ISBN: 3-540-64437-7
TITLE: Statistical Mechanics of Lattice Systems
AUTHOR: Lavis, David A.; Bell, George M.
TOC:

1. Introduction to Thermodynamics and Phase Transitions 1 
1.1 Thermodynamic Variables: Simple Fluids 1 
1.2 Change of Variable and Thermodynamic Potentials 3 
1.3 Response Functions and Thermodynamic Relations 4 
1.4 Magnetic Systems 6 
1.5 Stationary Properties of Thermodynamic Functions 7 
1.6 Phase Equilibrium in the Van der Waals Gas 9 
1.7 The Field-Extensive Variable Representation 
of Thermodynamics 12 
1.8 The Field-Density Representation of Thermodynamics 16 
1.9 General Theory of Phase Equilibrium 18 
1.9.1 A One-Component Fluid 19 
1.9.2 Azeotropy 22 
1.10 Classical Theory and Metastability 23 
1.10.1 Metastability in a One-Component Fluid 25 
1.10.2 The Experimental Situation 27 
Examples 28 
2. Statistical Mechanics 
and the One-Dimensional Ising Model 31 
2.1 The Canonical Distribution 31 
2.1.1 The Thermodynamic Limit 33 
2.1.2 Kinetic and Configuration Variables 33 
2.2 Distributions in General 35 
2.3 Particular Distributions 39 
2.3.1 The Constant Magnetic Field Distribution 39 
2.3.2 The Constant-Pressure (Isobaric) Distribution 39 
2.3.3 The Grand Distribution 40 
2.3.4 Restricted Distributions for Lattice Models 40 
2.4 Magnetism and the Ising Model 42 
2.4.1 The One-Dimensional Ferromagnet in Zero Field 44 
2.4.2 The One-Dimensional Ferromagnet in a Field 47 
2.5 Fluctuations and Entropy 49 
2.6 The Maximum-Term Method 53 
2.6.1 The One-Dimensional Ising Ferromagnet 54 
2.6.2 The General Distribution 56 
2.7 A One-Dimensional Model for DNA Denaturation 57 
Examples 63 
3. The Mean-Field Approximation, 
Scaling and Critical Exponents 67 
3.1 The Ising Model Ferromagnet 67 
3.1.1 Free Energy and Magnetization 68 
3.1.2 Fluctuations in Zero Field 71 
3.2 Interpretations of the Mean-Field Method 76 
3.2.1 Many-Neighbour Interactions 
and the LebowitzPenrose Theorem 76 
3.2.2 A Distance-Independent Interaction 77 
3.3 The Mean-Field Method for a More General Model 78 
3.4 Critical Points and Critical Exponents 79 
3.5 Scaling and Exponent Relations 84 
3.6 Classical Critical Exponents 86 
3.6.1 The Ising Model Ferromagnet: 
Mean-Field Approximation 87 
3.6.2 The Van der Waals Gas 88 
Examples 90 
4. Antiferromagnets and Other Magnetic Systems 93 
4.1 The One-Dimensional Antiferromagnet 93 
4.2 Antiferromagnetic Ising Models 94 
4.3 Mean-Field Theory 99 
4.3.1 The Paramagnetic State 100 
4.3.2 The Antiferromagnetic State 101 
4.3.3 The Simple Antiferromagnet 103 
4.4 Metamagnetism: Tricritical Points and Critical End-Points 105 
4.5 Ferrimagnetism: Compensation Points 110 
4.5.1 Zero Field 113 
4.5.2 Non-Zero Field 114 
Examples 117 
5. Lattice Gases 119 
5.1 Introduction 119 
5.2 The One-Dimensional Lattice Gas and the Continuous Limit 121 
5.2.1 The Lattice Gas 121 
5.2.2 The Continuum Limit 124 
5.3 The Simple Lattice Gas and the Ising Model 125 
5.4 Phase Separation in the Simple Lattice Gas 127 
5.5 A One-Dimensional Water-Like Model 128 
Examples 132 
6. Solid Mixtures and the Dilute Ising Model 135 
6.1 The Restricted Grand Partition Function 135 
6.2 Binary Mixtures 136 
6.2.1 The Equivalence to the Ising Model 137 
6.2.2 The Equivalence to a Lattice Gas 138 
6.3 OrderDisorder on Loose-Packed Lattices 138 
6.4 The Order Parameter and Landau Expansion 141 
6.5 First-Order Sublattice Transitions 144 
6.6 The Equilibrium Dilute Ising Model and Equivalent Models 147 
6.6.1 Model I: The Equilibrium Dilute Ising Model 147 
6.6.2 Model II: The Ising Model Lattice Gas 149 
6.6.3 Model III: The Symmetrical Ternary Solid Mixture 149 
6.6.4 Model IV: The Symmetrical Lattice Gas Mixture 150 
6.6.5 Other Models 151 
6.6.6 Applications 151 
6.7 Mean-Field Theory and the Dilute Ising Model 151 
6.8 Multicritical Points in the Dilute Ising Model 154 
6.9 Multicritical Phenomena 
with Additional Thermodynamic Dimension 158 
6.10 The Unsymmetrical and Completely Symmetrical Models 163 
6.11 Alternative Forms for the Dilute Ising Model 166 
6.11.1 Model A: Equilibrium Bond Dilution 166 
6.11.2 Model B: Random Site Dilution 166 
6.11.3 Model C: Random Bond Dilution 167 
6.11.4 Model D: Equilibrium Site Dilution epsilon < 0 168 
Examples 169 
7. Cluster Variation Methods 173 
7.1 Introduction 173 
7.2 A First-Order Method Using a General Site Group 174 
7.2.1 Equivalent Sites 174 
7.2.2 Sublattice Ordering 177 
7.3 The Pair Approximation and the Ising Model 178 
7.3.1 Zero Field 179 
7.3.2 The Critical Region 180 
7.3.3 The Linear Lattice 181 
7.4 Phase Transitions in Amphipathic Monolayers 181 
7.5 A Lattice Gas Model for Fluid Water 187 
7.6 1:1 Ordering on the Face-Centred Cubic Lattice 193 
7.7 Homogeneous Cacti 197 
Examples 200 
8. Exact Results for Two-Dimensional Ising Models 205 
8.1 Introduction 205 
8.2 The Low-Temperature Form and the Dual Lattice 206 
8.3 The High-Temperature Form and the Dual Transformation 208 
8.4 The Star-Triangle Transformation 211 
8.5 The Star-Triangle Transformation with Unequal Interactions 214 
8.6 A Linear Relation for Correlations 216 
8.7 Baxter and Enting's Transformation 
and the Functional Equation 218 
8.8 The Solution of the Functional Equation 220 
8.8.1 A Preliminary Result for f(K|k) 221 
8.8.2 Expressions for A(Infinity|k) and B(Infinity|k) 223 
8.8.3 An Expression for b(k) 224 
8.9 Critical Behaviour 226 
8.10 Thermodynamic Functions for the Square Lattice 229 
8.11 Thermodynamic Functions 
for the Triangular and Honeycomb Lattices 232 
8.12 The Antiferromagnet 235 
Examples 238 
9. Applications of Transform Methods 241 
9.1 The Decoration Transformation 241 
9.2 Dilute Decorated Models 244 
9.2.1 A Superexchange Model 244 
9.2.2 The Equilibrium Bond Dilute Ising Model 245 
9.3 Heat Capacity and Exponent Renormalization 246 
9.3.1 Three-Dimensional Lattices 247 
9.3.2 Two-Dimensional Lattices 250 
9.4 Fisher's Decorated Antiferromagnetic Model 251 
9.5 The Decorated Lattice Ferrimagnet 256 
9.6 The Kagom Lattice Ising Model 262 
9.7 A Modified Star-Triangle Transformation 
and Three-Spin Correlations 264 
9.8 The Star-Triangle Ferrimagnet 266 
9.9 The Unsymmetrical Ising Model 268 
9.10 A Competing Interaction Magnetic Model 271 
9.11 Decorated Lattice Mixtures of Orientable Molecules 273 
9.12 The Decorated Lattice Gas 278 
9.12.1 General Properties 278 
9.12.2 A Water-Like Model 281 
9.12.3 Maxithermal, Critical Double and Cuspoidal Points 283 
9.13 Decorated Lattice Gas Mixtures 289 
Examples 290 
10. The Six-Vertex Model 293 
10.1 Two-Dimensional Ice-Rule Models 293 
10.2 Parameter Space 295 
10.3 Graphical Representation and Ground States 297 
10.4 Free Energy and Transfer Matrices 299 
10.5 Transfer Matrix Eigenvalues 301 
10.5.1 The Case n = 0 302 
10.5.2 The Case n = 1 302 
10.5.3 The General n Case 304 
10.6 The Low-Temperature Frozen Ferroelectric State 306 
10.7 Wave-Number Density 307 
10.8 The Solution of the Integral Equation for -1 < Delta < 1 309 
10.9 The Free Energy of the Disordered State 311 
10.9.1 The KDP Model 312 
10.9.2 Square Ice 312 
10.9.3 Non-Zero Polarization 313 
10.10 The Ferroelectric Transition in Zero Field 313 
10.11 The Antiferroelectric State 316 
10.12 Field-Induced Transitions 
to the Completely Polarized State 319 
10.13 An Antiferroelectric in an Electric Field 322 
10.14 The Potts Model 325 
10.14.1 The Staggered Six-Vertex Model 326 
10.14.2 The Solvable Case xi_h xi_v = 1 329 
10.14.3 The Solvable Case xi_h xi_v = -1 330 
10.14.4 The Polarization and Internal Energy 331 
10.14.5 Critical Exponents 331 
Examples 332 
A. Appendices 335 
A.1 Regular Lattices 335 
A.2 Elliptic Integrals and Functions 337 
A.2.1 Elliptic Integrals 338 
A.2.2 Elliptic Functions 339 
A.2.3 Results Required for Chapter 8 340 
A.3 The Water Molecule and Hydrogen Bonding 342 
A.4 Results for the Six-Vertex Model 344 
A.4.1 The Proof of I 345 
A.4.2 The Proof of II 345 
A.4.3 The Proof of III 345 
A.4.4 The Proof of IV 348 
A.5 Fourier Transforms and Series 349 
A.5.1 Fourier Transforms 349 
A.5.2 Fourier Series 349 
References and Author Index 351 
Subject Index 365 
END
