Keywords:
Renormalization group.
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Critical phenomena (Physics).
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Mathematical physics.
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Electronic books.
Description / Table of Contents:
The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to therenormalization group and the calculation of critical exponents above and below the critical temperature.
Type of Medium:
Online Resource
Pages:
1 online resource (477 pages)
Edition:
1st ed.
ISBN:
9780191660566
URL:
https://ebookcentral.proquest.com/lib/geomar/detail.action?docID=1480944
DDC:
530.1/2
Language:
English
Note:
Cover -- Contents -- 1 Introduction -- 1.1 Continuous phase transitions and critical points -- 1.1.1 Divergences and critical exponents -- 1.1.2 Fluctuations and critical opalescence -- 1.2 The order parameter -- 1.2.1 Liquid-gas transition -- 1.2.2 Binary fluids -- 1.2.3 Ferromagnetic/paramagnetic transition -- 1.2.4 Anti-ferromagnetic/paramagnetic transition -- 1.2.5 Helium I/helium II transition -- 1.2.6 Conductor/superconductor transitions -- 1.2.7 Helium three -- 1.3 Correlation functions -- 1.4 Universality -- 1.5 Thermodynamic potentials -- 1.5.1 The Widom and Kadanoff scaling hypotheses -- 1.6 Why study phase transitions? -- Problems -- 2 Statistical mechanics -- 2.1 Thermodynamic quantities -- 2.2 Fluctuations and correlation functions -- 2.3 Metastability and spontaneous symmetry breaking -- 2.3.1 Metastability -- 2.3.2 Spontaneous symmetry breaking -- Problems -- 3 Models -- 3.1 Description of models -- 3.1.1 The Ising model -- 3.1.2 The lattice gas -- 3.1.3 ß-brass -- 3.1.4 The XY and Heisenberg models -- 3.1.5 Potts model -- 3.1.6 Gaussian and spherical models -- 3.1.7 Percolation model -- 3.2 Transfer matrices and the Ising ring -- 3.2.1 Solution of the Ising ring -- 3.2.2 Correlation functions -- 3.3 The partition function of the spherical model -- 3.4 High-temperature expansions and the Ising model -- 3.4.1 High-temperature expansions -- 3.4.2 The partition function of the Ising model -- 3.4.3 The correlation functions of the Ising model -- 3.4.4 Numerical evaluation of high-temperature expansions -- Problems -- 4 Numerical simulations -- 4.1 Direct evaluation of thermal averages -- 4.2 Sampling configurations -- 4.2.1 Importance sampling -- 4.2.2 General structure of numerical algorithms -- 4.3 Monte Carlo methods -- 4.3.1 The Metropolis algorithm -- 4.4 Molecular dynamics -- 4.4.1 Ergodicity and integrability.
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4.4.2 From microcanonical to canonical averages -- 4.5 Langevin equations -- 4.5.1 Comparison of the Langevin and molecular-dynamics methods -- 4.6 Independence of configurations -- 4.6.1 Correlations along the path -- 4.6.2 Critical slowing down -- 4.6.3 The Swendsen-Wang algorithm -- 4.6.4 The Wolff algorithm -- 4.7 Calculation of critical exponents from simulations -- Problems -- 5 Real-space renormalization -- 5.1 Renormalizing the lattice -- 5.2 Block variables -- 5.3 The renormalization of the Hamiltonian -- 5.3.1 Fixed points -- 5.3.2 The calculation of v -- 5.4 The renormalization of B, M, X and G[sub(c)] -- 5.4.1 The value of ω -- 5.4.2 Non-zero external field -- 5.4.3 The renormalization of M, χ and G[sub(c)] -- 5.4.4 Critical exponents for the renormalized model -- 5.5 The critical exponents for T = T[sub(c)] -- 5.5.1 The exponent η -- 5.5.2 The exponent δ -- 5.6 The critical exponents for T ≠ T[sub(c)] -- 5.6.1 The exponent β -- 5.6.2 The exponent γ -- 5.6.3 The exponent α -- 5.7 The scaling laws -- 5.8 Bond percolation in two dimensions -- 5.9 The Ising model -- 5.10 Monte Carlo renormalization -- Problems -- 6 Mean-field theory -- 6.1 Mean-field theory of the Ising model -- 6.2 Mean-field theory of percolation -- 6.3 Mean-field theory of the non-ideal gas -- 6.4 A variational derivation of mean-field theory -- 6.5 Correlation functions in mean-field theory -- 6.6 Infinite-range interactions -- 6.7 Critical exponents in mean-field theory -- 6.7.1 Calculating η from G[sup(2)][sub(c)](k) -- 6.8 What is missing from mean-field theory? -- Problems -- 7 The Landau-Ginzburg model -- 7.1 Formulation of the Landau-Ginzburg model -- 7.2 Landau theory -- Problems -- 8 Diagrammatic perturbation theory -- 8.1 The Gaussian partition function -- 8.1.1 Correlation functions in the Gaussian model.
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8.2 The partition function for the full Landau-Ginzburg model -- 8.2.1 The Feynman rules -- 8.2.2 The symmetry factor -- 8.3 The Helmholtz free energy of the Landau-Ginzburg model -- 8.3.1 Feynman rules in wavevector space -- 8.3.2 Vertex functions -- 8.4 The Gibbs free energy of the Landau-Ginzburg model -- 8.4.1 The rules for finding Γ[φ] -- 8.4.2 The loop expansion -- 8.4.3 The one-loop Gibbs free energy -- Problems -- 9 Renormalization -- 9.1 Mass renormalization -- 9.2 Field renormalization -- 9.3 Renormalizing the coupling constant -- 9.4 Renormalization at higher orders -- 9.5 More on field renormalization -- 9.6 The Ginzburg criterion -- Problems -- 10 The calculation of critical exponents for T ≥ Tc -- 10.1 Ultraviolet and infrared divergences -- 10.2 The calculation of γ -- 10.2.1 d = 4 and above -- 10.2.2 Below four dimensions -- 10.3 The calculation of η -- 10.3.1 d = 4 and above -- 10.3.2 Below four dimensions -- 10.4 The ε-expansion -- 10.4.1 Dimensional regularization -- 10.4.2 Calculating γ by dimensional regularization -- 10.4.3 Calculating η by dimensional regularization -- 10.4.4 Feynman parameters -- 10.4.5 The calculation of η again -- 10.4.6 Calculation of η by the ε-expansion -- Problems -- 11 The renormalization group -- 11.1 The renormalization group at T = T[sub(c)] -- 11.2 The exponents η and δ -- 11.2.1 The exponent η -- 11.2.2 The exponent δ -- 11.3 The calculation of β and γ[sub(1)] -- 11.3.1 The calculation of γ[sub(1)] to order ε[sup(2)] -- Problems -- 12 The renormalization group at T ≠ T[sub(c)] -- 12.1 Expansion about the critical temperature -- 12.1.1 Functional Taylor expansions -- 12.1.2 Diagrammatic representation of the Φ[sup(2)] correlation functions -- 12.1.3 Wavevector space -- 12.1.4 Vertex functions -- 12.1.5 Renormalization -- 12.1.6 Expanding the renormalized vertex functions.
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12.1.7 The validity of the expansion -- 12.2 The renormalization group equations -- 12.2.1 The exponent v -- 12.2.2 The exponent γ -- 12.2.3 The exponent α -- 12.3 The renormalization group below T[sub(c)] -- 12.3.1 The exponent β -- 12.4 Calculating γ[sub(2)] to one loop -- Problems -- 13 The lower critical dimension -- 13.1 Order below T[sub(c)] -- 13.1.1 The case D = 1 -- 13.1.2 Systems with more than one component -- 13.1.3 Goldstone modes -- 13.2 The non-linear σ-model -- 13.2.1 The two-point vertex function -- 13.2.2 The renormalization group equation -- 13.3 The Kosterlitz-Thouless transition -- 13.3.1 The two-dimensional Coulomb gas -- 13.3.2 General Remarks -- Problems -- 14 Universality -- 14.1 Perturbing the Gaussian Hamiltonian -- 14.1.1 The applicability of these results -- 14.2 Perturbing the Landau-Ginzburg Hamiltonian -- 14.2.1 The case of three dimensions -- 14.2.2 The case of two dimensions -- 14.3 Relevance and renormalizability -- Problems -- Appendices -- A: The magnetic scattering of neutrons -- B: The natural variables for thermodynamic potentials -- C: Magnetic energy -- D: Connected correlation functions and log Z[J] -- E: The Gibbs free energy -- F: Discrete Fourier transforms -- G: The method of steepest descent -- H: Counting closed loops on a square lattice -- I: Einstein's fluctuation theory -- J: The Gaussian transformation -- K: The Landau-Ginzburg model and the Ising model -- L: Functional differentiation and integration -- M: The Feynman rules for the vertex functions -- N: Feynman rules for generalized Landau-Ginzburg models -- Answers -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Z.
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