Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgements -- 1 The laws of thermodynamics -- 1.1 The thermodynamic system and processes -- 1.2 The zeroth law of thermodynamics -- 1.3 The thermal equation of state -- 1.4 The classical ideal gas -- 1.5 The quasistatic and reversible processes -- 1.6 The first law of thermodynamics -- 1.7 The heat capacity -- 1.8 The isothermal and adiabatic processes -- 1.9 The enthalpy -- 1.10 The second law of thermodynamics -- 1.11 The Carnot cycle -- 1.12 The thermodynamic temperature -- 1.13 The Carnot cycle of an ideal gas -- 1.14 The Clausius inequality -- 1.15 The entropy -- 1.16 General integrating factors -- 1.17 The integrating factor and cyclic processes -- 1.18 Hausen's cycle -- 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 -- 2.2 Maxwell relations -- 2.3 The open system -- 2.4 The Clausius-Clapeyron equation -- 2.5 The van der Waals equation -- 2.6 The grand potential -- 3 The ensemble theory -- 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 -- 3.5 The Gibbs paradox -- 3.6 Resolution of the Gibbs paradox: quantum ideal gases -- 3.7 Canonical ensemble -- 3.8 Thermodynamic relations -- 3.9 Open systems -- 3.10 The grand canonical distribution -- 3.11 The grand partition function -- 3.12 The ideal quantum gases -- 4 System Hamiltonians.

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 -- 4.8.1 The Fermi hole -- 4.8.2 The hydrogen molecule -- 4.9 The Heisenberg exchange Hamiltonian -- 4.10 The electron-phonon interaction in metal -- 4.11 The dilute Bose gas -- 4.12 The spin-wave Hamiltonian -- 5 The density matrix -- 5.1 The canonical partition function -- 5.2 The trace invariance -- 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 -- 7.5.2 Pair approximation -- 7.5.3 Tetrahedron approximation -- 7.5.4 Tetrahedron-plus-octahedron approximation -- 7.6 High temperature specific heat -- 7.6.1 Pair approximation -- 7.6.2 Tetrahedron approximation -- 7.6.3 Tetrahedron-plus-octahedron approximation -- 7.7 High temperature susceptibility -- 7.7.1 Mean-field approximation -- 7.7.2 Pair 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 -- 7.8.2 Pair 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 -- 8.2 The BCS Hamiltonian -- 8.3 The s-d interaction -- 8.4 The ground state of the Anderson model -- 8.5 The Hubbard 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 -- 9.5.1 Diamond lattice -- 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 -- Bond 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 -- 10.5.1 AgI -- 10.5.2 Ag2S -- 10.6 Oscillatory behavior of the radial distribution function -- 10.7 Summary -- 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 -- 11.2.1 Lattice terminals -- 11.2.2 The Pfaffian -- 11.3 Exact partition function -- 11.4 Critical exponents -- 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 -- References -- Bibliography -- Index.