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Equilibrium Statistical Mechanics of Lattice Models 2015 ed. [Hardback]

  • Formāts: Hardback, 793 pages, height x width: 235x155 mm, 101 Illustrations, black and white; XVII, 793 p. 101 illus., 1 Hardback
  • Sērija : Theoretical and Mathematical Physics
  • Izdošanas datums: 12-Feb-2015
  • Izdevniecība: Springer
  • ISBN-10: 9401794294
  • ISBN-13: 9789401794299
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Equilibrium Statistical Mechanics of Lattice Models 2015 ed.Palielināt
  • Formāts: Hardback, 793 pages, height x width: 235x155 mm, 101 Illustrations, black and white; XVII, 793 p. 101 illus., 1 Hardback
  • Sērija : Theoretical and Mathematical Physics
  • Izdošanas datums: 12-Feb-2015
  • Izdevniecība: Springer
  • ISBN-10: 9401794294
  • ISBN-13: 9789401794299
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9789401794299.html
Keywords:

Most interesting and difficult problems in equilibrium statistical mechanics concern models which exhibit phase transitions. For graduate students and more experienced researchers this book provides an invaluable reference source of approximate and exact solutions for a comprehensive range of such models.
Part I contains background material on classical thermodynamics and statistical mechanics, together with a classification and survey of lattice models. The geometry of phase transitions is described and scaling theory is used to introduce critical exponents and scaling laws. An introduction is given to finite-size scaling, conformal invariance and Schramm—Loewner evolution.
Part II contains accounts of classical mean-field methods. The parallels between Landau expansions and catastrophe theory are discussed and Ginzburg--Landau theory is introduced. The extension of mean-field theory to higher-orders is explored using the Kikuchi--Hijmans--De Boer hierarchy of approximations.
In Part III the use of algebraic, transformation and decoration methods to obtain exact system information is considered. This is followed by an account of the use of transfer matrices for the location of incipient phase transitions in one-dimensionally infinite models and for exact solutions for two-dimensionally infinite systems. The latter is applied to a general analysis of eight-vertex models yielding as special cases the two-dimensional Ising model and the six-vertex model. The treatment of exact results ends with a discussion of dimer models.
In Part IV series methods and real-space renormalization group transformations are discussed. The use of the De Neef—Enting finite-lattice method is described in detail and applied to the derivation of series for a number of model systems, in particular for the Potts model. The use of Pad\'e, differential and algebraic approximants to locate and analyze second- and first-order transitions is described. The realization of the ideas of scaling theory by the renormalization group is presented together with treatments of various approximation schemes including phenomenological renormalization.
Part V of the book contains a collection of mathematical appendices intended to minimise the need to refer to other mathematical sources.

Part I Thermodynamics, Statistical Mechanical Models and Phase Transitions
1 Thermodynamics
5(8)
1.1 Formulae and Variables
5(3)
1.2 The Field-Density Representation
8(1)
1.3 The Thermodynamic Limit
9(1)
1.4 Particular Response Functions
10(3)
1.4.1 Magnetic Systems
11(1)
1.4.2 Fluid Systems
11(2)
2 Statistical Mechanics
13(16)
2.1 Distributions
13(3)
2.1.1 Quantum Systems
14(1)
2.1.2 The Connection to Thermodynamics
15(1)
2.2 Variations of the Probability Function
16(1)
2.3 Coupling Representations
17(4)
2.3.1 The Case nf = 2
19(2)
2.4 Lattice Systems
21(2)
2.4.1 Site-Variable Models
22(1)
2.4.2 Edge-Variable Models
22(1)
2.5 Correlation Functions and Symmetry Properties
23(6)
2.5.1 A General Hamiltonian
23(1)
2.5.2 Correlation Functions
24(2)
2.5.3 Symmetry Properties
26(3)
3 A Survey of Models
29(60)
3.1 Upper and Lower Critical Dimensions
29(1)
3.2 The Quantum Heisenberg Model
30(3)
3.2.1 One-Dimensional Chains
31(2)
3.3 Classical Vector Models
33(1)
3.4 The Gaussian and Spherical Models
34(2)
3.5 Ising Models
36(9)
3.5.1 The Spin-1/3=2 Ising Model
36(8)
3.5.2 The Spin-1 Ising Model
44(1)
3.6 State-Difference Models
45(12)
3.6.1 The Classical XY Model
45(8)
3.6.2 The Ashkin--Teller Model
53(1)
3.6.3 Potts Models
54(1)
3.6.4 The Standard Potts Model
55(2)
3.7 Chirality
57(3)
3.7.1 Chiral Potts Models
58(1)
3.7.2 An Extended 3-State Potts Model on the Triangular Lattice
59(1)
3.8 Vertex Models
60(20)
3.8.1 The Eight-Vertex Model
61(11)
3.8.2 The Six-Vertex Model
72(8)
3.9 Dimer Models
80(9)
3.9.1 The Modified KDP Model Equivalence
83(2)
3.9.2 The Ising Model Equivalence
85(4)
4 Phase Transitions and Scaling Theory
89(80)
4.1 The Geometry of Phase Transitions
89(5)
4.1.1 A Two-Dimensional Phase Space
90(3)
4.1.2 A Three-Dimensional Phase Space
93(1)
4.2 Universality, Fluctuations and Scaling
94(6)
4.2.1 Universality
95(2)
4.2.2 Scaling for the Ising Model
97(3)
4.3 General Scaling Formulation
100(22)
4.3.1 The Kadanoff Scaling Hypothesis
100(4)
4.3.2 First-Order Transitions
104(1)
4.3.3 Effective Exponents
105(4)
4.3.4 The Nightingale--'T Hooft Scaling Hypothesis
109(1)
4.3.5 Constraints on Scaling
110(5)
4.3.6 Scaling Operators and Dimensions
115(3)
4.3.7 Correlation Functions
118(1)
4.3.8 Variable Scaling Exponents
119(2)
4.3.9 Densities and Response Functions
121(1)
4.4 Critical Point and Coexistence Curve
122(3)
4.4.1 Critical Exponents
124(1)
4.4.2 Exponent Inequalities
125(1)
4.5 Scaling for a Critical Point
125(11)
4.5.1 Scaling Fields for the Critical Point
126(2)
4.5.2 Approaches to the Critical Point
128(1)
4.5.3 Experimental Variables
129(1)
4.5.4 The Density and Response Functions
130(1)
4.5.5 Asymptotic Forms
131(1)
4.5.6 Critical Exponents and Scaling Laws
132(2)
4.5.7 Correlation Scaling at a Critical Point
134(2)
4.6 Tricritical Point
136(5)
4.7 Scaling for a Tricritical Point
141(5)
4.7.1 Scaling Fields for the Tricritical Point
141(2)
4.7.2 Tricritical Exponents and Scaling Laws
143(3)
4.8 Corrections to Scaling
146(2)
4.9 Scaling and Universality
148(4)
4.10 Finite-Size Scaling
152(7)
4.10.1 The Finite-Size Scaling Field
153(2)
4.10.2 The Shift and Rounding Exponents
155(2)
4.10.3 Universality and Finite-Size Scaling
157(2)
4.11 Conformal Invariance
159(10)
4.11.1 From Scaling to the Conformal Group
159(1)
4.11.2 Correlation Functions for d ≥ 2
159(2)
4.11.3 Universal Amplitudes for d = 2
161(2)
4.11.4 Schramm-Loewner Evolution
163(6)
Part II Classical Approximation Methods
5 Phenomenological Theory and Landau Expansions
169(36)
5.1 Classical Methods
169(6)
5.1.1 A First-Order Transition
171(3)
5.1.2 Metastability
174(1)
5.2 The Van der Waals Equation
175(1)
5.3 Landau Expansions with One Order Parameter
176(7)
5.3.1 The Spin-1/2 Ising Model
182(1)
5.4 Landau Expansions with Two Order Parameters
183(5)
5.4.1 The Spin-1 Ising Model
183(1)
5.4.2 The 3-State Potts Model
184(4)
5.5 Landau Theory for a Tricritical Point
188(5)
5.5.1 Tricritical Exponents
190(3)
5.6 Ginzburg-Landau Theory
193(12)
5.6.1 A Critical Point
193(1)
5.6.2 The Gaussian Approximation
194(3)
5.6.3 Gaussian Critical Exponents
197(8)
6 Mean-Field Theory
205(24)
6.1 The Ising Ferromagnet
205(7)
6.1.1 Mean-Field Fluctuations
209(3)
6.2 A Model for Metamagnetism
212(17)
6.2.1 The Paramagnetic State
215(1)
6.2.2 The Antiferromagnetic State
216(1)
6.2.3 A Neighbourhood of the Critical Curve
217(4)
6.2.4 The First-Neighbour Antiferromagnet: λ = 0
221(2)
6.2.5 The First-Order Transition
223(3)
6.2.6 A Neighbourhood of the Tricritical Point
226(3)
7 Cluster-Variation Methods
229(30)
7.1 Improving Mean-Field Theory
229(2)
7.2 The KHDeB Hierarchy of Approximations
231(8)
7.2.1 Distribution Numbers
231(2)
7.2.2 Extensive Quantities
233(2)
7.2.3 The Hamiltonian and Free Energy
235(1)
7.2.4 The Entropy
235(2)
7.2.5 Minimization
237(1)
7.2.6 Labelling Configurations
238(1)
7.3 The Bethe-Pair Approximation for the Ising Model
239(2)
7.4 Reduction to the Mean-Field Approximation
241(2)
7.5 3-State Potts Model on a Triangular Lattice
243(3)
7.6 A Lattice Model for Fluid Water
246(13)
Part III Exact Results
8 Algebraic Methods
259(24)
8.1 The Thermodynamic Limit
259(2)
8.2 The Infinite-System Approach
261(2)
8.3 Lower Bounds for Phase Transitions: The Peierls Method
263(8)
8.3.1 The Simple Lattice Fluid
269(2)
8.4 Grand Partition Function Zeros and Phase Transitions
271(12)
8.4.1 Ruelle's Theorem
273(3)
8.4.2 The Yang--Lee Circle Theorem
276(3)
8.4.3 Systems with Pair Interactions
279(4)
9 Transformation Methods
283(28)
9.1 Related Systems
283(1)
9.2 The Wegner Transformation
284(9)
9.2.1 Duality for the v-State Potts Model
286(4)
9.2.2 Duality for the Spin-1 Ising Model
290(1)
9.2.3 The Weak-Graph Transformation
291(2)
9.3 The Regular Square-Lattice Eight-Vertex Model
293(11)
9.3.1 Symmetry Properties and Transformations
294(3)
9.3.2 The Case of Region I
297(4)
9.3.3 Regions and Variables
301(3)
9.4 The Star-Triangle Transformation
304(7)
9.4.1 The v-State Potts Model
306(1)
9.4.2 The Spin-1/2 Ising Model
307(4)
10 Edge-Decorated Ising Models
311(34)
10.1 Primary and Secondary Sites
311(1)
10.2 Super-Exchange or Bond-Dilution
312(7)
10.2.1 Critical Properties and Exponent Renormalization
314(5)
10.3 A Ferrimagnet
319(5)
10.3.1 The Zero-Field Axis
320(2)
10.3.2 Non-Zero Field
322(2)
10.4 A Competing-Interaction Magnetic Model
324(2)
10.5 Decoration with Orientable Molecules
326(6)
10.6 A Decorated Lattice Fluid
332(13)
10.6.1 Case I: A Single Vapour/Liquid Transition
335(2)
10.6.2 Case II: A Water-Like Model
337(2)
10.6.3 Case III: Maxithermal, Critical Double and Cuspoidal Points
339(6)
11 Transfer Matrices: Incipient Phase Transitions
345(36)
11.1 The Transfer Matrix Formulation
345(8)
11.1.1 The Eigen Problem
346(1)
11.1.2 The Partition Function
347(1)
11.1.3 Correlation Functions and Lengths
348(5)
11.2 Incipient Phase Transitions
353(1)
11.3 Using Symmetry Properties
354(13)
11.3.1 Block-Diagonalization of the Transfer Matrix
355(4)
11.3.2 Applications
359(8)
11.4 Analysis in the Complex Plane: The Wood Method
367(14)
11.4.1 Evolution of Partition Function Zeros
367(2)
11.4.2 Connection Curves and Cross-Block Curves
369(3)
11.4.3 The Spin-1/2 Square-Lattice Ising Model
372(4)
11.4.4 Critical Points and Exponents
376(5)
12 Transfer Matrices: Exactly Solved Models
381(114)
12.1 A General Eight-Vertex Model
381(13)
12.1.1 A Generalized Star-Triangle Transformation
382(1)
12.1.2 The Solution to the GST Transformation and the Elliptic Variable Formulation
383(4)
12.1.3 Z-Invariance
387(4)
12.1.4 Edge Variables and Matrix Formulation
391(2)
12.1.5 Square-Lattice Models
393(1)
12.2 Square-Lattice Ising Models
394(37)
12.2.1 The Modified Checkerboard Ising Model
399(4)
12.2.2 Properties of the Transfer Matrices
403(4)
12.2.3 The Reduction to Regular Ising Models
407(2)
12.2.4 Transfer Matrix Eigenvectors
409(1)
12.2.5 Notational Changes
410(2)
12.2.6 Transfer Matrix Eigenvalues
412(9)
12.2.7 The Standard Model
421(10)
12.3 The Square-Lattice Eight-Vertex Model
431(64)
12.3.1 The Low-Temperature Zone RL(I)
431(2)
12.3.2 The Low-Temperature Zone RL(III)
433(1)
12.3.3 The Transfer Matrix
434(3)
12.3.4 Analysis in Terms of Pauli Matrices
437(5)
12.3.5 Analysis of the Transfer Matrix
442(5)
12.3.6 The VQ Equation
447(28)
12.3.7 The Free Energy and Magnetization
475(3)
12.3.8 Critical Behaviour
478(4)
12.3.9 The Coupling Form and the Ising Model Limit
482(3)
12.3.10 The Six-Vertex Model
485(6)
12.3.11 The Eight-Vertex Model and Universality
491(4)
13 Dimer Models
495(26)
13.1 The Dimer Partition Function
495(1)
13.2 Superposition Polynomials and Pfaffians
496(14)
13.2.1 The Square-Lattice Case
500(4)
13.2.2 The Honeycomb-Lattice Case
504(6)
13.3 Vertex and Ising Model Equivalences
510(4)
13.3.1 The Five-Vertex Model
510(2)
13.3.2 The Honeycomb-Lattice Anisotropic Ising Model
512(2)
13.4 K-Type and O-Type Transitions
514(7)
Part IV Series and Renormalization Group Methods
14 Series Expansions
521(46)
14.1 The Task and the Methods
521(3)
14.2 Moment Expansions
524(12)
14.2.1 At Low Temperatures
525(5)
14.2.2 At High Temperatures
530(4)
14.2.3 Duality for Graphs
534(2)
14.3 Cumulant Expansions
536(4)
14.3.1 The Low-Temperature Case
539(1)
14.3.2 The High-Temperature Case
539(1)
14.4 The Finite-Cluster Method
540(3)
14.5 The Finite-Lattice Method
543(12)
14.5.1 Block-Formation and Accuracy
544(4)
14.5.2 Constructing Block Partition Functions
548(3)
14.5.3 Calculating the Series
551(4)
14.6 The Analysis of Series: Second-Order Transitions
555(10)
14.6.1 Late-Term Analysis
556(1)
14.6.2 The Ratio Method
557(2)
14.6.3 Pade Approximants
559(3)
14.6.4 Differential and Algebraic Approximants
562(3)
14.7 The Analysis of Series: First-Order Transitions
565(2)
15 Real-Space Renormalization Group Theory
567(52)
15.1 The Basic Elements of the Renormalization Group
567(3)
15.2 RG Transformations and Weight Functions
570(4)
15.3 Fixed Points and the Linear Renormalization Group
574(3)
15.4 Free Energy and Densities
577(2)
15.5 Decimation for the Ising Model
579(9)
15.5.1 In One Dimension
579(6)
15.5.2 In Two Dimensions
585(3)
15.6 The Kosterlitz-Thouless Transition
588(6)
15.7 Upper-Bound and Lower-Bound Approximations
594(9)
15.7.1 An Upper-Bound Method
595(4)
15.7.2 A Lower-Bound Method
599(4)
15.8 Finite-Lattice Approximations
603(4)
15.9 Variational Approximations
607(2)
15.10 Phenomenological Renormalization
609(6)
15.10.1 The Square-Lattice Ising Model
611(2)
15.10.2 Other Models
613(1)
15.10.3 More Than One Coupling
614(1)
15.11 Other Renormalization Group Methods
615(4)
Part V Mathematical Appendices
16 Graphs and Lattices
619(40)
16.1 Graphs
619(5)
16.1.1 Introduction
619(2)
16.1.2 The Cyclomatic Number
621(1)
16.1.3 Triangulation of Graphs
622(1)
16.1.4 Oriented Graphs
622(1)
16.1.5 The Dual Graph
623(1)
16.2 Lattices
624(9)
16.2.1 Types of Regular Lattices
624(4)
16.2.2 Lattice Transformations
628(5)
16.3 Rapidity Graphs and Lattices
633(4)
16.4 Lattice Graphs
637(22)
16.4.1 Augmented Graphs and the Whitney Polynomial
638(1)
16.4.2 Hopping Matrices and the Canonical Flux Distribution
638(1)
16.4.3 Embeddings and Topologies
639(1)
16.4.4 Lattice Constants
640(4)
16.4.5 Partially-Ordered Sequences of Graphs and the T Matrix
644(3)
16.4.6 Generating the Partially-Ordered Sequence
647(4)
16.4.7 Incorporating Sublattices
651(3)
16.4.8 The Guggenheim--McGlashan Approach
654(2)
16.4.9 Further Results
656(3)
17 Algebra
659(44)
17.1 Catastrophe Theory
659(11)
17.1.1 Equivalence and Determinancy
659(3)
17.1.2 Critical Points, Codimension and Unfoldings
662(6)
17.1.3 Symmetry Considerations
668(2)
17.2 Matrix Algebra
670(6)
17.2.1 Diagonalizability
671(1)
17.2.2 Commutativity
672(1)
17.2.3 Reducibility
673(1)
17.2.4 Theorems of Perron and Frobenius
673(2)
17.2.5 Direct Products and Traces
675(1)
17.2.6 Defective Matrices
675(1)
17.2.7 Groups of Matrices
676(1)
17.3 Groups and Representations
676(15)
17.3.1 Representations
678(4)
17.3.2 Permutation Representations and Equivalence Classes
682(2)
17.3.3 Block Diagonalization Within an Equivalence Class...
684(3)
17.3.4 Symmetry Groups
687(4)
17.4 The Conformal Group
691(2)
17.5 Some Transformations in the Complex Plane
693(2)
17.6 Algebraic Functions
695(5)
17.7 Determinants of Cyclic Matrices
700(3)
18 Analysis
703(54)
18.1 Fourier Transforms in d Dimensions
703(8)
18.1.1 Discrete Finite Lattices
703(2)
18.1.2 A Continuous Finite Volume
705(2)
18.1.3 A Continuous Infinite Volume
707(1)
18.1.4 Integrals Involving Bessel Functions
708(2)
18.1.5 Lattice Green's Functions
710(1)
18.2 Doubly-Periodic and Quasi-Periodic Functions
711(3)
18.3 Elliptic Integrals and Functions
714(23)
18.3.1 Elliptic Integrals
714(3)
18.3.2 Jacobi Theta Functions
717(3)
18.3.3 Jacobi Elliptic Functions
720(4)
18.3.4 Transformations in the Elliptic Modulus
724(2)
18.3.5 The Modified Amplitude Function
726(1)
18.3.6 Nome Series
727(1)
18.3.7 Special Results and Functions for Chap. 12
728(3)
18.3.8 Baxter's Modified Theta Functions
731(6)
18.4 The Potts Delta Function
737(6)
18.4.1 The μ = 0 Case
740(2)
18.4.2 The μ ≠ 0 Case
742(1)
18.5 Pade, Differential and Algebraic Approximants
743(14)
18.5.1 Pade Approximants
743(5)
18.5.2 Dlog Pade Approximants
748(2)
18.5.3 Differential Approximants
750(4)
18.5.4 Algebraic Approximants
754(3)
References and Author Index 757(26)
Index 783