Chapter
1.12 The thermodynamic temperature
1.13 The Carnot cycle of an ideal gas
1.14 The Clausius inequality
1.16 General integrating factors
1.17 The integrating factor and cyclic processes
Process (i) Isothermal within each subsystem; adiabatic for the combined system
Process (ii) Adiabatic for each subsystem individually
Process (iii) Isothermal, within each subsystem; adiabatic for the combined system
Process (iv) Adiabatic for each system individually
1.19 Employment of the second law of thermodynamics
1.20 The universal integrating factor
2 Thermodynamic relations
2.1 Thermodynamic potentials
2.4 The Clausius–Clapeyron equation
2.5 The van der Waals equation
3.1 Microstate and macrostate
3.2 Assumption of equal a priori probabilities
3.3 The number of microstates
3.4 The most probable distribution
3.6 Resolution of the Gibbs paradox: quantum ideal gases
3.8 Thermodynamic relations
3.10 The grand canonical distribution
3.11 The grand partition function
3.12 The ideal quantum gases
4.1 Representations of the state vectors
4.1.1 Coordinate representation
4.1.2 The momentum representation
4.1.3 The eigenrepresentation
4.2 The unitary transformation
4.3 Representations of operators
4.4 Number representation for the harmonic oscillator
4.5 Coupled oscillators: the linear chain
4.6 The second quantization for bosons
4.7 The system of interacting fermions
4.8 Some examples exhibiting the effect of Fermi–Dirac statistics
4.8.2 The hydrogen molecule
4.9 The Heisenberg exchange Hamiltonian
4.10 The electron–phonon interaction in metal
4.12 The spin-wave Hamiltonian
5.1 The canonical partition function
5.3 The perturbation expansion
5.4 Reduced density matrices
5.5 One-site and two-site density matrices
5.5.1 One-site density matrix
5.5.2 Two-site density matrix
5.6 The four-site reduced density matrix
5.6.1 The reduced density matrix for a square cluster
5.6.2 The reduced density matrix for a regular tetrahedron cluster
5.7 The probability distribution functions for the Ising model
5.7.1 The one-site reduced distribution function
5.7.2 The two-site distribution function
5.7.3 The equilateral triangle distribution function
5.7.4 The four-site (square) distribution function
5.7.5 The four-site (tetrahedron) distribution function
5.7.6 The six-site (regular octahedron) distribution function [Exercise 5.7]
6 The cluster variation method
6.1 The variational principle
6.2 The cumulant expansion
6.3 The cluster variation method
6.4 The mean-field approximation
6.5 The Bethe approximation
6.6 Four-site approximation
6.7 Simplified cluster variation methods
6.8 Correlation function formulation
6.8.1 One-site density matrix
6.8.2 Two-site density matrix
6.9 The point and pair approximations in the CFF
6.10 The tetrahedron approximation in the CFF
7 Infinite-series representations of correlation functions
7.1 Singularity of the correlation functions
7.2 The classical values of the critical exponent
7.2.1 The mean-field approximation
7.2.2 The pair approximation
7.3 An infinite-series representation of the partition function
7.4 The method of Padé approximants
7.5 Infinite-series solutions of the cluster variation method
7.5.1 Mean-field approximation
7.5.3 Tetrahedron approximation
7.5.4 Tetrahedron-plus-octahedron approximation
7.6 High temperature specific heat
7.6.2 Tetrahedron approximation
7.6.3 Tetrahedron-plus-octahedron approximation
7.7 High temperature susceptibility
7.7.1 Mean-field approximation
7.7.3 Tetrahedron approximation
7.7.4 Tetrahedron-plus-octahedron approximation
7.8 Low temperature specific heat
7.8.1 Mean-field approximation
7.8.3 Tetrahedron approximation
7.8.4 Tetrahedron-plus-octahedron approximation
7.9 Infinite series for other correlation functions
8 The extended mean-field approximation
8.1 The Wentzel criterion
8.4 The ground state of the Anderson model
8.6 The first-order transition in cubic ice
9 The exact Ising lattice identities
9.1 The basic generating equations
9.2 Linear identities for odd-number correlations
9.3 Star-triangle-type relationships
9.4 Exact solution on the triangular lattice
9.5 Identities for diamond and simple cubic lattices
9.5.2 Simple cubic lattice
9.6 Systematic naming of correlation functions on the lattice
9.6.1 Characterization of correlation functions
Site-number representation
Occupation-number representation
Vertex-number representation
9.6.2 Is the vertex-number representation an over-characterization?
9.6.3 Computer evaluation of the correlation functions
10 Propagation of short range order
10.1 The radial distribution function
10.2 Lattice structure of the superionic conductor AlphaAgI
10.3 The mean-field approximation
10.4 The pair approximation
10.5 Higher order correlation functions
10.6 Oscillatory behavior of the radial distribution function
11 Phase transition of the two-dimensional Ising model
11.1 The high temperature series expansion of the partition function
11.2 The Pfaffian for the Ising partition function
11.3 Exact partition function
Appendix 1 The gamma function
A1.1 The Stirling formula
A1.2 Surface area of the N-dimensional sphere
Appendix 2 The critical exponent in the tetrahedron approximation
Appendix 3 Programming organization of the cluster variation method
A3.1 Characteristic matrices
A3.2 Properties of characteristic matrices
A3.3 Susceptibility determinants
Appendix 4 A unitary transformation applied to the Hubbard Hamiltonian
Appendix 5 Exact Ising identities on the diamond lattice
A5.1 Definitions of some correlation functions
A5.1.1 Even correlation functions
A5.1.2 Odd correlation functions
A5.2 Some of the Ising identities for the odd correlation functions
Methods of Statistical Physics
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