Lectures On Phase Transitions And The Renormalization Group [Pehme köide]

Preface		1	(5)
Introduction		5	(18)
Scaling and Dimensional Analysis		5	(2)
Power Laws in Statistical Physics		7	(3)
Liquid Gas Critical Point		7	(2)
Magnetic Critical Point		9	(2)
Superfluid Transition in 4He		11	(1)
Self-Avoiding Walk		12	(1)
Dynamic Critical Phenomena		13
Some Important Questions		10	(4)
Historical Development		14	(9)
Exercises		20	(3)
How Phase Transitions Occur In Principle		23	(62)
Review of Statistical Mechanics		23	(2)
The Thermodynamic Limit		25	(4)
Thermodynamic Limit in a Charged System		26	(1)
Thermodynamic Limit for Power Law Interactions		27	(2)
Phase Boundaries and Phase Transitions		29	(3)
Ambiguity in the Definition of Phase Boundary		29	(1)
Types of Phase Transition		30	(1)
Finite Size Effects and the Correlation Length		30	(2)
The Role of Models		32	(1)
The Ising Model		33	(2)
Analytic Properties of the Ising Model		35	(5)
Convex Functions		38	(1)
Convexity and the Free Energy Density		38	(2)
Symmetry Properties of the Ising Model		40	(3)
Time Reversal Symmetry		40	(1)
Sub-Lattice Symmetry		41	(2)
Existence of Phase Transitions		43	(7)
Zero Temperature Phase Diagram		44	(1)
Phase Diagram at Non-Zero Temperature: d = 1		45	(2)
Phase Diagram at Non-Zero Temperature: d = 2		47	(2)
Impossibility of Phase Transitions		49	(1)
Spontaneous Symmetry Breaking		50	(5)
Probability Distribution		52	(2)
Continuous Symmetry		54	(1)
Ergodicity Breaking		55	(16)
Illustrative Example		57	(2)
Symmetry and Its Implications for Ergodicity Breaking		59	(3)
Example of Replica Symmetry Breaking: Rubber		62	(3)
Order Parameters and Overlaps in a Classical Spin Glass		65	(3)
Replica Formalism for Constrained Systems		68	(3)
Fluids		71	(3)
Lattice Gases		74	(5)
Lattice Gas Thermodynamics from the Ising Model		75	(2)
Derivation of Lattice Gas Model from the Configurational Sum		77	(2)
Equivalence in Statistical Mechanics		79	(1)
Miscellaneous Remarks		80	(5)
History of the Thermodynamic Limit		80	(1)
Do Quantum Effects Matter?		81	(1)
Exercises		82	(3)
How Phase Transitions Occur in Practice		85	(32)
Ad Hoc Solution Methods		85	(3)
Free Boundary Conditions and H = 0		86	(1)
Periodic Boundary Conditios and H = 0		86	(1)
Recursion Method for H = 0		87	(1)
Effect of Boundary Conditions		88	(1)
The Transfer Matrix		88	(3)
Phase Transitions		91	(1)
Thermodynamic Properties		92	(3)
Spatial Correlations		95	(6)
Zero Field: Ad Hoc Method		95	(3)
Existence of Long Range Order		98	(1)
Transfer Matrix Method		99	(2)
Low Temperature Expansion		101	(3)
d > 1		103	(1)
d = 1		103	(1)
Mean Field Theory		104	(13)
Weiss' Mean Field Theory		105	(3)
Spatial Correlations		108	(3)
How Good Is Mean Field Theory?		111	(1)
Exercises		112	(5)
Critical Phenomena in Fluids		117	(18)
Thermodynamics		117	(2)
Thermodynamic Potentials		117	(1)
Phase Diagram		118	(1)
Landau's Symmetry Principle		119	(1)
Two-Phase Coexistence		119	(3)
Fluid at Constant Pressure		119	(1)
Fluid at Constant Temperature		120	(1)
Maxwell's Equal Area Rule		121	(1)
Vicinity of the Critical Point		122	(1)
Van der Waals' Equation		123	(5)
Determination of the Critical Point		123	(1)
Law of Corresponding States		124	(1)
Critical Behaviour		125	(3)
Spatial Correlations		128	(3)
Number Fluctuations and Compressibility		128	(1)
Number Fluctuations and Correlations		129	(1)
Critical Opalescence		130	(1)
Measurement of Critical Exponents		131	(4)
Definition of Critical Exponents		131	(1)
Determination of Critical Exponents		132	(2)
Exercises		134	(1)
Landau Theory		135	(32)
Order Parameters		136	(2)
Heisenberg Model		136	(1)
XY Model		137	(1)
3He		137	(1)
Common Features of Mean Field Theories		138	(1)
Phenomenological Landau Theory		139	(4)
Assumptions		140	(1)
Construction of ℒ		141	(2)
Continuous Phase Transitions		143	(2)
Critical Exponents in Landau Theory		144	(1)
First Order Transitions		145	(2)
Inhomogeneous Systems		147	(7)
Coarse Graining		147	(2)
Interpretation of the Landau Free Energy		149	(5)
Correlation Functions		154	(13)
The Continuum Limit		154	(1)
Functional Integrals in Real and Fourier Space		154	(2)
Functional Differentiation		156	(1)
Response Functions		157	(1)
Calculation of Two-Point Correlation Function		158	(4)
The Coefficient γ		162	(1)
Exercises		163	(4)
Fluctuations and the Breakdown of Landau Theory		167	(22)
Breakdown of Microscopic Landau Theory		167	(2)
Fluctuations Away from the Critical Point		168	(1)
Fluctuations Near the Critical Point		169	(1)
Breakdown of Phenomenological Landau Theory		169	(5)
Calculation of the Ginzburg Criterion		170	(2)
Size of the Critical Region		172	(2)
The Gaussian Approximation		174	(7)
One Degree of Freedom		174	(1)
N Degrees of Freedom		175	(1)
Infinite Number of Degrees of Freedom		176	(3)
Two-Point Correlation Function Revisited		179	(2)
Critical Exponents		181	(8)
Exercises		185	(4)
Anomalous Dimensions		189	(12)
Dimensional Analysis of Landau Theory		189	(4)
Dimensional Analysis and Critical Exponents		193	(3)
Anomalous Dimensions and Asymtotics		196	(1)
Renormalisation and Anomalous Dimensions		197	(4)
Exercises		199	(2)
Scaling in Static, Dynamic and Non-Equilibrium Phenomena		201	(28)
The Static Scaling Hypothesis		202	(4)
Time-Reversal Symmetry		204	(1)
Behaviour as h ← 0		204	(1)
The Zero-field Susceptibility		204	(1)
The Critical Isotherm and a Scaling Law		205	(1)
Other Forms of the Scalling Hypothesis		206	(3)
Scaling Hypothesis for the Free Energy		206	(1)
Scaling Hypothesis for the Correlation Function		206	(1)
Scaling and the Correlation Length		207	(2)
Dynamic Critical Phenomena		209	(7)
Small Time-Dependent Fluctuations		209	(2)
The Relaxation Time		211	(2)
Dynamic Scaling Hypothesis for Relaxation Times		213	(1)
Dynamic Scaling Hypothesis for the Response Function		214	(1)
Scaling of the Non-linear Response		214	(2)
Scaling in the Approach to Equilibrium		216	(10)
Growth of a Fluctuating Surface		217	(2)
Spinodal Decomposition in Alloys and Block Copolymers		219	(7)
Summary		226	(3)
Appendix 8 The Fokker-Planck Equation		226	(3)
The Renormalisation Group		229	(58)
Block Spins		230	(6)
Thermodynamics		230	(4)
Correlation Functions		234	(1)
Discussion		235	(1)
Basic Ideas of the Renormalisation Group		236	(6)
Properties of Renormalisation Group Transformations		236	(4)
The Origin of Singular Behaviour		240	(2)
Fixed Points		242	(7)
Physical Significance of Fixed Points		242	(1)
Local Behaviour of RG Flows Near a Fixed Point		243	(3)
Global Properties of RG Flows		246	(3)
Origin of Scaling		249	(7)
One Relevant Variable		249	(3)
Diagonal RG Transformation for Two Relevant Variables		252	(2)
Irrelevant Variables		254	(1)
Non-diagonal RG Transformations		255	(1)
RG in Differential Form		256	(1)
RG for the Two Dimensional Ising Model		257	(11)
Exact Calculation of Eigenvalues from Onsager's Solution		258	(1)
Formal Representation of the Coarse-Grained Hamiltonian		259	(1)
Perturbation Theory for the RG Recursion Relation		260	(3)
Fixed Points and Critical Exponents		263	(1)
Effect of External Field		264	(1)
Phase Diagram		265	(2)
Remarks		267	(1)
First Order Transitions and Non-Critical Properties		268	(2)
RG for the Correlation Function		270	(1)
Crossover Phenomena		271	(7)
Small Fields		271	(2)
Crossover Arising from Anisotropy		273	(3)
Crossover and Disorder: the Harris Criterion		276	(2)
Corrections to Scaling		278	(1)
Finite Size Scaling		279	(8)
Exercises		283	(4)
Anomalous Dimensions Far From Equilibrium		287	(48)
Introduction		287	(2)
Similarity Solutions		289	(4)
Long Time Behaviour of the Diffusion Equation		289	(1)
Dimensional Analysis of the Diffusion Equation		290	(1)
Intermediate Asymptotics of the First Kind		291	(2)
Anomalous Dimensions in Similarity Solutions		293	(9)
The Modified Porous Medium Equation		293	(3)
Dimensional Analysis for Barenblatt's Equation		296	(2)
Similarity Solution with an Anomalous Dimension		298	(3)
Intermediate Asymptotics of the Second Kind		301	(1)
Renormalisation		302	(16)
Renormalisation and its Physical Interpretation		303	(2)
Heuristic Calculation of the Anomalous Dimension α		305	(1)
Renormalisation and Dimensional Analysis		306	(3)
Removal of Divergences and the RG		309	(4)
Assumption of Renormalisability		313	(1)
Renormalisation and Physical Theory		314	(2)
Renormalisation of the Modified Porous Medium Equation		316	(2)
Perturbation Theory for Barenblatt's Equation		318	(7)
Formal Solution		318	(1)
Zeroth Order in ε		319	(1)
First Order in ε		320	(1)
Isolation of the Divergence		320	(2)
Perturbative Renormalisation		322	(2)
Renormalised Perturbation Expansion		324	(1)
Origin of Divergence of Perturbation Theory		325	(1)
Fixed Points		325	(3)
Similarity Solutions as Fixed Points		326	(2)
Universality in the Approach to Equilibrium		328	(1)
Conclusion		328	(7)
Appendix 10 Method of Characteristics		329	(3)
Exercises		332	(3)
Continuous Symmetry		335	(16)
Correlation in the Ordered Phase		336	(9)
The Susceptibility Tensor		337	(2)
Excitations for T < Tc: Goldstone's Theorem		339	(4)
Emergence of Order Parameter Rigidity		343	(1)
Scaling of the Stiffness		344	(1)
Lower Critical Dimension		344	(1)
Kosterlitz-Thouless Transition		345	(6)
Phase Fluctuations		345	(1)
Phase Correlations		346	(2)
Vortex Unbinding		348	(2)
Universal Jump in the Stiffness		350	(1)
Critical Phenomena Near Four Dimensions		351	(38)
Basic Idea of the Epsilon Expansion		353	(1)
RG for the Gaussian Model		354	(7)
Integrating Out the Short Wavelength Degrees of Freedom		355	(2)
Rescaling of Fields and Momenta		357	(1)
Analysis of Recursion Relation		357	(1)
Critical Exponents		358	(1)
A Dangerous Irrelevant Variable in Landau Theory		359	(2)
RG Beyond the Gaussian Approximation		361	(9)
Setting Up Perturbation Theory		362	(3)
Calculation of ⟨V⟩0: Strategy		365	(1)
Correlation Functions of σl: Wick's Theorem		366	(2)
Evaluation of ⟨V⟩0		368	(2)
Feynman Diagrams		370	(7)
Feynman Diagrams to O(V)		370	(3)
Feynman Diagrams for ⟨V2⟩0 - ⟨V⟩20		373	(3)
Elimination of Unnecessary Diagrams		376	(1)
The RG Recursion Relations		377	(7)
Feynman Diagrams for Small ε = 4 - d		378	(2)
Recursion Relations to O(ε)		380	(1)
Fixed Points to O(ε)		381	(1)
RG Flows and Exponents		382	(2)
Conclusion		384	(5)
Appendix 12 The Linked Cluster Theorem		385	(1)
Exercises		386	(3)
Index		389

Nigel Goldenfeld is Professor of Physics at the University of Illinois at Urbana-Champaign. He received his Ph.D. from the University of Cambridge (U.K.) in 1982, and for the years 1982-1985 was a postdoctoral fellow at the Institute for Theoretical Physics, University of California at Santa Barbara. He is a University Scholar of the University of Illinois, a Fellow of the American Physical Society, a recipient of the Xerox Award for research, a member of the Editorial Board of the International Journal of Theoretical and Applied Finance, and was an Alfred P. Sloan Foundation Fellow from 1987-1991. David Pines is research professor of physics at the University of Illinois at Urbana-Champaign. He has made pioneering contributions to an understanding of many-body problems in condensed matter and nuclear physics, and to theoretical astrophysics. editor of Perseus' Frontiers in Physics series and former editor of American Physical Society's Reviews of Modern Physics, Dr. Pines is a member of the National Academy of Sciences, the American Philosophical Society, a foreign member of the USSR Academy of Sciences, a fellow of the American Academy of Arts and Sciences, and of the American Association for the Advancement of Science. Dr. Pines has received a number of awards, including the Eugene Feenberg Memorial Medal for Contributions to Many-Body Theory the P.A.M. Dirac Silver Medal for the Advancement of Theoretical Physics and the Friemann Prize in Condensed Matter Physics.