Renormalization methods: a guide for beginners
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Oxford Univ. Press
2008
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| Physical Description: | XVIII, 330 S. graph. Darst. |
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| 100 | 1 | |a McComb, William D. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Renormalization methods |b a guide for beginners |c W. D. McComb |
| 250 | |a Repr. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2008 | |
| 300 | |a XVIII, 330 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Kritisches Phänomen |0 (DE-588)4165788-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Renormierung |0 (DE-588)4128419-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Renormierung |0 (DE-588)4128419-7 |D s |
| 689 | 0 | 1 | |a Kritisches Phänomen |0 (DE-588)4165788-3 |D s |
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| adam_text | CONTENTS
Normalization of Fourier integrals
]
Notation for Chapters
1-4
and
7-10 2
Notation for Chapters
5,6,
and
11 4
I WHAT IS RENORMALIZATION?
1
The bedrock problem: why we need renormalization methods
7
1.1
Some practical matters
8
1.1.1
Presumed knowledge: what you need to know before starting
8
1.1.2
The terminology minefield
9
1.1.3
The formalism minefield
1 ()
1.2
Quasi-particles and renormalization
10
1.2.1
A first example of renormalization: an electron
in an electrolyte
10
1.2.2
From micro to macro: a quantum system
11
1.2.3
A simple model of a gas
13
1.2.4
A more realistic model of a gas
15
1.2.5
Example of a coupled system: lattice vibrations 1
7
1.2.6
The general approach based on quasi-particles
20
1.2.7
Control parameters: how to weaken the coupling
20
1.2.8
Perturbation theory
21
1.3
Solving the relevant differential equations by perturbation theory
21
1.3.1
Solution in series
22
1.3.2
Green s functions
24
1.3.3
Example: Green s function in electrostatics
26
1.3.4
Example: Green s function for the diffusion equation
26
1.3.5
Simple perturbation theory: small
λ
29
1.3.6
Example: a slightly anharmonic oscillator
31
1.3.7
When
λ
is not small
33
1.4
Quantum field theory: a first look
33
1.4.1
What is quantum mechanics?
34
1.4.2
A simple field theory: the Klein-Gordon equation
35
1.4.3
The Klein-Gordon equation with interactions
36
1.4.4
Infrared and ultraviolet divergences
37
1.4.5
Renormaiized perturbation theory
38
CONTENTS
1.5
What is the renormalization group?
38
1.5.1
Magnetic models and spin
39
1.5.2
RG: the general idea
39
1.5.3
The problem of many scales
41
1.5.4
Problems with a few characteristic scales
41
1.5.5
Geometrical similarity, fractals, and self-similarity
42
1.5.6
Fixed points for a ferromagnet
44
1.5.7
Form
invariance
and scale
invariance: RG
and
the partition function
46
1.6
Discrete dynamical systems: recursion relations and fixed points
47
1.6.1
Example: repaying a loan at a fixed rate of interest
47
1.6.2
Definition of a fixed point
48
1.6.3
Example: a dynamical system with two fixed points
48
1.7
Revision of statistical mechanics
48
Further reading
51
1.8
Exercises
51
Easy applications of Renormalization Group to simple models
53
2.1
A one-dimensional magnet
53
2.1.1
The model
55
2.1.2
Coarse-graining transformation
56
2.1.3
Renormalization of the coupling constant
57
2.1.4
The free energy
F
58
2.1.5
Numerical evaluation of the recursion relation
59
2.2
Two-dimensional percolation
60
2.2.1
Bond percolation in two dimensions: the problem
61
2.2.2
The correlation length
62
2.2.3
Coarse-graining transformation
63
2.2.4
The recursion relation
64
2.2.5
Calculation of the fixed point: the critical probability
65
2.2.6
General remarks on RG applied to percolation
66
2.3
A two-dimensional magnet
66
2.3.1
The partition function
67
2.3.2
Coarse-graining transformation
68
2.4
Exercises
69
Mean-field theories for simple models
71
3.1
The Weiss theory of ferromagnetism
71
.1
The ferro-paramagnetic transition: theoretical aims
71
.2
The mean magnetization
72
.3
The molecular field B
72
.4
The self-consistent assumption:
B <x M
73
.5
Graphical solution for the critical temperature Tc
75
3.2
The Debye-Hiickel theory of the electron gas
76
3.2.1
The mean-field assumption
76
CONTENTS xiii
3.2.2
The self-consistent approximation
78
3.2.3
The screened potential
78
3.2.4
Validity of the continuum approximation
79
3.3
Macroscopic mean-field theory: the Landau model for phase transitions
79
3.3.1
The theoretical objective: critical exponents
79
3.3.2
Approximation for the free energy
F
80
3.3.3
Values of critical exponents
83
3.4
Exercises
84
II RENORMALIZED PERTURBATION THEORIES
4
Perturbation theory using a control parameter
87
4.1
High-temperature expansions
88
4.2
Application to a one-dimensional magnet
88
4.2.1
High-temperature expansion for the Ising ring
88
4.2.2
Formulation in terms of diagrams
91
4.2.3
Behavior near the critical point
92
4.3
Low-density expansions
93
4.4
Application to a slightly imperfect gas
94
4.4.1
Perturbation expansion of the configuration integral
96
4.4.2
The density expansion and the Virial coefficients
98
4.4.3
The two-particle cluster
100
4.4.4
The three-particle cluster
100
4.4.5
The four-particle cluster
101
4.4.6
Calculation of the second virial coefficient Bj
101
4.5
The Van
der Waals
equation
103
4.6
The Debye-Hiickel theory revisited
105
Further reading
107
4.7
Exercises
107
5
Classical nonlinear systems driven by random noise
110
5.1
The generic equation of motion
110
5.1.1
The Navier-Stokes equation: NSE
112
5.1.2
The Burgers equation
112
5.1.3
The KPZ equation
112
5.2
The moment closure problem
113
5.3
The pair-correlation tensor
113
5.4
The zero-order model system
114
5.5
A toy version of the equation of motion
115
5.6
Perturbation expansion of the toy equation of motion
115
5.6.1
The iterative calculation of coefficients
115
5.6.2
Explicit form of the coefficients
116
5.6.3
Equation for the exact correlation
117
CONTENTS
5.6.4 Factorizing
the zero-order moments
1
Π
5.7
Renormalized transport equations for the correlation function
118
5.7.1
Introduction of an exact response function
119
5.7.2
RPT equations for the exact correlation and response functions
120
5.8
Reversion of power series
120
5.9
Formulation in Wyld diagrams
121
5.9.1
Diagrams in the expansion for the exact correlation
123
5.9.2
The renormalized response function
124
5.9.3
Vertex renormalization
125
5.9.4
Renormalized expansions for the exact correlation and
response functions
126
5.9.5
Comparison with quantum field theory
126
Application of renormalized perturbation theories to turbulence and
related problems
129
6.1
The real and idealized versions of the turbulence problem
130
6.1.1
Stationary
isotropie
turbulence
130
6.1.2
Freely-decaying
isotropie
turbulence
131
6.1.3
Length scales for
isotropie
turbulence
131
6.1.4
Numerical simulation of turbulence
132
6.2
Two turbulence theories: the
DIA
and LET equations
133
6.2.1
DIA
and LET as mean-field theories
135
6.3
Theoretical results: free decay of turbulence
136
6.3.1
The energy spectrum E(k, t)
136
6.3.2
The energy transfer spectrum T(k, t)
137
6.4
Theoretical results: stationary turbulence
140
6.5
Detailed energy balance in wave number
140
6.5.1
Scale-invariance and the Kolmogorov spectrum
141
6.5.2
Theoretical results for the energy spectrum at large Reynolds
numbers
142
6.6
Application to other systems
144
HI RENORMALIZATION GROUP (RG)
7
Setting the scene: critical phenomena
147
7.1
Some background material on critical phenomena
147
7.1.1
Critical exponents
147
7.1.2
Correlation functions
149
7.1.3
The correlation length
150
7.1.4
Summary of critical exponents
151
7.2
Theoretical models
151
7.2.1
Generic Hamiltonian for
ű-dimensional
spins on a
¿f-dimensional lattice
151
CONTENTS xv
7.2.2
Examples of models of magnetism
152
7.2.3
The Ising model
152
7.3
Scaling behavior
] 53
7.3.1
Generalized homogeneous functions
153
7.3.2
The static scaling hypothesis
154
7.3.3
Relations among the critical exponents
154
7.3.4
Relationship between
β, γ,
and
S
156
7.3.5
Magnetic equation of state
157
7.3.6
Kadanoff
s
theory of block spins
157
7.4
Linear response theory
160
7.4.1
Example: spins on a lattice
161
7.5
Serious mean-field theory
162
7.5.1
The Bogoliubov variational theorem
162
7.5.2
Proof of the Bogoliubov inequality
163
7.5.3
Mean-field theory of the Ising model
164
7.5.4
The variational method
165
7.5.5
The optimal free energy
167
7.6
Mean-field critical exponents
α, β, γ,
and
S
for the Ising model
168
7.6.1
Exponenta
168
7.6.2
Exponent
β
169
7.6.3
Exponents
y
and
S
170
7.7
The remaining mean-field critical exponents for the Ising model
171
7.7.1
The connected correlation function
Gl¡¡
171
7.7.2
The discrete Fourier transform of eqn
(7.89) 172
7.7.3
The connected correlation function in Fourier space
174
7.7.4
Critical exponents
ν
and
η
175
7.8
Validity of mean-field theory
176
7.8.1
The model
176
7.8.2
A trick to evaluate the partition sum
177
7.8.3
The thermodynamic limit
178
7.9
Upper critical dimension
178
Further reading
179
7.10
Exercises
179
Real-space Renormalization Group
182
8.1
A general statement of the RG transformation
182
8.1.1
Invariance
of expectation values 1
83
8.2
RG transformation of the Hamiltonian and its fixed points
184
8.2.1
Linearization of the RGT about the fixed point: critical indices
185
8.2.2
System-point flows in parameter space
187
8.2.3
Scaling fields
188
8.3
Relations between critical exponents from RG
190
8.3.1
Application to magnetic systems
191
8.3.2
The critical exponent or
192
8.3.3
The critical exponent
ν
192
xvi CONTENTS
8.4
Applications of the linearized
RGT 193
8.4.1
Example: two-dimensional percolation
193
8.4.2
Example: two-dimensional magnet
194
Further reading
195
8.5
Exercises
196
9
Momentum-space Renormalization Group
197
9.1
Overview of this chapter
197
9.2
Statistical field theory
198
9.2.1
The continuum limit
198
9.2.2
Densities
199
9.2.3
The Ginsburg-Landau model
200
9.2.4
Consistency with the Landau model
201
9.3
Renormalization group transformation in wave number space
202
9.4
Scaling dimension: anomalous and normal
203
9.4.1
Anomalous dimension
204
9.4.2
Normal dimension
205
9.5
Restatement of our objectives: numerical calculation of the
critical exponents
206
9.6
The Gaussian zero-order model
206
9.6.1
Functional integration
207
9.6.2
The Gaussian functional integral
208
9.7
Partition function for the Gaussian model
210
9.8
Correlation functions
213
9.8.1
Example: two-point connected Gaussian correlation
213
9.9
Fixed points for the Gaussian model
214
9.9.1
The RG equations
214
9.9.2
The fixed points
216
9.9.3
Normal dimension of coupling constants
217
9.10
Ginsburg-Landau (GL) theory
218
9.10.1
Perturbative implementation of the RGT
218
9.10.2
The Gaussian fixed point for
d
> 4 220
9.10.3
Non-Gaussian fixed points for
d
< 4 222
9.10.4
The beta-function
223
9.10.5
The marginal case:
d
= 4 224
9.10.6
Critical exponents to order
e
224
Further reading
226
9.11
Exercises
226
10
Field-theoretic Renormalization Group
228
10.1
Preliminary remarks
229
10.1.1
Changes of notation
229
10.1.2
Regularization versus renormalization
229
10.2
The Ginsburg-Landau model as a quantum field theory
230
CONTENTS xvii
10.3
Infrared and ultraviolet divergences
230
10.3.1
Example: the photon propagator
231
10.3.2
Dimensional regularization
231
10.3.3
Examples of regularization
232
10.4
Renormalization
invariance
234
10.4.1
The Callan-Symanzik equations
234
10.4.2
Example: the beta-function for
φ4
theory in dimension
d
< 4 236
10.5
Perturbation theory in x-space
237
10.5.1
The generating functional for correlations
237
10.5.2
Gaussian
η
-point
correlations
238
10.5.3
Wick s theorem: evaluation of the 2^-point Gaussian
correlation
239
10.6
Perturbation expansion in x-space
240
10.6.1
Evaluation of the exact n-point correlation
240
10.6.2
Example: the two-point correlation
240
10.6.3
Feynman diagrams
243
10.6.4
Vacuum fluctuations or bubbles
244
10.7
Perturbation expansion in ¿-space
245
10.7.1
Connected and disconnected diagrams
246
10.7.2
Reducible and irreducible diagrams
246
10.7.3
The self-energy £(fc)
247
10.7.4
Vertex functions
248
10.7.5
Generating functional for vertex functions
Γ[ψ]
248
10.8
The UV divergence and renormalization
248
10.8.1
Mass renormalization: mo
—>■
m at one-loop order
249
10.8.2
Coupling constant renormalization:
λο -> λ
at one-loop order
250
10.8.3
Field renormalization:
Γ
(2) to two-loop order
251
10.9
The
IR
divergence and the e-expansion
254
10.9.1
Modified coupling constant
254
10.9.2
Calculation of r/
255
10.9.3
Values of the critical exponents
258
10.10
The pictorial significance of Feynman diagrams
258
Further reading
259
11
Dynamical Renormalization Group applied to classical nonlinear system
260
11.1
The dynamical RG algorithm
260
11.2
Application to the Navier-Stokes equation
262
11.2.1
The RG transformation: the technical problems
263
11.2.2
Overview of perturbation theory
266
11.2.3
The application of RG at small wave numbers
267
11.2.4
The application of RG at large wave numbers
268
11.3
Application of RG to stirred fluid motion with asymptotic freedom as
к
-> 0 270
11.3.1
Differential RG equations
272
11.3.2
Application to other systems
274
xviii CONTENTS
11.4
Relevance of RG to the large-eddy simulation of turbulence
274
11.4.1
Statement of the problem
276
11.4.2
Conservation equations for the explicit scales
k
<kc
278
11.5
The conditional average at large wave numbers
280
11.5.1
The asymptotic conditional average
282
11.6
Application of RG to turbulence at large wave numbers
284
11.6.1
Perturbative calculation of the conditional average
286
11.6.2
Truncation of the moment expansion
287
11.6.3
The RG calculation of the effective viscosity
287
11.6.4
Recursion relations for the effective viscosity
288
Further reading
291
IV APPENDICES
A Statistical ensembles
295
A.
1
Statistical specification of the iV-body assembly
295
A.2 The basic postulates of equilibrium statistical mechanics
296
A.3 Ensemble of assemblies in energy contact
297
A.4 Entropy of an assembly in an ensemble
298
A.5 Principle of maximum entropy
300
A.6
Variatíonal
method for the most probable distribution
301
В
From statistical mechanics to thermodynamics
304
B.I The canonical ensemble
304
В Л
. 1
Identification of the
Lagrange
multiplier
305
B.I
.2
General thermodynamic processes
306
B.
1.3
Equilibrium distribution and the bridge equation
307
B.2 Overview and summary
308
B.2.1 The canonical ensemble
309
С
Exact solutions in one and two dimensions
310
C.I The one-dimensional Ising model
310
C.2 Bond percolation in
d
= 2 312
D
Quantum treatment of the Hamiltonian
Л
-body
assembly
ЗІЗ
D.I The density matrix pmn
314
D.2 Properties of the density matrix
315
D.3 Density operator for the canonical ensemble
316
E
Generalization of the Bogoliubov
variatíonal
method to a spatially varying
magnetic field
318
References
320
Index
323
This is a unique book, occupying a gap between final-year undergraduate texts on
critical phenomena and advanced texts on quantum field theory. It covers a range
of renormalization methods with a clear physical interpretation (and motivation),
including mean-field theories and high-temperature and low-density expansions.
It then proceeds by easy steps to the famous
epsilon-expansion,
ending up with the
first-order corrections to critical exponents beyond mean-field theory. Nowadays
there is widespread interest in applications of renormalization methods to various
topics ranging over soft condensed matter, engineering dynamics, traffic queiieing
and fluctuations in the stock market Hence macroscopic systems are also included,
with particular emphasis on the archetypal problem of fluid turbulence. The book
is unique in making this material accessible to readers other than theoretical
physicists, as it requires only the basic physics and mathematics which should
be known to most scientists, engineers and mathematicians.
W. D. McComb is Professor of Statistical Physics at the
University of Edinburgh, UK.
|
| any_adam_object | 1 |
| author | McComb, William D. |
| author_facet | McComb, William D. |
| author_role | aut |
| author_sort | McComb, William D. |
| author_variant | w d m wd wdm |
| building | Verbundindex |
| bvnumber | BV035988611 |
| classification_rvk | UO 4020 |
| classification_tum | PHY 025f |
| ctrlnum | (OCoLC)633378933 (DE-599)BVBBV035988611 |
| discipline | Physik |
| edition | Repr. |
| format | Book |
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| id | DE-604.BV035988611 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:09:05Z |
| institution | BVB |
| isbn | 9780198506942 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018881373 |
| oclc_num | 633378933 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | XVIII, 330 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| spelling | McComb, William D. Verfasser aut Renormalization methods a guide for beginners W. D. McComb Repr. Oxford [u.a.] Oxford Univ. Press 2008 XVIII, 330 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Kritisches Phänomen (DE-588)4165788-3 gnd rswk-swf Renormierung (DE-588)4128419-7 gnd rswk-swf Renormierung (DE-588)4128419-7 s Kritisches Phänomen (DE-588)4165788-3 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018881373&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018881373&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | McComb, William D. Renormalization methods a guide for beginners Kritisches Phänomen (DE-588)4165788-3 gnd Renormierung (DE-588)4128419-7 gnd |
| subject_GND | (DE-588)4165788-3 (DE-588)4128419-7 |
| title | Renormalization methods a guide for beginners |
| title_auth | Renormalization methods a guide for beginners |
| title_exact_search | Renormalization methods a guide for beginners |
| title_full | Renormalization methods a guide for beginners W. D. McComb |
| title_fullStr | Renormalization methods a guide for beginners W. D. McComb |
| title_full_unstemmed | Renormalization methods a guide for beginners W. D. McComb |
| title_short | Renormalization methods |
| title_sort | renormalization methods a guide for beginners |
| title_sub | a guide for beginners |
| topic | Kritisches Phänomen (DE-588)4165788-3 gnd Renormierung (DE-588)4128419-7 gnd |
| topic_facet | Kritisches Phänomen Renormierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018881373&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018881373&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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