states themselves are completely different from each other.
3.2.2.3 Harmonic Oscillator in three dimension
By analogy to the three-dimensional box, the energy levels for the 3D harmonic oscillator are simply
E w(n n n ),where n ,n ,n 0,1,2,... (3.22) nx ,ny ,nz x y z x y z
51
Again, because the energies for each dimension are simply additive, the 3D partition can be simply written as the product of three one-dimensional partition
3 functions, i.e 3D (1D ) .
Because almost all thermodynamic quantities are related to
3 ln(3D ) ln(Z1D ) 3ln(Z1D ) , almost all quantities will simply be multiplied by
a factor of 3. For example, U 3D 3NK BT 3U1D and Cv (3D) 3NK B 3Cv (1D).
One can think of atoms in a crystal as N point masses connected to each other with springs. To a first approximation, we can think of the system as N harmonic oscillators in three dimensions.
In fact, for most crystals, the specific heat is measured experimentally to be
2.76NK B at room temperature, accounting for 92% of this simple classical picture.
It is interesting to consider the expression for the specific heat at low temperatures. At low temperature, the mean energy goes to
U 3Nexp( w KBT) , so that the specific heat approaches
3Nw w w 2 w Cv 2 (w)exp( ) 3Nk ( ) exp( ) . (3.23) K BT K BT K BT K BT
U and Cv both vanish exponentially if T 0 , which means the motion is frozen in.
52
CONCLUSION
In this chapter we define the partition function and the canonical partition function in classical and quantum (discrete, continuous) systems as single particle in box and harmonic oscillator.
53
CHAPTER FOUR
Applications of Fractional Dimensional Space in Statistical
Mechanics
45
Chapter 4
Applications of Fractional Dimensional Space in Statistical
Mechanics
4.1 Classical statistical mechanics
Ideal gas
The canonical partition function of a classical ideal gas composed of N-particles occupying a volume V at a temperature T is given by[58]:
(Z ) N Z 1 (4.1) N N!
where Z1 is the individual partition function of a free particle in the canonical ensemble[58]:
d d d pd x ( p, x ) Z e (4.2) 1 h d
where , the Hamiltonian is given by[59]:
( p, x) D | p | (4.3)
1 (K BT) ,(K B is Boltzmann constant and T is the tempreture ) and ( D is a
constant to choose preserving the dimension of the energy for ( p, x), is a parameter greater than one and less than or equal to two (1˂α≤2).
44
For α=2 the Eq. (4.3) must represent the free particle Hamiltonian, then
1 D2 (2m) . In quantum mechanics | p | becomes a fractional derivative. As
( p, x) given by (4.3) doesn't depend on the position x , the integral over x
in (4.2) gives the system volume V. The integral over p in (4.2) is performed by using the spherical co-ordinates, we find that[58]:
2 d / 2V Z1 d D / (d /) (4.4) (d / 2)h (D ) where (x) is Euler Gamma function. Note here that N.Laskin[60] has
calculated Z1 only in one dimension. The last result (4.4) can be transformed into
V Z1 3 , where we have defined a novel generated thermal g wavelength[61]:
1 d d ( )h d(D ) d / 2 . g d 2 d / 2( ) Finally, we get the classical partition function for the ideal gas whose fractional
Hamiltonian presents a non-integer power of the momentum p [62].
N d / 2 1 2 V d 1 V N Z N ( ) ( ) (4.5) N! d d d / N! ( )h (D ) g 2
45
It is easy to see that when putting α=2 and D=3 in Eqs. (4.4,4.5) we retrieve the
usual thermal wave-length th h / 2 mK BT and the usual partition function
1 V N Z N ( 3 ) . N! th
Using Stirling formula we deduce the free energy or Helmholtz energy of the ideal gas
V F K T logZ NK T log 1 (4.6) B N B g N
When the partition function Z N and the Helmholtz's energy F are known it is easy to compute all the remaining thermodynamical quantities of the system.
(1) Internal energy and specific heat
The internal energy for the ideal gas is the mean value of its kinetic energy
Ec :
logZ d U N NK T (4.7) B
and the specific heat at constant volume:
U d Cv NK B (4.8) T v
(2) Entropy
The entropy of the system in the canonical ensemble is given by[59]:
45
F V d S NK log 1 (4.9) B T v g N
(3) State equation
The canonical pressure of the system is given by:
F NK T P B (4.10) V T V we remark here that the state equation is independent of the parameter α wh- ereas all other thermodynamical quantities depend. This remark holds also when we use the micro-canonical ensemble.
4.2 Fractional oscillators system
Consider now a system composed of N classical independent three-dimensional fractional oscillators. The Hamiltonian of this system is given by[63],[64]:
N 2 , (D | p | q | ri | ) (4.11) i1 where q is a constant with dimensionq energy1/ 2 length / 2 , α and are parameters such as 1˂α≤2,1˂ ≤2.
The canonical partition function for the classical system is given by[65]:
1 Z Z N (4.12) N N! 1
45
Then,
N 1 2 Z d d pd d re (D | p| q |r| ) N (4.13) N! 0
Where,
D d d 4 d d (D | p| q2 |r| ) d pd re 0 2 d d d / 2 d / h V (D ) (q ) 2
Then[65],
N d d 4 d 1 Z N (4.14) N! 2 d d d / 2 d / h V (D ) (q ) 2
If we express the Helmholtz energy (using Stirling formula),
d d 4 d F K T logZ NK T log1 B N B (4.15) 2 d d d / 2 d / h V (D ) (q ) 2
4.2.1 Energy and specific heat
The internal energy in the canonical ensemble is given by:
logZ N d d U NK BT (4.16) and the specific heat at constant volume:
45
U d d Cv NK B (4.17) T v
4.3 Quantum statistical mechanics
Ideal Bose Gas
An ideal Bose gas is a quantum mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Pauli exclusion principle with antisymmetric wave function[66].
We will consider N particles with zero spin (bosons) without interaction, enclosed in a volume V. We assume that the system is in thermodynamical equilibrium with a thermostat at temperature T. We will study some properties of the ideal gas of bosons in the fractional case. To address the problem of an ideal gas of bosons, we use the grand canonical partition function D(z,V ,T )
[66]:
1 D(z,V ,T) (4.18) i i 1 ze
where, i is the energy of the single particle. We assume that i is given by:
i D | p | (4.19) and z is the fugacity of the gas which is related to the chemical potential through the formula:
z e (4.20)
56 or equivalently, the grand partition function is:
1 D(z,V,T) (4.21) D |P| p 1 ze
Using this expression we can derive some thermodynamical properties of the
Bose gas system in the case of fractional quantum mechanics.
State equation
In the grand canonical ensemble of equilibrium mechanics, the state equation is given by:
PV logD(z,V ,T) log(1 zeD |P| ) log(1 z) K BT 1 log(D(z,V ,T) zeD p z N z D p (4.22) z 1 1 ze 1 z
Where P and N respectively the pressure and the total number average of particles of the system.
The last sums can be extended to an integral[66]:
d / 2 PV 2 V p d 1dplog(1 zeD p ) log(1 z) K T d B h d 0 2 (4.23) 2 d / 2V zeD p z N p d 1dp D p d 1 z h d 0 1 ze 2
Performing the integrals we find:
56
d 2 d / 2 P 1 g d (z) log(1 z) K T d 1 V B d d / h (D ) 2 (4.24) d 2 d / 2 N z g d (z) V d V (1 z) d d / h (D ) 2
Where[66],
z l g (z) r r (4.25) l0 l
The state equation can be obtained by eliminating z from the two following coupled equations:
P 1 1 g d (z) log(1 z) K T 1 V B g (4.26) N 1 z g d (z) V V (1 z) g
where g is the generalized thermal wave-length. Now, look at the equations
(4.26) when we perform the thermodynamical limit (N and V going to infinity keeping the ratio N/V equal to a constant): a) when z˂1 then (4.26) becomes:
P 1 g d (z) K T 1 B g (4.27) N 1 g d (z) V g
56 these coupled equations correspond to the state equation for the case z˂1. b) when z→1,
z N the term 0 , becomes the significative fraction of the total particles V (1 z) V
1 1 density, whereas the term log(1 z) log(N 1) is negligible. V V 0
In this case the coupled state equations become:
P 1 g d (1) K T 1 B g d / (4.28) N 1 N 1 N T g (1) 0 g (1) 1 d d V V V T g g c
where N 0 is the mean number of bosons occupying the state of zero energy (k=0)
and Tc is the Bose temperature that will be defined in the next subsection. We remark here that in the case of z=1, the pressure (4.28) is independent of the volume V. This case obviously corresponds to situation where the temperature
T is less than TB .
when T TB , we immediately find, the internal energy of the system:
g d (z) 1 D(z,V,T) d U K BTV (4.29) V ,z g
And the specific heat at constant volume and mean number N[66]:
56
U Cv (4.30) T V ,N
Using the properties:
g (z) z n g (z) (4.31) z n1 and,
z dz g (z) d / (4.32) T V ,N T gd / 1 (z) we find:
C V (d d 2 ) g (z) d 2 g (z) v (d / 1) d / (4.33) 2 d 2 NK B N g(d / 1) (z)
And the Helmholtz energy:
g (z) F N PV NK Tlogz V (d / 1) (4.34) B d
And the entropy of the system of bosons:
U F d g (z) S VK (d / 1) NK logz (4.35) T B d B
We mention here, that all above results lead to the well- known standard results when we put α=2 and d=3.
55
Ideal Fermi gas
An ideal Fermi gas is a phase of matter which is an ensemble of a large number of non-interacting fermions. Fermions are particles that obey Fermi-Dirac statistics, like electrons, photons and neutrons. In general, particles with half- integer spin and didn't obey Pauli exclusion principle with symmetric wave function[66].
For the case of Fermi gas, the same procedure is to be followed except that the energy level cannot be occupied by more than one particle (fermion). Then the
Grand-partition function is given by:
D(z,V ,T ) (1 zeD p ) (4.36) p
This leads us to write:
PV logD(z,V ,T) log(1 zeD p ) K BT 1 (4.37) logD(z,V ,T) zeD p N z D p z 1 1 ze
Using the same demarche as the bosons, we find at first the two coupled state equations:
P 1 f d (z) K T 1 B g (4.38) N 1 f d (z) V g
where f n (z) is the Fermi function defined by:
54
z l f (z) (1)l r r (4.39) l1 l
All the thermo-dynamical properties can be found easily, and we list them here:
Internal energy
f d (z) 1 d U K BTN (4.40) f d (z) the specific heat at constant volume
2 2 Cv d d f (d / 1) (z) d f d / (z) 2 2 (4.41) NK B f d / (z) f (d / 1) (z)
The Helmhholtz's energy and the entropy[59]:
f (z) (d / 1) F NK BTlogz f d / (z) (4.42) d f (d / 1) (z) S NK B NK B logz f d /
4.4 The Non- Relativistic ideal gas
The classical Hamiltonian for an ideal non-relativistic gas[67] takes the form:
P 2 H i (4.43) i 2m
55 and use that a classical system is characterized by the set of three dimensional
momenta and positions pi , xi .This suggests to write the partition sum as
p 2 Z exp( i ) (4.44) pi ,xi i 2m for practical purposes it is completely sufficient to approximate the sum by an integral, i.e to write
px dpdx f ( p, x) f ( p, x) f ( p, x) (4.45) p,x px p,x px
Here px is the smallest possible unit it makes sense to discretize momentum and energy in. It's value will only be an additive constant to the partition function, i.e will only be an additive correction to the free energy 3NT logpx.The most sensible choice is certainly to use
px (4.46) with Plank's quantum h 6.62607551034 J.s . Below, we consider ideal quantum gases, we will perfume the classical limit, and demonstrate explicity that this choice for is the correct one. It is remarkable, that there seems to be no natural way to avoid quantum mechanics in the description of a classical statistical mechanics problem. Right away we will encounter another of quantum physics when we analyze the entropy of the ideal classical gas with the above choice for follows[67]:
55
N N d d p d d x p 2 N d d p p 2 V Z i i exp( i ) V N i exp( i ) (4.47) d d d i1 h 2m i1 h 2m where we used,
dpexp(p 2 ) (4.48) and introduced
h2 (4.49) 2m which is the thermal de-Broglie wave length. It is the wave length obtained via
2 2 k 2 K BT C ,and k with C some constant of order unity. It follows which 2m
Ch 2 Ch 2 1 gives ,i.e C . 2m 2mK BT
For the free energy it follows:
V F(V,T) K BT logZ NK BT log (4.50) 3 using dF SdT pdV (4.51) gives the pressure
F NK T P B (4.52) V T V
55 which is the well-known equation of state of the ideal gas. Next we determine the entropy
F V log V 3 S NK B log 3 3NK BT NK B log 3 NK B (4.53) T V T 2 it gives,
3 U F TS NK T (4.54) 2 B
Thus, we recover both equations of state, which were the starting point of our ealier thermodynamic considerations.
Nevertheless, there is a discrepancy between our results obtained within statistical mechanics. The entropy is not extensive but has a term of the form N log V which overestimates the number of states.
The issue is that there are physical configurations, where (for some values PA
and PB ):
P1 PA , & P2 PB (4.55) as well as
P2 PA , P1 PB (4.56) i.e., we counted them both. Assuming indistinguishable particles however, all
what matters are whether there is some particle with momentum PA and some
particle with PB , but not which particles are in those states. Thus we assumed particles to be distinguishable. There are however N! ways to relabel the particles
55 which are all identical. We simply counted all these configurations. The proper
expression for the partition sum is therefore[67]:
N 1 N d d p d d x p 2 N d d p p 2 1 V Z i i exp( i ) V N i exp( i ) d d d N! i1 h 2m i1 h 2m N! using log N! N log(N 1) , valid for large N gives
V V F NK T log NK T log(N 1) NK T1 log . (4.57) B 3 B B 3 N
This result differs from the ealier one by a factor NK BT log(N 1) which is why it is obvious that the equation of state for the pressure is unchanged.
However, for the entropy follows now.
V 5 S NK B log NK B (4.58) N3 2 which gives again
3 U F TS NK T (4.59) 2 B
The reason that the internal energy is correct is due to the fact it can be written as an expectation value[67]:
1 N d d p d d x p 2 p 2 i i i exp( i ) N! h d 2m 2m U i1 i (4.60) 1 N d d p d d x p 2 i i exp( i ) d N! i1 h 2m and the factor N! does not change the value of U. The prefactor N! is called
Gibbs correction factor.
56
The general partition function of a classical real gas or liquid with pair potential
V(xi x j ) is characterized by the partition function[67]:
1 N d d p d d x N p 2 N Z i i exp i V x x (4.61) d i j N! i1 h i1 2m i, j1
4.5 Relativistic ideal gas
The classical Hamiltonian for an ideal relativistic gas[68] takes the form:
2 N p 2 mc 2 1 i 1 , pi pi (4.62) i1 mc where m is the rest mass and where the rest energy has been substracted so that the Hamiltonian represents the total kinetic energy of N-particles, and c is the velocity.
But we know the canonical partition function as:
N Z1 Z N (4.63) N!
where Z1 defined as:
d d p d d x Z i i e ( p, x ) (4.64) 1 h d
Which is the individual particle function of a free particle canonical ensemble.
Then by using Eqs. (4.58) and (4.60), we have
56
N p2 N 2 N 2 i 2 pi 2 d d mc 1 1 d d mc 1 mc d p d x mc d p d x mc Z i i e i1 i i e i1 i1 1 h d h d
For m˂˂ pi : .
We can deduce the series to the first three terms, to get
N 2 2 2 2 1 pi pi N mc 1 2 d d mc d p d x i1 2 mc mc i i i1 Z1 e .e h d
2 N N p2 N p2 N d d mc2 / 2 i i mc2 d p d x mc mc i i e i1 i1 i1 i1 h d 2 N p2 N p2 d d / 2 i i d p d x mc mc i i e i1 i1 h d
2 N p2 p2 d / 2 / 2 i d / 2 i d p mc d p mc V N i .e i1 i .e h d / 2 h d / 2 2 N p2 p2 d / 2 / 2 i d / 2 i d p mc d p mc V N i e i1 i e h d / 2 h d / 2 2N N 1 1 V N d / 2 d / 2 g1 g2 N 2N V 1 d / 2 d / 2 g2 g1 where we used Eq. (4.43) which say:
N N d d p p 2 V Z V N i exp( i ) d d i1 h 2m g
56
h2 and introduced , which is the thermal wavelength. It is the wave- 2m
2 2 k 2 length obtained via K BT and, k , where c is constant. mc
2 h2 It follows which gives, . mc cmK BT
There are however N! ways to relabel the particles are all identical.
The proper expression for the partition sum is therefore:
N 2N 1 V 1 Z N N! d / 2 d / 2 (4.65) g2 g1 using Stirling's formula we deduce the free energy or Helmholtz energy of the relativistic ideal gas:
N 2N 1 V 1 F K T logZ K T log (4.66) B N B N! d / 2 d / 2 g2 g1
When the partition function Z N and the Helmhotz's energy F are known it is easy to compute all the remaining thermos-dynamical quantities of the system.
Internal energy and specific heat
The internal energy for the relativistic ideal gas is the mean value of its kinetic
energy Ec :
logZ U N (4.67)
56 to calculate it recall the wave-length od an ideal gas in fractional dimensional space:
d d / 4 d / 2 d h / 2 D 4 4 d / 2 g1 d 2 d / 4 2 and,
d d / 4 d / 2 d h D 4 4 d / 2 g2 d 2 d / 4 2 implies that
N 2N d d 2V d / 4 2 d / 2 1 2 2 ZN . N! d d / 4 d / 2 d d d / 4 d / 2 d h D h / 2 D 4 4 4 4 so,
2N N d d 2 d / 4 2V d / 2 1 d 2 d 2 U NK BT d / 2 . NK BT . d d / 2 d N! 2 d d / 4 d 4 d / 4 h D h D 4 2 4 4 4
Rearrange we have,
N 2N 1 d V 1 d U N 2 K 2T 2 N 2 K 2T 2 Z 2 B d / 2 d / 2 2 B N (4.68) N! 2 2 g2 g1
and the specific heat at constant volume
55
U D 2 2 Cv 2 N K BTZ N (4.69) T V
Entropy
The entropy of the system in the canonical ensemble is given by:
F S K B logZ N T V
State equation
The canonical pressure of the system is given by:
F NK T P B V T V
As before, we noticed that the state equation is independent of the parameter α whereas all other thermodynamical quantities depend.
4.6 Ultra-relativistic ideal gas
A gas of N-ultra-relativistic particles having Hamiltonian[69]:
N 2 H c pi , m˂˂ pi (4.70) i1
N Z1 recall that the canonical partition function is Z N , where Z1 is defined N! as
d d d p d x p, x Z i i e (4.71) 1 h d
54 substitute from Eq. (4.70) into Eq. (4.71), to get
N N N d d 2 d d 2 d p d x cpi d p d x c pi V Z i i e i1 i i e i1 1 d d d (4.72) h h g
Ch2 where ,and C is constant. g
So, the usual partition function is:
N 1 V Z (4.73) N d N! g using Stirling formula we deduce the free energy or Helmholtz energy of the ideal gas:
V F K T logZ NK T log 1 (4.74) B N B d g N
Internal energy and specific heat
The internal energy for the ideal gas is the mean value of its energy Ec :
logZ d U N NK T (4.75) B and the specific heat at constant volume:
U d Cv NK B (4.76) T V
55
Entropy
The entropy of the system in the canonical ensemble is given by:
F V d S NK log 1 (4.77) B d T V g N
State equation
The canonical pressure of the system is given by:
F NK T B (4.78) V T V we notice that all thermodynamical quantities are the same as a non-relativistic ideal gas , but differ in the wave length.
55
4.7 Dirac Hamiltonian
Originally, the Klein-Gordan equation[70] was thought to be the relativistic version of the Schrodinger equation- that is, an equation for the wave function
(x,t) for one relativistic particle. But pretty soon this interpretation run into trouble with bad probabilities (negative, or ˃1 ) when a particle travels through high potential barriers or deep potential wells. There were also troubles with
relativistic causality, and a few other things.
Paul Adrien Maurice Dirac had thought that the source of all those troubles was
2 2 the ugly form of relativistic Hamiltonian p m in the co-ordinate basis, and that he could solve all the problems with the Klein-Gordan equation by rewriting the Hamiltonian as a first-order differential operator[70]:
H p. m (4.79) but the Dirac equation is[70]:
i i. m (4.80) t
where1 , 2 , 3 , are matrices acting on a multi-component wave function.
Specifically, all four of these matrices are Hermitian, square to 1, and antico- mmute with each other,
2 2 2 i , j 2ij , i , 0 i j 1 (4.81)
87 because ordinary numbers cannot satisfy this expression, Dirac assumed that these coefficients are 4×4 matrices having the commutation properties:
0 j j (4.82) i j j i 0i j
1 0 0 0 0 0 0 1 0 0 0 i 0 1 0 0 0 0 1 0 0 0 i 0 , , , 0 0 1 0 1 0 1 0 0 2 0 i 0 0 0 0 0 1 1 0 0 0 i 0 0 0
0 0 1 0 0 0 0 1 . (4.83) 3 1 0 0 0 0 1 0 0
the j quantities can be expressed interms of the Pauli Matrices as:
0 j j (4.84) i 0 where each entry in the matrix is itself a 2×2 matrix. Similarly, we have
1 0 (4.85) 0 1 while we're at it, we may as well show you how the Dirac equation automatically includes fermion spin.
We introduced Dirac Hamiltonian operator as[70]:
2 H mc c j p j (4.86) where
87
p i . j x j
1 Here we will replace the matrix to ablude (K BT ) which is Boltzmann constant. so, the canonical partition function is:
N Z1 Z N , N!
and remember that from Eq. (4.2)
d d p d d x Z i i e ( p, x ) (4.87) 1 h d
Let us find Z1 for the Dirac Hamiltonian, substitute from Eq. (4.16) into Eq.
(4.17) to get:
d d d d p d x 2 d p 2 Z i i e (mc ci pi ) V N i emc .eci pi 1 h d h d
Now, substitute in Eq. (4.3) for α=1, we get
H1 p, x D1 | p | Dp
Our equation becomes according to Eq. (4.4) for D C i
So, we can find that:
78
N d / 2 2 2V c Z e mc i (4.88) 1 d d d h c i 2 this implies that:
N N d / 2 1 2 2V c 1 V Z emc i (4.89) N N! d d N! d d / 2h c i
N d d d h c i 2 2 where, d emc , is the wave-length of Dirac 2V d / 2c i
Hamiltonian.
For m×m symmetric positive definite matrices[71]:
m m(m1) / 4 j 1 m 1 m (a) a , R(a) j1 2 2
Now,using Stirling's formula we deduce the free energy or Helmhotz energy of the Dirac Hamiltonian is:
V F K BT logZ N NK BT log 1 (4.90) d N
Internal energy and specific heat
The internal energy for Dirac-Hamiltonian is the mean value of its kinetic energy:
logZ U N dNK T (4.91) B
78 and the specific heat at constant volume:
U Cv dNK B (4.92) T V
Entropy
The Entropy of the system in the canonical ensemble is given by:
F V S NK B log d d 1 (4.93) T V
State equation
The canonical pressure of the system is given by:
F NK T P B (4.94) V T V
78
CONCLUSION
In this thesis, we have studied some systems of statistical mechanics by considering the fractional classical and quantum mechanics developed recently by Nick Laskin. More specifically we have presented at first the concept of fractal and its dimension, applications in world data, Fractal Antenna as one of the most important application, partition function, a scaling method and its applications to problems in fractional dimensional space, and D-dimensional statistical mechanics in fractional classical and quantum mechanics. In this context we have obtain the equations describe an ideal gas and its thermodynamical properties, and in the context we derive the types of an ideal gas and Dirac Hamiltonian and it's thermodynamical properties. We have proved that the thermodynamical properties are the same in the general description, but are differ in the wave-length.
78
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