Statistical physics: an introduction

206 Pages·2007·1.34 MB·english

by  Daijiro Yoshioka

Statistical Physics
Daijiro Yoshioka
Statistical
Physics
An Introduction
With71Figuresand7Tables
123
ProfessorDaijiroYoshioka
DepartmentofBasicScience
TheUniversityofTokyo
3-8-1Komaba,Meguro
Tokyo,153-8902
Japan
e-mail:[email protected]
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Preface
“More is different” is a famous aphorism of P.W. Anderson, who contributed
rather a lot to the development of condensed-matter physics in the latter
half of the 20th century. He claimed, by this aphorism, that macroscopic sys-
temsbehaveinawaythatisqualitativelydifferentfrommicroscopicsystems.
Therefore, additional rules are needed to understand macroscopic systems,
rules additional to the fundamental laws for individual atoms and molecules.
An example is provided by the various kinds of phase transitions that occur.
The state of a sample of matter changes drastically at a transition, and sin-
gular behavior is observed at the transition point. Another good example in
whichquantitybringsaboutaqualitativedifferenceisthebrain.Abraincon-
sists of a macroscopic number of neural cells. It is believed that every brain
cell functions like an element of a computer. However, even the most sophis-
ticated computer consists of only a limited number of elements and has no
consciousness. The study of the human brain is still developing.
Ontheotherhand,theparadigmformacroscopicmatter,namelythermo-
dynamics and statistical physics, has a long history of investigation. The first
and second laws of thermodynamics and the principle of equal probability in
statisticalphysicshavebeenestablishedaslawsthatgovernsystemsconsisting
ofamacroscopicnumberofmolecules,suchasliquids, gases,andsolids(met-
als, semiconductors, insulators, magnetic materials, etc.). These laws belong
to a different hierarchy from the laws at the microscopic level, and cannot be
deduced from the latter laws, i.e. quantum mechanics and the laws for forces.
Therefore, a “theory of everything” is useless without these thermodynamic
and statistical-mechanical laws in the real world. The purpose of this book is
to explain these laws of the macroscopic level to undergraduate students who
are learning statistical physics for the first time.
In this book, we start from a description of a macroscopic system. We
then investigate ideal gases kinematically. Following on from the discussion
of the results, we introduce the principle of equal probability. In the second
and third chapters we explain the general principles of statistical physics on
the basis of this principle. We start our discussion by defining entropy. Then
VI Preface
temperature, pressure, free energy, etc. are derived from this entropy. This
concludes Part I of the book. In Part II, from Chap. 4 onwards, we apply
statistical physics to some simple examples. In the course of this application,
we show that entropy, temperature, and pressure, when defined statistical-
mechanically, coincide with the corresponding quantities defined thermody-
namically. We consider only thermal-equilibrium states in this book. Most of
our examples are simple systems in which interaction between particles is ab-
sent.Interaction,however,isessentialforphasetransitions.Foranillustration
of how a phase transition occurs, we consider a simple ferromagnetic system
in Chap. 7. At this point, readers will be able to obtain a general idea about
statistical physics: how a system in equilibrium is treated, and what can be
known. In Part III, some slightly more advanced topics are treated. First, we
consider first- and second-order phase transitions in Chaps. 8 and 9. Then,
in Chap. 10, we return to our starting point of the ideal gas, and learn what
happens at low temperature, when the density becomes higher.
Physics is one of the natural sciences, and the starting point of an inves-
tigation is the question “Why does nature behave like this?” Therefore, it is
a good attitude to ask “why?” This question should be aimed only at natural
phenomena, though. In this book, we give an explanation, for example, for
various strange characteristics of rubber. However, it is often useless to ask
“why?”aboutthemethodsusedforsolvingthesequestions,orhowanideaor
conceptusedtotreataproblemwasobtained.Forexample,itisnotfruitfulto
ask how the definition of entropy was derived. The expression for the entropy
was obtained by a genius after trial and error, and it cannot be obtained as
a consequence of logical deduction. Logical deduction can be done by a com-
puter. Great discoveries in science are not things that can be deduced. They
are rushes of ideas to the head. Some students stumble over these whys and
hows of the methods, and fail to proceed. We hope that you will accept the
various concepts that geniuses have introduced into science, and enjoy the
beauty of the physics developed by the application of such concepts.
Tokyo,
October 2005 Daijiro Yoshioka
Contents
Part I General Principles
1 Thermal Equilibrium and the Principle
of Equal Probability ....................................... 3
1.1 Introduction to Thermal and Statistical Physics ............. 3
1.2 Thermal Equilibrium .................................... 4
1.2.1 Description of a System in Equilibrium............... 4
1.2.2 State Variables, Work, and Heat .................... 5
1.2.3 Temperature and the Zeroth Law of Thermodynamics.. 7
1.2.4 Heat Capacity and Specific Heat .................... 8
1.3 Kinetic Theory of Gas Molecules .......................... 9
1.3.1 The Spatial Distribution of Gas Molecules............ 10
1.3.2 Velocity Distribution of an Ideal Gas ................ 15
1.3.3 The Pressure of a Gas ............................. 18
1.4 The Principle of Equal Probability......................... 20
2 Entropy ................................................... 23
2.1 The Microcanonical Distribution .......................... 23
2.2 Number of States and Density of States .................... 26
2.3 Conditions for Thermal Equilibrium ....................... 28
2.3.1 Equilibrium Condition
when only Energy is Exchanged ..................... 28
2.3.2 Equilibrium Condition when Molecules are Exchanged . 30
2.3.3 Equilibrium Condition when Two Systems
Share a Common Volume........................... 31
2.4 Thermal Nonequilibrium and Irreversible Processes .......... 32
3 The Partition Function and the Free Energy............... 35
3.1 A System in a Heat Bath................................. 35
3.1.1 Canonical Distribution ............................. 36
3.1.2 Application to a Molecule in Gas.................... 37
VIII Contents
3.2 Partition Function....................................... 38
3.3 Free Energy ............................................ 39
3.4 Internal Energy ......................................... 41
3.5 Thermodynamic Functions
and Legendre Transformations ............................ 42
3.6 Maxwell Relations ....................................... 43
Part II Elementary Applications
4 Ideal Gases ................................................ 47
4.1 Quantum Mechanics of a Gas Molecule..................... 47
4.2 Phase Space and the Number of Microscopic States.......... 49
4.3 Entropy of an Ideal Gas .................................. 51
4.4 Pressure of an Ideal Gas:
Quantum Mechanical Treatment .......................... 54
4.5 Statistical-Mechanical Temperature and Pressure ............ 55
4.6 Partition Function of an Ideal Gas......................... 56
4.7 Diatomic Molecules...................................... 58
4.7.1 Decomposition of the Partition Function ............. 58
4.7.2 Center-of-Gravity Part: Z(CG) ...................... 60
4.7.3 Vibrational Part: Z(V) ............................. 61
4.7.4 Rotational Part: Z(R) .............................. 64
5 The Heat Capacity of a Solid,
and Black-Body Radiation................................. 67
5.1 Heat Capacity of a Solid I – Einstein Model ................ 67
5.2 Heat Capacity of a Solid II – Debye Model ................. 70
5.2.1 Collective Oscillations of the Lattice
and the Internal Energy............................ 70
5.2.2 Heat Capacity at High Temperature ................. 73
5.2.3 Heat Capacity at Low Temperature.................. 74
5.2.4 Heat Capacity at Intermediate Temperature .......... 74
5.2.5 Physical Explanation for the Temperature Dependence. 75
5.3 Black-Body Radiation.................................... 76
5.3.1 Wien’s Law and Stefan’s Law ....................... 76
5.3.2 Energy of Radiation in a Cavity..................... 77
5.3.3 Spectrum of Light Emitted from a Hole .............. 78
5.3.4 The Temperature of the Universe.................... 80
6 The Elasticity of Rubber .................................. 83
6.1 Characteristics of Rubber ................................ 83
6.2 Model of Rubber ........................................ 84
6.3 Entropy of Rubber ...................................... 85
6.4 Hooke’s Law ............................................ 86
Contents IX
7 Magnetic Materials ........................................ 89
7.1 Origin of Permanent Magnetism........................... 89
7.2 Statistical Mechanics of a Free Spin System................. 91
7.2.1 Model and Entropy................................ 91
7.2.2 Free Energy, Magnetization, and Susceptibility........ 93
7.2.3 Internal Energy and Heat Capacity .................. 95
7.3 Ising Model – Mean-Field Approximation................... 97
7.3.1 Links ............................................ 97
7.3.2 Mean-Field Approximation ......................... 99
7.3.3 Solution of the Self-Consistent Equation..............100
7.3.4 Entropy and Heat Capacity.........................103
7.3.5 Susceptibility .....................................105
7.3.6 Domain Structure .................................106
7.4 The One-Dimensional Ising Model .........................106
7.4.1 Free Energy ......................................106
7.4.2 Entropy and Heat Capacity.........................108
7.4.3 Magnetization and Susceptibility ....................110
Part III More Advanced Topics
8 First-Order Phase Transitions .............................115
8.1 The Various Phases of Matter.............................115
8.2 System in a Heat Bath at Fixed P and T ...................119
8.3 Coexistence of Phases....................................121
8.4 The Clausius–Clapeyron Law .............................123
8.5 The Critical Point .......................................126
8.6 The van der Waals Gas ..................................128
8.6.1 Coexistence of Gas and Liquid ......................130
9 Second-Order Phase Transitions ...........................133
9.1 Various Phase Transitions and Order Parameters ............133
9.2 Landau Theory .........................................134
9.2.1 Free Energy ......................................137
9.2.2 Entropy, Internal Energy, and Heat Capacity .........138
9.2.3 Critical Phenomena................................139
9.3 The Two-Dimensional Ising Model.........................140
10 Dense Gases – Ideal Gases at Low Temperature ...........147
10.1 The Phase Space for N Identical Particles ..................147
10.2 The Grand Canonical Distribution.........................149
10.3 Ideal Fermi Gases and Ideal Bose Gases ....................151
10.3.1 Occupation Number Representation .................151
10.3.2 Thermodynamic Functions .........................154
X Contents
10.4 Properties of a Free-Fermion Gas..........................154
10.4.1 Properties at T =0................................157
10.4.2 Properties at Low Temperature .....................160
10.5 Properties of a Free-Boson Gas............................169
10.5.1 The Two Kinds of Bose Gas ........................169
10.5.2 Properties at Low Temperature .....................170
10.6 Properties of Gases at High Temperature...................178
Part IV Appendices
A Formulas Related to the Factorial Function ................185
A.1 Binomial Coefficients and Binomial Theorem................185
A.2 Stirling’s Formula .......................................185
A.3 n!!.....................................................186
B The Gaussian Distribution Function .......................187
B.1 The Central Limit Theorem ..............................187
B.1.1 Example .........................................188
B.2 Gaussian Integrals.......................................188
B.3 The Fourier Transform of a Gaussian Distribution Function...189
C Lagrange’s Method
of Undetermined Multipliers...............................191
C.1 Example ...............................................192
C.2 Generalization ..........................................192
D Volume of a Hypersphere..................................193
E Hyperbolic Functions ......................................195
F Boundary Conditions ......................................197
F.1 Fixed Boundary Condition ...............................197
F.2 Periodic Boundary Condition .............................198
G The Riemann Zeta Function ...............................201
References.....................................................203
Index..........................................................205