Thermodynamics
and
Statistical Physics
WS 2013/14
Roser Valentı́
Institut für Theoretische Physik
Universität Frankfurt
Contents
1 Introduction
1.1 Literature . . . . . . . . . .
1.2 What is thermodynamics?
What is statistical physics?
1.3 Historical development . . .
1.4 Concept of effective theory .
. . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
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2
3
6
2 Basic concepts in thermodynamics
2.1 Thermodynamic systems . . . . . . . . .
2.2 Thermodynamic variables . . . . . . . .
2.3 Thermodynamic limit . . . . . . . . . . .
2.4 Notion of temperature . . . . . . . . . .
2.5 Thermal equilibrium and thermodynamic
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
transformations .
3 Work, heat and the first law of thermodynamics
3.1 Mechanical work . . . . . . . . . . . . . . . . . .
3.2 Heat transfer . . . . . . . . . . . . . . . . . . . .
3.3 First law of thermodynamics.
Internal energy . . . . . . . . . . . . . . . . . . .
3.4 Energy and heat capacity for various processes . .
3.4.1 Isochoric process . . . . . . . . . . . . . .
3.4.2 Isobaric process . . . . . . . . . . . . . . .
3.4.3 Isothermal process . . . . . . . . . . . . .
3.4.4 Free expansion of an ideal gas . . . . . . .
3.4.5 Adiabatic process . . . . . . . . . . . . . .
3.5 Heat equations . . . . . . . . . . . . . . . . . . .
3.6 Magnetic systems . . . . . . . . . . . . . . . . . .
4 Entropy and second law of thermodynamics
4.1 Carnot cycle . . . . . . . . . . . . . . . . . .
4.2 Second law of thermodynamics . . . . . . .
4.3 Absolute temperature . . . . . . . . . . . . .
4.4 Temperature as integrating factor . . . . . .
4.5 Entropy . . . . . . . . . . . . . . . . . . . .
4.6 The energy equation . . . . . . . . . . . . .
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34
-2
CONTENTS
4.7
4.8
4.9
Useful coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Entropy and disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Third law of thermodynamics (Nernst law) . . . . . . . . . . . . . . . . . . 38
5 Thermodynamic potentials
5.1 Internal energy U . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Legendre transformation . . . . . . . . . . . . . . . . . . . . .
5.3 Helmholtz free energy . . . . . . . . . . . . . . . . . . . . . . .
5.4 Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Gibbs enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Homogeneity relations . . . . . . . . . . . . . . . . . . . . . .
5.7 Grand canonical potential . . . . . . . . . . . . . . . . . . . .
5.8 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . .
5.8.1 Isolated system . . . . . . . . . . . . . . . . . . . . . .
5.8.2 Closed system in a reservoir without work interchange
5.9 Summary of thermodynamic potentials . . . . . . . . . . . . .
6 Phase transitions
6.1 Concept of phase . . . . . . . . . . . . . .
6.2 First-order phase transition . . . . . . . .
6.3 Condition for phase coexistence . . . . . .
6.4 Clapeyron equation . . . . . . . . . . . . .
6.5 Ehrenfest classification of phase transitions
6.6 Van der Waals equation of state . . . . . .
6.6.1 Virial expansion . . . . . . . . . . .
6.6.2 Critical point . . . . . . . . . . . .
6.6.3 Maxwell construction . . . . . . . .
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7 Statistical physics in equilibrium
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The statistical approach . . . . . . . . . . . . . . . . .
7.3 Phase space . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Thermal equilibrium from the microscopic view . . . .
7.5 Distribution function . . . . . . . . . . . . . . . . . . .
7.6 Ergodic hypothesis . . . . . . . . . . . . . . . . . . . .
7.7 Fundamental postulate of equally a priori probabilities
7.8 Statistical ensemble . . . . . . . . . . . . . . . . . . . .
7.9 Liouville theorem . . . . . . . . . . . . . . . . . . . . .
8 Microcanonical ensemble
8.1 Definition . . . . . . . . . . . . . . . . . . . .
8.2 Entropy . . . . . . . . . . . . . . . . . . . . .
8.3 Calculations with the microcanonical ensemble
8.4 The classical ideal gas . . . . . . . . . . . . .
8.5 Harmonic oscillators . . . . . . . . . . . . . .
8.6 Fluctuations and correlation functions . . . .
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-1
CONTENTS
8.7
The central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.7.1 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9 Canonical ensemble
9.1 System in contact with a heat reservoir
9.2 Interpretation of the exponential:
Boltzmann factor . . . . . . . . . . . .
9.3 Canonical partition function . . . . . .
9.3.1 Thermodynamic properties . . .
9.4 Comparison between canonical
and microcanonical ensembles . . . . .
9.5 Additivity of F (T, V, N ) . . . . . . . .
9.6 Ideal gas in the canonical ensemble . .
9.7 Energy fluctuations . . . . . . . . . . .
9.8 Paramagnetism . . . . . . . . . . . . .
103
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. . . . . . . . . . . . . . . . . . . . 110
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111
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116
10 Grand canonical ensemble
10.1 Grand canonical partition function . . . . . . . . .
10.2 Fugacity . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Particle distribution function . . . . . . . . . . . . .
10.4 Grand canonical potential . . . . . . . . . . . . . .
10.5 Particle fluctuations . . . . . . . . . . . . . . . . .
10.6 Example: ideal gas in the grand canonical ensemble
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121
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131
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. 133
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. 148
. 150
12 Quantum gases
12.1 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Partition function of the ideal quantum gases . . . . . . . . . . . . . . .
153
. 153
. 155
. 157
11 Quantum statistics
11.1 Thermal wavelength . . . . . . . . . . . .
11.2 Fundamentals of quantum statistics . . . .
11.3 Statistical operator: density matrix . . . .
11.3.1 Pure state . . . . . . . . . . . . . .
11.3.2 Mixed state . . . . . . . . . . . . .
11.4 Properties of ρ̂ . . . . . . . . . . . . . . .
11.5 Statistical operator in thermal equilibrium
11.6 Correspondence principle . . . . . . . . . .
11.7 Microcanonical ensemble . . . . . . . . . .
11.8 Third law of thermodynamics . . . . . . .
11.9 Example in the microcanonical ensemble: .
11.10Canonical ensemble . . . . . . . . . . . . .
11.11Example in the canonical ensemble: . . . .
11.12Grand canonical ensemble . . . . . . . . .
11.13Entropy and the density matrix operator .
11.14Information theory . . . . . . . . . . . . .
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0
CONTENTS
12.4 Thermodynamical properties of ideal quantum gases . . . . . . . . . . . . . 159
12.5 Particle distribution function for bosons and fermions . . . . . . . . . . . . 160
13 Ideal Fermi gas
13.1 Equation of state . . . . . .
13.2 Classical limit . . . . . . . .
13.3 Fermi energy . . . . . . . .
13.4 Ground state . . . . . . . .
13.5 Fermi temperature . . . . .
13.6 Low temperature properties
13.7 Particles and holes . . . . .
13.8 Density of states . . . . . .
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14 Ideal Bose gas
14.1 Equation of state . . . . . . . . . . . .
14.2 Classical limit . . . . . . . . . . . . . .
14.3 Bose-Einstein condensation . . . . . . .
14.4 Thermodynamics for the ideal Bose gas
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163
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174
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177
. 177
. 180
. 182
. 187
Chapter 1
Introduction
1.1
Literature
1) K. Huang, ”Introduction to Statistical Physics”, CRC Press, 2001.
2) E. M. Lifschitz and L. P. Pitajewski, ”Statistical Physics”, London, Landau Lifschitz
Band 5.
3) S. K. Ma, ”Statistical Mechanics”, World Scientific Publishing, 1985.
4) W. Nolting, ”Grundkurs Theoretische Physik”, 2. Halfte Band 4: Thermodynamik,
Band 5: Statistische Physik.
5) F. Reif, ”Fundamentals of Statistical and Thermal Physics”, Mc Grow-Hill, Singapore, 1964.
6) R. Baierlein, ”Thermal Physics”, Cambridge, 1999.
7) F. Mandl, ”Statistical Physics”, Wiley, N. Y, 2nd Edition 1988.
8) M. Toda, R. Kubo, N. Saito, ”Statistical Physics I, Equilibrium Statistical Mechanics”, Springer, 2nd Edition 1992.
9) P. Kopietz, ”Thermodynamik und Statistische Mechanik” Script 2007.
10) Harrey Gould and Jan Tobochnik, ”Statistical and Thermal Physics with computer
applications”, Princeton University Press (2010).
Acknowledgements I would like to thank Peter Kopietz for providing his 2007 script.
Some of the chapters in the present manuscript contain a revised version of the contents
in his 2007 script. And I would also like to thank Kateryna Foyevtsova for her help in
typing the present manuscript.
1
2
CHAPTER 1. INTRODUCTION
1.2
What is thermodynamics?
What is statistical physics?
Thermodynamics : phenomenological theory of the macroscopic properties of systems
with a large number of constituents (particles, degrees of freedom). This theory describes
the many-body system in terms of a few experimentally measurable parameters like
- Volume V ,
- Pressure P ,
- Temperature T ,
- Density ρ.
Phenomenological theory means that it does not provide the microscopic origin of a phenomenon.
A large number of constituents means that the fluctuation effects of the constituents are
negligible at the scale at which we are studying the system. A macroscopic object (1 cm3 )
has usually
N ≃ NAvogadro = 1 mole ≃ 6.022 × 1023 particles.
In such a system, the fluctuation effects of a macroscopic observable O are of the order:
∆O
1
∼ √ ≈ 10−12 ,
O
N
which is a negligible effect if we study the system at the meter scale but will gain in
importance if we consider shorter scales as would be required, for instance, for present
electronic devices, which have sizes of 1 nm=10−9 m.
Statistical physics provides a description of the macroscopic properties of a many-body
system in terms of the microscopic structure of the system.
Statistical physics considers the basic equations of Classical Mechanics (classical statistical
physics) or Quantum Mechanics (quantum statistical physics) and describes the macroscopic quantities in terms of the microscopic behavior. Statistical physics is more general
than thermodynamics and the laws of thermodynamics can be derived from statistical
physics.
Classical statistical physics. As a reminder, the basic equations in classical mechanics for
a system of N particles are given in the Hamilton formalism as
q̇i =
∂H
,
∂pi
ṗi = −
∂H
,
∂qi
i = 1, 2, . . . , 3N,
(1.1)
with H = H(q1 . . . q3N , p1 . . . p3N , t) being the Hamilton function and qi and pi the generalized position and momentum coordinates.
1.3. HISTORICAL DEVELOPMENT
3
Our objective in this course will be to relate equations (1.1) to such concepts as ”temperature”, ”pressure”, ”internal energy”, i.e., to establish relations
microscopic equations ←→ macroscopic equations.
Quantum statistical physics. In quantum mechanics, the basic equation for a system with
N particles is the Schrödinger equation:
" N
#
X −h̄2
∂Ψ
2
(~r1 σ1 , ~r2 σ2 , . . . , ~rN σN , t) =
∇ + V (~r1 , . . . , ~rN , t) Ψ(~r1 σ1 , ~r2 σ2 , . . . , ~rN σN , t),
ih̄
∂t
2m i
i=1
where Ψ(~r1 σ1 , ~r2 σ2 , . . . , ~rN σN , t) is the wave-function of the N-particle system.
We have in quantum mechanics two types of particles:
Bosons: particles with integer spin. Their wave-function is symmetric with respect
to interchange of two particles:
Ψ(~r1 σ1 , . . . , ~ri σi , . . . , ~rj σj , . . . , ~rN σN ) = Ψ(~r1 σ1 , . . . , ~rj σj , . . . , ~ri σi , . . . , ~rN σN );
Fermions: particles with half-integer spin. Their wave-function is antisymmetric
with respect to interchange of two particles:
Ψ(~r1 σ1 , . . . , ~ri σi , . . . , ~rj σj , . . . , ~rN σN ) = −Ψ(~r1 σ1 , . . . , ~rj σj , . . . , ~ri σi , . . . , ~rN σN ).
As in the classical statistical physics, our aim in quantum statistical physics is to obtain,
out of the microscopic equations, a description of the macroscopic properties, which can
be related to experiment.
1.3
Historical development
I. Thermodynamics
Thermodynamics was born out of a few unresolved important questions like:
- what is heat?
- what is temperature?
- why do some reactions happen and others do not?
- is it possible to convert heat in useful energy?
Chronology:
1600’s - Concept of heat: already in the seventeenth century some scientists thought
that heat is associated with the motion of microscopic constituents of matter but
the idea was too revolutionary at that time.
4
CHAPTER 1. INTRODUCTION
1700’s - there was a general belief that heat is a separate fluid-like substance.
1840’s - experiments of James Joule invalidated the theory of a separate fluid.
1850’s - it became accepted that heat is a form of energy.
1824 - Sadi Carnot (French engineer and physicist) showed that the work produced
by a steam engine is proportional to the heat transferred from the boiler to the
condenser and that, in general, work could only be gained from heat by a transfer
from a warmer to a colder body (Carnot’s law).
1842 - Julius Robert von Mayer (German physicist) formulated the general law
of conservation of energy and defined heat as a form of energy.
1843 - James Joule (British physicist) formulated the Joule’s laws and determined
the mechanical equivalent of heat.
– The heat generated by the current flowing through a conductor is Q = I 2 Rt.
– The internal energy of an ideal gas depends only on its temperature.
1850 - Rudolf Clausius (German physicist) formulated the second law of thermodynamics and introduced the entropy concept (1865) (in the same year Maxwell
postulated the Maxwell equations in electrodynamics).
The second law of thermodynamics reads:
”Heat cannot be transformed from a colder to a warmer body”.
Ideal Gas
→ In the nineteenth century, there were also a few advances concerning the behavior
of the gas phase:
1807 - Joseph Gay-Lussac (French chemist and physicist) investigated the changes
in volume of a gas at different temperatures and showed that at constant pressure
V
V0
= ,
T
T0
(1.2)
where T is absolute temperature in K (remember T = 273.15 K + t (in Celsius)).
At constant temperature V P = V0 P0 .
1811 - Amadeo Avogadro (Italian physicist and chemist) postulated the Avogadro’s law:
”The same volume of different gases under same conditions (T , P ) contains the
same number of particles”.
This postulate together with (1.2) can be summarized in the equation of state for
an ideal gas, which we will analyze in detail in the coming chapters:
P V = N kB T,
5
1.3. HISTORICAL DEVELOPMENT
where N is number of particles and kB =1.381×10−23 Joule/K is the Boltzmann
constant.
II. Statistical physics
1860 - James Maxwell (British physicist) laid the foundations of the modern
electrodynamic theory of light. He was also involved in the development of the
kinetic gas theory.
1870’s - Ludwig Boltzmann (Austrian physicist). His contributions were fundamental. He was the first scientist to provide a statistical interpretation of the 2nd
law of thermodynamics:
S = kB ln Γ,
namely the entropy is proportional to the logarithm of the number of available states
given by Γ.
The fact that he used the concept of atoms in motion in order to explain the concept
of heat provoked very strong criticisms at that time (Ernst Mach, Wilhelm Otswald).
1902 - Josiah Willard Gibbs (US American physicist and mathematician) formulated the ”Gibbs phase rule” which allows to calculate thermodynamical properties
out of microscopic quantities.
Twentieth Century
→ The twentieth century is the century of quantum mechanics and quantum statistical mechanics.
1900 - Max Planck (German physicist) postulated that the energy exchange of the
radiation with the walls of a black body does not take place continuously but in
discrete units of h̄ω,
εn = nh̄ω
←
quantization of the radiation energy,
so that the energy density of radiation of a black body is given by
u(ω, T ) =
kB T ω 2 h̄ω/kB T
.
π 2 c3 eh̄ω/kB T − 1
For his derivation, Planck used concepts of electrodynamics, quantum mechanics
(quantization of energy) and statistical physics, following Boltzmann.
1925 - Satyendranath Bose (Indian physicist) and Albert Einstein (German
physicist)
• derived the statistics (Bose-Einstein statistics) for photons and, in general, for
a system of bosons.
6
CHAPTER 1. INTRODUCTION
• predicted the phenomenon of ”Bose-Einstein condensation” which was first
corroborated experimentally 70 years later by W. Ketterle, E. Cornell and C.
E. Wieman (Nobel Prize 2001).
1926 - Enrico Fermi (Italian physicist) and Paul Dirac (British physicist) derived
the quantum statistics for fermions (Fermi-Dirac statistics).
1940: Wolfgang Pauli (Austrian physicist) stated the spin-statistics theorem which
associates the value of the spin of a particle to the nature of the particle.
Bosons: integer spin
Fermions: half-integer spin.
Important developments were done after that regarding, for instance,
1 - treatment of phase transitions, both macroscopically and microscopically (Landau, Ginzburg, Bardeen, Cooper, Schrieffer, ...).
Important phase transitions are of type
”disordered” phase → ”ordered” phase, which include
-
gas-liquid-solid,
paramagnetic-ferromagnetic (antiferromagnetic),
liquid-superfluid (He4 , He3 ) transitions,
metal-superconductor.
2 - treatment of interacting many-particle systems, especially if the interactions
are strong. Areas of application are
- solid state theory: correlated systems (strong Coulomb repulsion),
- high-energy physics: quantum chromodynamics (QCD).
3 - systems in equilibrium ←→ non-equilibrium systems.
1.4
Concept of effective theory
We have defined thermodynamics as a phenomenological theory which was developed
to describe phenomena at the scale of our naked-eye observations (length-scales ≈ 1cm,
time-scales ≈ 1s). At these scales the knowledge of the atomic constituents is irrelevant.
Thermodynamics is an effective theory and, in general, effective theories are useful for the
understanding of observations at a given length- and time-scale.
Statistical physics is, on the other hand, a fundamental theory since it intends to describe
macroscopic phenomena out of the laws governing the microscopic constituents (atoms,
electrons) of the systems.