List of Units
Part I Atomic and mesoscopic discussions
Section 1. Introduction
1.1 Atomisms, ancient and modern
Mach and atomism [This color indicates a footnote with a title]
Atomism and humanism
1.2 What was beyond philosophers' grasp?
1.3 How numerous are atoms and molecules?
1.4 Why are molecules so small?
1.5 Our world is lawful to the extent of allowing the evolution of intelligence
1.6 Microscopic world is unpredictable
1.7 Why our macroscopic world is lawful: the law of large numbers
Random variables
Landau symbol o
1.8 We live in a rather gentle world
An outline of the book follows:
1.9 Thermodynamics, statistical mechanics and phase transition
1.10 Mesoscopic world
1.11 Large deviation and fluctuation
Q1.1 Big numbers
Q1.2 Atoms
Section 2. Atomic picture of gases
2.1 Aristotelian physics and Galileo's struggle
Archimedean mechanic was crucial
2.2 Boyle: the true pioneer of kinetic theory of heat
Thermal physics history reading
2.3 Discovery of atmospheric pressure and vacuum
2.4 Daniel Bernoulli's modern dynamic atomism and 100 year hiatus
From the preface to Principia
Acceptance of molecular theory
2.5 Between Bernoulli and Maxwell
2.6 Daniel Bernoulli's kinetic theory
2.7 Equipartition of kinetic energy
Q2.1 Elementary questions
Carboniferous atmospheric oxygen
Q2.2 Effusion and Graham's law
Q2.3 Dalton's law by Bernoulli's theory
Q2.4 Relative velocities
Section 3. Introduction to Probability
Introduction by Kolmogorov
measure
3.1 Probability is a measure of confidence level
Evolution or phylogenetic learning
3.2 Events and sets
3.3 Probability
3.4 Additivity
3.5 Probability measure"
3.6 Relation to combinatorics
3.7 Objectivity of subjective probability
3.8 Conditional probability
3.9 Statistical independence
3.10 Stochastic variables
Probability space
3.11 Expectation value
Operator
3.12 Variance
3.13 Indicator
3.14 Independence of stochastic variables
3.15 Independence and correlation
3.16 Stochastic process
Appendix 3A: Rudiments of combinatorics
3A.1 Sequential arrangement (without repetition) of objects from n distinguishable objects: nPr
3A.2 Selection (without repetition) of r objects from n distinguishable objects: binomial coefficient
3A.3 Binomial theorem
3A.4 Multinomial coefficient
3A.5 Multinomial theorem
3A.6 Arrangement of indistinguishable objects into distinguishable boxes
Q3.1 Elementary problems
Q3.2 Fun problems
Q3.3 Independence of multiple events: pairwise independence is not enough
Q3.4 Bayes' theorem and P-value paradox
Q3.5 Interatomic space distribution
Q3.6 Bertrand's paradox
Section 4. Law of large numbers
4.1 How can we measure probability?
4.2 Precise statement of the law of large numbers
Two kinds of law of large numbers
4.3 Why is the law of large numbers plausible?
4.4 Chebyshev's inequality and a proof of LLN
4.5 Almost no fluctuation of internal energy, an example
4.6 Monte Carlo integration
4.7 Central limit theorem
Symbol O
Central limit theorem references
4.8 There is no further refinement of LLN
Q4.1 Very elementary questions
Q4.2 Law of large numbers does not hold, if the distribution is too broad
Q4.3 St. Petersburg Paradox by Daniel Bernoulli
Section 5. Maxwell's distribution
5.1 Density distribution function
n-object
5.2 Maxwell's derivation of Maxwell's distribution function
Equilibrium
How to solve Cauchy's functional equation
5.3 Gaussian integral
5.5 Moment generating function
d: spatial dimensionality notation
5.6 Bernoulli revisited
Important Remark
5.7 How to derive the Boltzmann factor
5.8 Elementary derivation of Maxwell's distribution
Q5.1 Doppler observation of Maxwell's distribution
Q5.2 The mode of the molecular speed
Q5.3 Density distribution of relative velocity, an elementary approach
Q5.4 Probability of the kinetic energy larger than the average
Section 6. Delta function and density distribution functions
6.1 What average gives the density distribution?
`formally'
6.2 Introducing delta-function
d-object
6.3 Formal expression of density distribution
6.4 Distribution function: 1D case
6.5 Density distribution function: 1D case
6.6 delta-function
Theory of distribution
6.7 Key formula 1
6.8 Key formula 2
6.9 Some practice
6.10 How to compute density distribution functions
6.11 Kinetic energy density distribution of 2D ideal gas
Appendix 6A. Extreme rudiments of distributions
6A.1 Motivation of theory: delta function as linear functional
6A.2 Generalized function
6A.3 Test functions
On test function sets
6A.4 Equality of generalized functions
6A.5 delta-function: an official definition
6A.6 Differentiation of generalized functions
6A.7 All the ordinary rules for differentiation survive
6A.8 Value of generalized function at each point is meaningless
6A.9 Convergence of generalized function
6A.10 Definition of generalized function using limits
Q6.1 Density distribution of speed
Q6.2 Density distribution of kinetic energy in 3-space
Q6.3 Relative velocity distribution revisited
Section 7. Mean free path
7.1 Mean free path
Darwin and Boltzmann
Mean free path in air
7.2 Effect of molecular collisions
7.3 Spatial spread of random walks
7.4 Mixing process of two distinct gases: each molecule walks randomly
7.5 Characterizing the equilibrium macrostate
Microstate
Macrostate
7.6 Equilibrium macrostate is most probable
Approximation for N!
7.7 Equilibrium macrostate is stable
7.8 `Entropic force'
Entropic force o each molecule
7.9 How we can observe entropic force
Q7.1 Velocity dependence of the mean-free path
Q7.2 Distribution of free path for a constant speed particle
Section 8. Introduction to transport phenomena
8.1 What is a (linear) transport phenomenon?
Linear functional
8.2 Density
8.3 Flux
8.4 Linear transport law
8.5 Gradient
8.6 Conservation law and divergence
Notation for boundary
8.7 Local expression of conservation law
8.8 Diffusion equation
8.9 The meaning of the Laplacian
8.10 Intuitive computation of transport coefficient
8.11 Diffusion constant
8.12 Shear viscosity
8.13 Maxwell estimated the molecular size
Definition of Avogadro's constant
8.14 Thermal conductivity
8.15 Dimensional analysis of transport coefficients
Introduction to dimensional analysis
8.16 Significance of J propto grad x
Q8.1 Estimating Avogadro's constant from diffusion
Section 9. Brownian motion
9.1 How mesoscopic particles behave
9.2 Brown discovered a universal motion (now) called the Brownian motion
Who was R. Brown?
9.3 General properties of Brownian motion
Why no founders
9.4 Langevin's explanation of the Brownian motion
9.5 Relation to random walk
1905
9.6 Einstein's theory of Brownian particle flux
9.7 Einstein's formula
Relevant hydrodynamics
9.8 The Einstein-Stokes relation: dimensional analysis
9.9 Displacement of particles by diffusion
9.10 Einstein's fundamental idea: summary
9.11 Summary of the Boltzmann constant kB
Appendix 9A: How to obtain (9.23)
Q9.1 Perrin reproduced
Q9.2 2D lattice random walk
Section 10. Langevin equation, uctuation-dissipation relation and large deviation
10.1 Overdamped Langevin equation
10.2 Langevin noise: qualitative features
Warning
10.3 Langevin noise
10.4 Smoluchowski equation
10.5 Relation between the Langevin noise and D: step (ii)
10.6 Smoluchowski equation and fluctuation-dissipation relation step (iii)
10.7 Significance of the fluctuation-dissipation relation
10.8 Mesoscopics and large deviation principle
Rate function summary
10.9 Three `infinitesimal times, ¥D t, ¥d t , dt
Time scale ratio: we are almost eternal
10.10 Time averaging to get mesoscopic results
10.11 Langevin equation as a result of large deviation theory
10.12 Langevin equation: practical summary
Q10.1 Brownian motion of a harmonic oscillator, basic questions
Q10.2 Diffusing globular proteins
Q10.3 Large deviation for binomial distribution
Part II Statistical Thermodynamics: Basics
Section 11. Macrosystems in equilibrium
11.1 How to describe a macroscopic system in mechanics
11.2 Conservation of mechanical energy and the first law of thermodynamics
Mayer, Joule and Helmholtz
Is Helmholtz's Mechanische Weltanschauung empirical?
11.3 What mechanics?
11.4 Additivity of energy
11.5 Why we are interested in additive quantities
11.6 The fourth law of thermodynamics
The forth law
Superextensive quantities?
The Fourth Law
11.7 Time-reversal symmetry of mechanics
11.8 Irreversibility from mechanics?
Physicists were philosophical those days
Poincare's theorem on recurrence
Zermelo and statistical mechanics
11.9 Boltzmann equation
We do not dwell on the Boltzmann equation
11.10 What is the lesson?
11.11 Purely mechanical toy model and irreversibility
11.12 Law of large numbers likely to win
Q11.1 Existence of ground state energy density
Section 12. Thermodynamic description
12.1 What is phenomenology?
Implication of pure mechanics
What is macroscopic?
12.2 Equilibrium states
Equilibrium: another possible characterization
12.3 External disturbance: mechanics vs. thermodynamics
Purely mechanical macrosystems are fictitious
12.4 Thermodynamic coordinates, a privileged set of variables
How can we choose thermodynamic coordinates?
12.5 Thermodynamic space
Very Important Warning
12.6 Quasistatic processes
Why can we generally realize a reversible process by slowing down?
Warning
12.7 State quantities/functions
12.8 Simple system
12.9 Compound system
12.10 Why thermodynamics can be useful
12.11 Partitioning-rejoining invariance
12.12 Thermal contact
12.13 The zeroth law of thermodynamics"
Equivalence relation
Section 13. Basic thermodynamics
13.1 The first law of thermodynamics
Mechanical vs. thermal energy
Work-heat distinction
13.2 Sign convention
13.3 Volume work
13.4 Magnetic work
13.5 Convention for electromagnetic field
13.6 Prehistory of the second law
Industry was far ahead
Joule and Thomson meet
Carnot was totally forgotten
Clapeyron
Thermodynamics is established by Clausius
13.7 The second law of thermodynamics
Adiabatic wall and adiabatic system
13.8 Planck's principle, Kelvin's principle and Clausius' principle are equivalent
13.9 Clausius recognized entropy
Entropy and Gibbs
13.10 Entropy exists
13.11 Entropy stratifies thermodynamic space
S is extensive
13.12 Entropy and heat
Identification of absolute temperature
On the use of the ideal gas
13.13 Entropy principle
13.14 Gibbs relation
Conjugate pairs wrt energy and wrt entropy
Q13.1 Equivalence of heat and work: extremely elementary question
Q13.2 Newcomen vs. Watt
Q13.3 Free expansion and Joule-Thomson processes are irreversible
Q13.4 Magnetic work using magnetic dipoles
Section 14. Thermodynamics: General consequences
14.1 Summary of basic principles
Nernst's joke on the three principles
Standard state function symbols
14.2 Entropy maximization principle
Remark on entropy max principle
14.3 Entropy is concave
14.4 Internal energy minimization principle
14.5 Internal energy is convex
14.6 Extension to non-adiabatic systems
14.7 Clausius' inequality
14.8 Equilibrium conditions between two systems
Section 15. Entropy through examples
15.1 Mayer's relation
15.2 Poisson's relation
15.3 Reversible engine: Carnot's theorem
15.4 The original Carnot's argument using the Carnot cycle of an ideal gas
Carnot's original used the caloric theory
15.5 Fundamental equation of ideal gas
15.6 Adiabatic heat exchange between two blocks at different temperatures
Irreversible case
Reversible case
Heat transfer, however slow, is reversible only across infinitesimal temperature difference
15.7 Sudden doubling of volume
15.8 Entropy and information: preview
Bit
15.9 Mixing entropy
15.10 Entropy changes due to phase transition
Q15.1 Basic questions
Q15.2 Entropy change due to heating, very elementary question
Q15.3 Gas fridge
Q15.4 Explosion in a box
Q15.4 If ¥D S = 0, there ought to be a reversible process
Section 16. Free energy and convex analysis
16.1 Isothermal system
16.2 A by an irreversible process
What does `isothermal' mean?
16.3 Clausius' inequality and work principle
16.4 Free energy minimum principle
16.5 Gibbs relation for A
16.6 Gibbs free energy
16.7 Enthalpy
16.8 Legendre transformation
Involution
16.9 Convex function
16.10 Geometrical meaning of Legendre transformation
Convex functions are continuous
16.11 f* is also convex and f(x) = f**(x) = max_a{ax-f(x)}
16.12 Legendre transformation applied to E
Q16.1 Basic problems
Q16.2 Joule-Thomson effect
Q16.3 Gas under weights
Section 17. Introduction to statistical mechanics
17.1 Power and limitation of thermodynamics
17.2 Why statistical mechanics?
17.3 What do we really need?
17.4 Translation of entropy: Boltzmann's principle
Equivalence relation and equivalence class
17.5 Counting microstates
Why we need the leeway
17.6 Statistical Mechanics is completed
17.7 Classic approximation to classic ideal gas
Why h^{3N}
17.8 Quantum mechanical study of classical ideal gas
17.9 Projection and dimension
17.10 Derivation of Boltzmann's formula: key observations
17.11 Typical states in ? w(E,X) give the identical equilibrium state
17.12 Derivation of Boltzmann's formula
Q17.1 Einstein model with microcanonical formalism
Q17.2 How Boltzmann reached his entropy formula
Q17.3 Boltzmann's principle does not contradict thermodynamics
Q17.4 High dimensional volume is near its skin
Q17.5 The volume of a d-ball
Appendix 17A Linear algebra in Dirac's notation
17A.1 Ket and bra
17A.2 Bracket product
17A.3 Orthonormal basis
17A.4 Componentwise representation
17A.5 Resolution of unity
17A.6 Changing basis
17A.7 Unitary operator
17A.8 Linear operator on V
17A.9 Unitary transformation of linear operator
17A.10 Eigenvalue problem
17A.11 Spectral decomposition of linear operator
17A.12 Functions of self-adjoint operators
17A.13 Commuting matrices are diagonalizable simultaneously
17A.14 Normality is equivalent to unitary diagonalizability
Section 18. Statistical mechanics of isothermal systems
18.1 Schottky defects
18.2 Entropy of the crystal with Schottky defects
18.3 Stirling's formula
18.4 Schottky type specific heat
18.5 Thermostat
18.6 Canonical formalism
18.7 The Gibbs-Helmholtz equation
18.8 Schottky defects revisited
18.9 Factorization of canonical partition function
18.10 Schottky defect by canonical formalism (continued)
18.11 Frenkel defect and ensemble equivalence
18.12 Equivalence of microcanonical and canonical ensembles
18.13 Frenkel defect, microcanonical approach
18.14 Review: ensemble equivalence
18.15 Analogy between equilibrium statistical mechanics and large deviation theory
Q18.1 Einstein's derivation of canonical formalism from thermodynamics
Q18.2 Elementary quizzes
Q18.3 Correct thermodynamic potential
Q18.4 Elementary lattice problem
Selection 19. Canonical density operator
19.1 Principle of equal probability: do we need it?
19.2 What can we do it we assume principle of equal probability?
19.3 Density operator
Positive definite operator
19.4 Microcanonical density operator
19.5 Canonical density operator
19.6 Classical case: canonical distribution
19.7 Maxwell's distribution revisited
19.8 Equilibrium ensemble and time-independence: classical case
19.9 Liouville's theorem
19.10 Equilibrium ensemble and time-independence: quantum case
19.11 Jarzynski's equality
19.12 Demonstration of Jarzynski's equality
Planck's principle from quantum mechanics
19.13 Demonstration of Jarzynski's equality: Quantum case
19.14 Jarzynski's equality and rare fluctuations
(Doubly) stochastic matrix
19.15 Key thermodynamic facts
19.16 Equilibrium state is equivalent to a collection of statistically independent subsystems
19.17 Law of large numbers implies principle of equal probability
19.18 Summary; when we can use the principle of equal probability
Q19.1 Density matrix for a spin system: elementary spin review
Q19.2 Density operator for free particles: perhaps an elementary QM review
Q19.3 Variational principle for free energy (classical case)
Q19.4 Gibbs-Bogoliubov inequality (quantum case)
Demo of Klein's inequality
Q19.5 Variational principle for density operators
Q19.6 Jarzynski's equality
Section 20. Classical canonical ensemble
20.1 Classical ideal gas via canonical approach: single particle
20.2 Classical ideal gas: Gibbs paradox
20.3 Origin of N!
Not chemically identical particles
20.4 Classical partition function of particle systems
20.5 Classical microcanonical partition function for a particle system
20.6 Generalization of equipartition of kinetic energy
20.7 Specific heat of gases, computed classically
Q20.1 Gas under a weight; pressure ensemble
Q20.2 Magnetic phenomena are all quantum statistical
Review of 4-potential
Q20.3 Quartic internal motion
Section 21. Information and entropy
21.1 Gibbs-Shannon formula of entropy
Textbook of information theory
21.2 How about Boltzmann's formula?
21.3 How to quantify information
21.4 Shannon formula
Claude Shannon
21.5 Average surprisal
21.6 Entropy vs Information
21.7 `Thermodynamic unit' of information
21.8 Subjective and objective information
21.9 How to quantify the amount of knowledge (gained)
21.10 Information per letter of English
21.11 What is 1/2 (or any fraction of a) question?
21.12 Summary of information vs entropy
21.13 Statistical mechanics from information theory?
21.14 Sanov's theorem and entropy maximization
Q21.1 Elementary question
Q21.2 Information-based microscopic guess
Section 22. Information and thermodynamics
22.1 Converting information into work
22.2 Second law with supplied information
22.3 Clausius' inequality with supplied information
22.4 What is measurement?
22.5 Conditional entropy
22.6 What information can we get from measurement?
22.7 Cost of handling acquired information
Heat required by memory erasure
22.8 Use of information cannot circumvent the second law
22.9 Significance of information thermodynamics
Q22.1 Elementary questions
Q22.2 Bound for mutual information
Selection 23. Oscillators at low temperatures
23.1 Ideal gas with internal vibration
23.2 Classical statistical mechanics of a harmonic oscillators
23.3 Quantum statistical mechanics of a harmonic oscillators
23.4 Systems consisting of identical harmonic oscillators
23.5 Einstein's explanation of small specific heat of solid at low temperatures
23.6 Classically evaluated entropy must have a serious problem
23.7 The third law of thermodynamics
23.8 Some consequences of the third law
23.9 Empirical low temperature heat capacity of solids
23.10 Real 3D crystal: Debye model
23.11 Debye model; thermodynamics
Q23.1 Sackur-Tetrode equation
Q23.2 Specific heats near T = 0
Part II Statistical Thermodynamics: developments
Section 24. How to manipulate partial derivatives
24.1 Rubber band experiment
24.2 Polymer chain is just as kids playing hand in hand
24.3 Freely jointed polymer chain
24.4 Freely-jointed polymer entropy
24.5 Ideal rubber band
24.6 Partial derivative review
24.7 Maxwell's relations
Young's theorem
24.8 Remarks on the notation of partial derivatives
24.9 Jacobian expression of partial derivatives
What is independent, what is dependent
24.10 Chain rule in terms of Jacobians
`Formal'
24.11 Expression of heat capacities
24.12 (Unified) Maxwell's relation
24.13 Rubber band thermodynamics
24.14 Adiabatic cooling with rubber band
24.15 Cooling via adiabatic demagnetization
24.16 Ideal magnetic system
Q24.1 Basic problems
Q24.2 Negative temperature is very hot
Q24.3 Ideal rubber model example
Section 25. Thermodynamic stability
25.1 Need for stability criteria
25.2 Two kinds of inequality in thermodynamics
25.3 Universal stability criterion
25.4 Universal stability criterion in terms of internal energy
25.5 Stability and convexity
25.6 Le Chatelier's principle
25.7 Le Chatelier-Braun's principle
K. F. Braun
25.8 2 2 stability criterion
Principal minors
Q25.1 Elementary problems related to Le Chatelier-Braun
Q25.2 More stability-related questions
Section 26. Fluctuation and response
26.1 Importance of fluctuations
26.2 Generalized Gibbs free energy
26.3 Generalized canonical formalism
26.4 Fluctuation-response relation
26.5 Three key observation about fluctuation-response relations
26.6 Extension to multivariable cases
Note on the quantum version
26.7 Onsager's regression hypothesis and the Green-Kubo formula
26.8 Mesoscopic fluctuations: introduction to Einstein's theory
26.9 Einstein's fundamental formula for small fluctuations
26.10 Practical form of fluctuation probability
26.11 Fluctuation and reversible work needed to create it
26.12 Too rapid fluctuations
26.13 How to use the practical formula
Fluctuation in the T-> 0 limit
26.14 Multivariate Gaussian distribution
Q26.1 Internal energy fluctuation
Q26.2 Easy questions about fluctuations
Q26.3 Thermodynamic fluctuations, general questions
Q26.4 Fluctuation and Le Chatelier-Braun principle
Q26.5 Not `clever' choice of variables
Q26.6 Fluctuation and spring constant
Appendix 26A. Onsager's theory of irreversible processes
26.15 Small deviation from equilibrium
26.16 Mesoscopic description of phenomenological laws
26.17 Large deviation formalism of Onsager's regression hypothesis
26.18 Fluctuation-dissipation relation of the first kind
26.19 Fluctuation-dissipation relation
26.20 Green-Kubo relation
26.21 Large deviation formalism of Onsager's theory:summary
26.22 Onsager reciprocity
26.23 Spatially nonuniform case: transport phenomena
26.24 Green-Kubo formula for transport coefficients
Section 27 Chemical potential
27.1 Open systems
27.2 Mass action and chemical potential
Independent chemicals
27.3 Gibbs relation with math action
27.4 Gibbs-Duhem relation
The energy zero
27.5 Gibbs relation for densities
27.6 Equilibrium condition
27.7 Phase equilibria
Continuous and discontinuous phase transitions
27.8 Clapeyron-Clausius equation
27.9 Chemical potential of ideal gas
27.10 Chemical potential of ideal solutions
27.11 Raoult's law
27.12 Osmotic pressure
27.13 Colligative properties
27.14 Chemical reactions
Mole
Elementary reactions
Fugacity
27.15 Equilibrium condition for reactions: the law of mass action
27.16 Shift of chemical equilibrium
Q27.1 Euler's theorem about homogeneous functions
Q27.2 Dilution limit
Q27.3 Vapor pressure of silicon, an elementary question
Q27.4 Making diamond, an elementary question
Q27.5 Colligative properties continued
Q27.6 Ammonia synthesis
Q27.7 Saha equation for ionization
Section 28. Grand canonical ensemble
28.1 Grand canonical formalism
28.2 Example: adsorption
28.3 Microstates for non-interacting indistinguishable particle systems
28.4 Grand partition function of indistinguishable particle system
28.5 Bosons and fermions
28.6 Ideal boson systems
28.7 Ideal fermion systems
Fermi energy and Fermi level
28.8 Classical limit
28.9 Elementary illustrations
28.10 Pressure of ideal systems (same N)
28.11 Pressure of ideal systems (same )
28.12 Universal P-E relation: introduction
28.13 Density of states
28.14 Universal P-E relation: demonstration
Dimensional analytic approach
28.15 Virial equation of state
Q28.1 Ideal gas with the aid of grand canonical ensemble
Q28.2 Poisson distribution
Q28.3 Shannon formula works
Q28.4 Adsorption
Q28.5 Equivalence of canonical and grand canonical ensembles
Q28.6 Very elementary problem about ideal particles
Q28.7 Elementary problem on ideal particles
Q28.8 T required for classic behavior
Q28.9 Occupation number fluctuation
Q28.10 Fermi level of intrinsic semiconductor
Q28.11 Conducting electrons in n-type semiconductor
Section 29. Ideal quantum systems
29.1 Noninteracting fermion pressure at T = 0
29.2 Low temperature specific heat of fermions (intuitively)
29.4 Low temperature entropy of fermion systems
29.5 Low temperature behavior of chemical potential
29.6 Einstein condensation for noninteracting bosons
29.7 Non-condensate population
29.8 Einstein condensation does not occur in 2- and 1-space
29.9 Continuum approximation is always valid for E and P
29.10 Low temperature heat capacity of boson systems
Let us build our intuition
29.11 Isothermal compression
29.12 Adiabatic free expansion
29.13 Adiabatic reversible expansion
Q29.1 Density of one-particle state
Q29.2 Do we have only to treat the ground state as special below Tc?
Q29.3 2D harmonic trap
Q29.4 Peculiar features of condensate
Q29.5 Volume increase of ideal quantum particles
Section 30 Photons and internal motions
30.1 Quantization of harmonic degrees of freedom
30.2 Warning: grand partition function with = 0 is only a gimmick
30.3 Expectation number of photons
30.4 Internal energy of photon systems
30.5 Planck's distribution, or radiation formula
The Sun
30.6 Statistical thermodynamics of black-body radiation
30.7 Black-body equation of state
30.8 Internal degrees of freedom of classical ideal gas
30.9 Rotation and vibration
Q30.1 Einstein's A and B
Part III Elements of phase transition
Section 31. Phases and phase transitions
31.1 What is a phase?
31.2 Statistical thermodynamic key points relevant to phase transitions
31.3 Phase coexistence condition: two phases
31.4 Phase coexistence condition
31.5 Gibbs phase rule
31.6 How G behaves at the phase boundaries?
31.7 Classification of phase transitions
31.8 Typical example of second order phase transition
31.9 Ising model
31.10 Fundamental questions about phase transitions
31.11 Necessity of thermodynamic limit: densities and fields
31.12 Statistical mechanics in the thermodynamic limit
Q31.1 Phase transition: basic questions
Q31.2 Thermodynamic space of 3-Ising model
Q31.3 Melting heat for tetrachlorocarbon
Section 32. Phase transition in d-space
32.1 Order parameter
32.2 Spatial dimensionality is crucial
Entropic effect: defect locations
32.3 There is an ordered phase in 2-space
Entropic effect: boundary fluctuations
Spin dimension
32.4 Interaction `range' is also crucial
32.5 Van der Waals model: `derivation'
van der Waals' biography
32.6 Liquid-gas phase transition described by the van der Waals model
32.7 Kac potential and van der Waals equation of state
32.8 1D Kac potential system may be computed exactly
32.9 What are the strategies to study phase transitions?
32.10 Magnets, liquids and binary mixtures share some common features
32.11 Ising model in d-space: a brief review
32.12 Fluctuation can be crucial in d < 4
Q32.1 van der Waals equation of state
Q32.2 Thermodynamic justification of Maxwell's rule
Q32.3 Phase transition in 1D long-range system
Q32.4 Demonstration of Peierls' theorem
Section 33. Critical phenomena and renormalization
33.1 Typical second order phase transition and correlation length
33.2 Divergence of susceptibilities and critical exponents
33.3 Critical exponent (in)equalities
33.4 Proof of Rushbrooke's inequality: a good review of thermodynamics
33.5 Fluctuation and universality
33.6 Universality, trivial and nontrivial
33.7 What is statistical mechanics for?
33.8 Kadanoff construction
33.9 Scaling law
33.10 Critical index equality
33.11 Renormalization group transformation
33.12 Renormalization group fixed point
33.13 Renormalization group flow
33.14 Central limit theorem and renormalization
33.15 Detailed illustration of real space renormalization group calculation
Triangular lattice 2-Ising coarse-graining
How to specify block spins
Spin-block spin relation
K -> K’ relation
H -> H’ relation
R has been constructed; its fixed points
Fixed point of R
Fixed point: linear approximation to R
Q33.1 Simple 1D renormalization
Selection 34. Mean field and transfer matrix
34.1 Mean field idea
34.2 Quantitative formulation of mean field theory
34.3 The crudest version of the mean-field theory
34.4 Improving mean field approach
34.5 When is the mean field approach reliable?
34.6 Transfer matrix method
34.7 How to compute the product of matrices
Matrix diagonalization
34.8 Why there is no phase transition in 1-space
34.9 Onsager obtained the exact free energy of 2-Ising model
Onsager's biography
Q34.1 Another derivation of mean field theory
Q34.2 Gibbs-Bogoliubov inequality and mean field
Q34.3 Mean field on diamond lattice
Q34.4 Improving 33.3
Q34.5 Transfer matrix exercise
Selection 35. Symmetry breaking
35.1 How to describe the symmetry
35.3 Group and subgroup
35.4 Spontaneous breaking of symmetry
35.5 Symmetry breaking in Heisenberg magnet
35.6 Consequences of symmetry breaking: rigidity
35.7 Nambu-Goldstone bosons: a consequence of breaking of continuous symmetry
Equilibrium states are stable against localized perturbations
35.7 NG bosons do not exist for long-range interaction systems
35.9 Summary of symmetry breaking
35.10 Symmetry breaking requires big systems
Symmetry breaking and size
35.11 What actually selects a particular symmetry broken state?
Section 36. First order phase transition
36.1 First order phase transition example: nematic-isotropic liquid crystal transition
36.2 Caricature model of first order phase transition
36.3 Bifurcations exhibited by the caricature model
36.4 Metastable and unstable states
36.5 Phase ordering kinetics: nucleation and spinodal decomposition
36.6 First order phase transition due to external field change
36.7 Statistical thermodynamic study of first order phase transitions
36.8 Phase transition and internal energy singularity
36.9 Internal energetic description of first order phase transition
36.10 Legendre transformation and phase transition
Q36.1 The slope pf coexisting curves, an elementary question
Q36.2 Lattice gas
Q36.3 Legendre transformation in convex analysis
Appendix: Introduction to mechanics
A.1 Empirical basis of mechanics
A.2 Remarks for those who are familiar with introductory mechanics
A.3 Introduction to Fourier series
A.4 Fourier integral representation of delta-function
A.5 Fourier transformation
A.6 f(x) as position representation of ket(f)
A.7 Fourier analysis in Dirac notation
A.8 Momentum ket
A.9 Matter waves
de Broglie wave
A.10 How potential energy is involved
A.11 Conservation of energy = constancy of frequency of matter wave
A.12 Double slit experiment and superposition principle
Postulate 1. The totality of states of a mechanical system is a vector space.
Postulate 1'. Physical states correspond to rays.
A.13 Evolution of matter wave for an infinitesimal time
A.14 Derivation of Schrodinger's equation
A.15 Schrodinger equation in kets
Postulate 2. [Unitarity of time evolution] The
A.17 Time evolution of many-body systems
A.18 Heisenberg's equation of motions
A.19 Mechanics preserves information/entropy
A.20 Mechanical observables are linear operators
A.21 Pure state and mixed state
A.22 Measurement process and decoherence
Postulate 3. [Composition postulate]
Postulate 4. [Repeatability]
Respectable observables
Postulate 5. Every observable is associated with a Hermitian operator
Quantum jump
A.24 Energy eigenstates and time-independent Schrodinger equation
A.25 Commutativity and simultaneous observation
A.26 Probability of observation: quantum mechanics and probability
Postulate 6. [Born's rule]
A.27 Expectation value of observable
A.28 Uncertainty relation
A.29 Harmonic oscillator
A.30 Spins
A.31 Perturbation theory
A.32 Minimax principle for eigenvalues
A.33 Path integral representation
A.34 Macroscopic particle: classical approximation
A.35 Classical energy
A.36 Hamilton's equation of motion
A.37 Commutator and Poisson bracket
A.38 Canonical coordinates
A.39 Canonical quantization