The field theoretic renormalization group in critical behavior theory and stochastic dynamics
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| 100 | 1 | |a Vasilʹev, Aleksandr N. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Kvantovopolevaja renormgruppa v teorii kritičeskogo povedenija i stochastičeskoj dinamike |
| 245 | 1 | 0 | |a The field theoretic renormalization group in critical behavior theory and stochastic dynamics |c A. N. Vasil'ev. Transl. by Patricia A. de Forcrand-Millard |
| 264 | 1 | |a Boca Raton [u.a.] |b Chapman & Hall/CRC Press |c 2004 | |
| 300 | |a XVI, 681 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Aus dem Russ. übers. - Bibliogr. S. 661 - 673 | ||
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| 650 | 7 | |a Teoria quântica de campo |2 larpcal | |
| 650 | 4 | |a Critical phenomena (Physics) | |
| 650 | 4 | |a Renormalization group | |
| 650 | 4 | |a Statistical physics | |
| 650 | 4 | |a Stochastic processes | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014878739&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| _version_ | 1819638786302148608 |
|---|---|
| adam_text | Titel: The field theoretic renormalization group in critical behavior theory and stochastic dynamics
Autor: Vasilʹev, Aleksandr N
Jahr: 2004
Contents
PREFACE xiii
CHAPTER 1 Foundations of the Theory of Critical
Phenomena 1
1.1 Historical review....................................................1
1.2 Generalized homogeneity............................................13
1.3 The scaling hypothesis (critical scaling) in thermodynamics . . 15
1.4 The Ising model and thermodynamics of a ferromagnet.....17
1.5 The scaling hypothesis for the uniaxial ferromagnet..............19
1.6 The On-symmetric classical Heisenberg ferromagnet......22
1.7 The classical nonideal gas: the model and thermodynamics . . 23
1.8 The thermodynamical scaling hypothesis for the critical point
of the liquid-gas transition ........................................27
1.9 The scaling hypothesis for the correlation functions.......31
1.10 The functional formulation ........................................35
1.11 Exact variational principle for the mean field..........37
1.12 The Landau theory..................................................40
1.13 The fluctuation theory of critical behavior............41
1.14 Examples of specific models........................................44
1.15 Canonical dimensions and canonical scale invariance......47
1.16 Relevant and irrelevant interactions. The logarithmic
dimension............................................................49
1.17 An example of a two-scale model: the uniaxial ferroelectric . . 52
1.18 Ultraviolet multiplicative renormalization........................54
1.19 Dimensional regularization. Relation between the exact and
formal expressions for one-loop integrals..........................58
1.20 The renormalization problem in dimensional regularization . . 62
1.21 Explicit renormalization formulas..................................66
1.22 The constants Z in the minimal subtraction scheme ......68
1.23 The relation between the IR and UV problems.........69
1.24 The differential RG equations......................................70
1.25 The RG functions in terms of the renormalization constants . . 72
1.26 Relations between the residues of poles in Z of various order
in e. Representation of Z in terms of RG functions.......74
1.27 Relation between the renormalized and bare charges............75
1.28 Renormalization and RG equations for T TC .........77
1.29 Solution of the linear partial differential equations.......78
v
CONTENTS
1.30 The RG equation for the correlator of the p4 model m zero
field.................................
1.31 Fixed points and their classification ...............
1.32 Invariant charge of the RG equation for the correlator .....
1.33 Critical scaling, anomalous critical dimensions, scaling function
of the correlator ..........................
1.34 Conditions for reaching the critical regime. Corrections to
scaling................................
1.35 What is summed in the solution of the RG equation?......
1.3G Algorithm for calculating the coefficients of e expansions of
critical exponents and scaling functions .............
1.37 Results of calculating the e expansions of the exponents of the
On (p4 model in dimension d = 4 — 2e ..............
1.38 Summation of the e expansions. Results.............
1.39 The RG equation for F(o) (the equation of state)........
1.40 Subtraction-scheme independence of the critical exponents
and normalized scaling functions.................
1.41 The renormalization group in real dimension ..........
1.42 Multicharge theories........................
1.43 Logarithmic corrections for e = 0.................
1.44 Summation of the gins contributions at £ — 0 using the RG
equations..............................
CHAPTER 2 Functional and Diagrammatic Technique
of Quantum Field Theory 115
2.1 Basic formulas ...........................115
2.2 The universal graph technique...................119
2.3 Graph representations of Green functions ............124
2.4 Graph technique for spontaneous symmetry breaking (t 0) . 127
2.5 One-irreducible Green functions..................129
2.6 Graph representations of T(a) and the functions Tn ......131
2.7 Passage to momentum space....................134
2.8 The saddle-point method. Loop expansion of V(A) ......137
2.9 Loop expansion of T(q)......................139
2.10 Loop calculation of T(a) in the On ip4 model..........141
2.11 The Schwinger equations......................145
2.12 Solutions of the equations of motion...............147
2.13 Green functions with insertion of composite operators.....149
2.14 Summary of definitions of various Green functions.......152
2.15 Symmetries, currents, and the energy-momentum tensor .... 153
2.16 Ward identities....................
2.17 The relation between scale and conformal invariance......164
2.18 Conformal structures for dressed propagators and triple
vertices...................
2.19 The large-n expansion in the On p4 model for T Tc.....168
81
82
84
86
89
90
92
93
96
100
101
104
107
109
112
CONTENTS vii
2.20 A simple method of constructing the large-n expansion.....174
2.21 The large-n expansion of the functionals W and F for
A^a^n1/2............................176
2.22 The solution for arbitrary A, T in leading order in 1/n.....178
2.23 The A —? 0 asymptote. Singularity of the longitudinal
susceptibility for T TC......................181
2.24 Critical behavior in leading order in 1/n.............182
2.25 A simplified field model for calculating the large-n
expansions of critical exponents..................184
2.26 The classical Heisenberg magnet and the nonlinear a model . . 187
2.27 The large-n expansion in the nonlinear a model.........189
2.28 Generalizations: the CPn_1 and matrix a models........191
2.29 The large-n expansion for ( £2)3-type interactions........192
2.30 Systems with random admixtures.................193
2.31 The replica method for a system with frozen admixtures .... 196
CHAPTER 3 Ultraviolet Renormalization 199
3.1 Preliminary remarks........................199
3.2 Superficially divergent graphs. Classification of theories
according to their rcnormalizability................201
3.3 Primitive and superficial divergences...............202
3.4 Renormalization of the parameters r and g in the one-loop
approximation ...........................204
3.5 Various subtraction schemes. The physical meaning of the
parameter r.............................205
3.6 The two-loop approximation....................208
3.7 The basic action and counterterms................210
3.8 The operators L, R, and Rr....................212
3.9 The Bogolyubov-Parasyuk R operation.............215
3.10 Recursive construction of L in terms of the subtraction
operator K.............................217
3.11 The commutativity of L, R , and R with dT-type operators . . 219
3.12 The basic statements of renormalization theory.........220
3.13 Remarks about the basic statements...............222
3.14 Proof of the basic combinatorial formula for the R operation . 225
3.15 Graph calculations in arbitrary dimension............232
3.16 Dimensional regularization and minimal subtractions......236
3.17 Normalized functions........................238
3.18 The renormalization constants in terms of counterterms in
the MS scheme...........................242
3.19 The passage to massless graphs..................243
3.20 The constants Z in three-loop order in the MS scheme for
the On p4 model..........................247
3.21 Technique for calculating the ..................250
CONTENTS
viii
3.22 Nonmultiplicativity of the renormalization in analytic
regularization............................
3.23 The inclusion of composite operators...............
3.24 The renormalized composite operator...............
3.25 Renormalization of the action and Green functions of the
extended model...........................
3.26 Structure of the operator counterterms..............266
3.27 An example of calculating operator counterterms........269
3.28 Matrix multiplicative renormalization of families of operators . 273
3.29 UV finiteness of operators associated with the renormalized
action and conserved currents...................275
3.30 The On (p4 model: renormalization of scalar operators with
d*F = 2,3,4.............................278
3.31 Renormalization of conserved currents..............280
3.32 Renormalization of tensor operators with d*F = 4 in the On p 4
model................................281
3.33 The Wilson operator expansion for short distances.......283
3.34 Calculation of the Wilson coefficients in the one-loop
approximation ...........................288
3.35 Expandability of multiloop counterterms L^7 in p and r ... . 291
3.36 Renormalization in the case of spontaneous symmetry
breaking...............................293
CHAPTER 4 Critical Statics 299
4.1 General scheme for the RG analysis of an arbitrary model . . . 299
4.2 The On p4 model: the constants Z, RG functions, and
4 — £ expansion of the exponents.................301
4.3 Renormalization and the RG equations for the renormalized
functional Wr{A) including vacuum loops............305
4.4 The On p4 model: renormalization and the RG equation for
the free energy...........................308
4.5 General solution of the inhomogeneous RG equation for the
free energy of the ip4 model and the amplitude ratio A+j A— in
the specific heat ..........................399
4.6 RG equations for composite operators and coefficients of the
Wilson operator expansion.....................312
4.7 Critical dimensions of composite operators............314
4.8 Correction exponents u associated with IR-irrelevant
composite operators................................319
4.9 Example: the system F = {l,y 2} in the On tp4 model.....319
4.10 Second example: scalar operators with d*F = 4..........321
4.11 Determination of the critical dimensions of composite operators
following Sec. 3.29 ....................................324
4.12 The On ip4 model: calculation of the 1- and 2-loop graphs of
the renormalized correlator in the symmetric phase.......325
CONTENTS ix
4.13 £ expansion of the normalized scaling function..........329
4.14 Analysis of the r — 0 asymptote using the Wilson operator
expansion..............................332
4.15 Goldstone singularities for T TC ................337
4.16 The two-charge p4 model with cubic symmetry.........344
4.17 RG functions and critical regimes.................348
4.18 The Ising model (uniaxial magnet) with random impurities.
e1/2 expansions of the exponents.................350
4.19 Two-loop calculation of the e expansions of the exponents for
a uniaxial ferroelectric.......................352
4.20 The rap4 interaction (modified critical behavior)........356
4.21 The pQ model in dimension d = 3 — 2e..............357
4.22 The p4 + p6 model.........................362
4.23 RG analysis of the tricritical asymptote in the p4 + p6 model . 364
4.24 Renormalization of the p3 model in dimension d = 6 — 2e ... 369
4.25 RG equations for the p3 model including vacuum loops .... 373
4.26 The 2 + e expansion in the nonlinear a model: multiplicative
renormalizability of low-temperature perturbation theory . . . 376
4.27 Calculation of the constants Z and the RG functions in the
one-loop approximation......................379
4.28 The Goldstone and critical asymptotes. 2 + e expansion of the
critical exponents..........................381
4.29 The 1/n expansion of the critical exponents of the On p4 and
a models...............................385
4.30 Calculation of 1/n expansions of the exponents in terms of the
RG functions of the p4 model...................387
4.31 The analog of dimensional regularization and nonmultiplicative
renormalization of the massless a model.............388
4.32 Critical scaling. Calculation of the critical dimensions from the
Green functions...........................392
4.33 Calculation of the dimensions of fields and composite operators
using counterterms of graphs in first order in 1/n........394
4.34 Examples..............................399
4.35 Calculation of the principal exponents using the self-consistency
equations for the correlators....................405
4.36 The technique for calculating massless graphs..........410
4.37 Calculation of r i..........................425
4.38 Generalization of the self-consistency equations to the case of
correction exponents........................428
4.39 Calculation of V2 and U ......................433
4.40 Calculation of 773 in the cr model by the conformal bootstrap
technique..............................435
4.41 Conformal invariance in the critical regime ...........444
4.42 Generalization to composite operators..............449
4.43 Examples..............................454
CONTENTS
4.44 The chiral phase transition in the Gross-Neveu model.....451
4.45 Two-loop calculation of the RG functions of the GN model in
dimension 2 . .........................*
4 46 The multiplicatively renormalizable two-charge GN model with
c , , .......469
a field.........................
4.47 Proof of critical conformal invariance...............472
4.48 1/n expansion of the critical exponents of the GN model .... 476
4.49 Use of the 1/n expansions of exponents to calculate RG
functions ..............................
CHAPTER 5 Critical Dynamics 487
5.1 Standard form of the equations of stochastic dynamics.....487
5.2 Iterative solution of the stochastic equations...........490
5.3 Reduction of the stochastic problem to a quantum field model . 491
5.4 Some consequences of retardation.................495
5.5 Stability criterion for a system in stochastic dynamics.....497
5.6 Equations for equal-time correlation functions of the field (p . . 498
5.7 The Fokkcr-Planck equation for the equal-time distribution
function of the field (p.......................500
5.8 Relation between dynamics and statics for the stochastic
Langevin equation.........................501
5.9 General principles for constructing models of critical dynamics.
The intermode interaction.....................503
5.10 Response to an external field...................506
5.11 The fluctuation-dissipation theorem...............507
5.12 Examples of actual models of critical dynamics.........509
5.13 The physical interpretation of models A-J............512
5.14 Canonical dimensions in dynamics................515
5.15 Analysis of the UV divergences and counterterms in dynamics . 517
5.16 Models A and B.........................521
5.17 Model C (slow heat conduction): statics.............527
5.18 Model C: dynamics....................................533
5.19 Model D........................................333
5.20 Models F and E..................................333
5.21 Model G..................
5.22 Model J....................................3^
5.23 Model H: determination of dynamical variables.........546
5.24 Model H: IR irrelevance of the sound modes in the regime
^ ^ V...................................
5.25 Model H: renormalization and RG analysis in the regime
w ~ P ................................555
5.26 Sound propagation near the critical point..............534
CONTENTS xi
CHAPTER 6 Stochastic Theory of Turbulence 581
6.1 The phenomenon of turbulence..................581
6.2 The stochastic Navier-Stokes equation. The Kolmogorov
hypotheses .............................582
6.3 Choice of the random-force correlator ..............586
6.4 UV divergences, renormalization, and RG equations of the
quantum field model........................589
6.5 General solution of the RG equations. IR scaling for fixed
parameters go and ........................592
6.6 IR scaling at fixed parameters W and v$. Viscosity
independence and the freezing of dimensions for e 2.....596
6.7 Renormalization of composite operators. Use of the Schwinger
equations and Galilean invariance.................599
6.8 Renormalization of composite operators in the energy and
momentum conservation laws...................603
6.9 Study of the m — 0 asymptote of the scaling functions of the
pair velocity correlator using the SDE..............608
6.10 Summation of the contributions of dangerous operators and
d[(pn in the dynamical velocity correlator............613
6.11 The problem of singularities for e — 2 in the massless model,
e-expansion of the Kolmogorov constant.............616
6.12 Deviations from Kolmogorov scaling for composite operators . 621
6.13 Turbulent mixing of a scalar passive admixture.........625
6.14 Stochastic magnetic hydrodynamics (MHD)...........629
6.15 Critical dimensions in MHD....................636
6.16 The turbulent dynamo in gyrotropic MHD............640
6.17 Critical dimensions in the dynamo regime............644
6.18 Two-dimensional turbulence....................647
6.19 Langmuir turbulence of a plasma.................649
ADDENDUM 657
BIBLIOGRAPHY 661
SUBJECT INDEX
675
|
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| author | Vasilʹev, Aleksandr N. |
| author_facet | Vasilʹev, Aleksandr N. |
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| id | DE-604.BV021664277 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T19:28:09Z |
| institution | BVB |
| isbn | 0415310024 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014878739 |
| oclc_num | 54349621 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-83 DE-11 |
| owner_facet | DE-19 DE-BY-UBM DE-83 DE-11 |
| physical | XVI, 681 S. graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Chapman & Hall/CRC Press |
| record_format | marc |
| spellingShingle | Vasilʹev, Aleksandr N. The field theoretic renormalization group in critical behavior theory and stochastic dynamics Mecânica estatística larpcal Processos estocásticos larpcal Teoria quântica de campo larpcal Critical phenomena (Physics) Renormalization group Statistical physics Stochastic processes |
| title | The field theoretic renormalization group in critical behavior theory and stochastic dynamics |
| title_alt | Kvantovopolevaja renormgruppa v teorii kritičeskogo povedenija i stochastičeskoj dinamike |
| title_auth | The field theoretic renormalization group in critical behavior theory and stochastic dynamics |
| title_exact_search | The field theoretic renormalization group in critical behavior theory and stochastic dynamics |
| title_full | The field theoretic renormalization group in critical behavior theory and stochastic dynamics A. N. Vasil'ev. Transl. by Patricia A. de Forcrand-Millard |
| title_fullStr | The field theoretic renormalization group in critical behavior theory and stochastic dynamics A. N. Vasil'ev. Transl. by Patricia A. de Forcrand-Millard |
| title_full_unstemmed | The field theoretic renormalization group in critical behavior theory and stochastic dynamics A. N. Vasil'ev. Transl. by Patricia A. de Forcrand-Millard |
| title_short | The field theoretic renormalization group in critical behavior theory and stochastic dynamics |
| title_sort | the field theoretic renormalization group in critical behavior theory and stochastic dynamics |
| topic | Mecânica estatística larpcal Processos estocásticos larpcal Teoria quântica de campo larpcal Critical phenomena (Physics) Renormalization group Statistical physics Stochastic processes |
| topic_facet | Mecânica estatística Processos estocásticos Teoria quântica de campo Critical phenomena (Physics) Renormalization group Statistical physics Stochastic processes |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014878739&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT vasilʹevaleksandrn kvantovopolevajarenormgruppavteoriikriticeskogopovedenijaistochasticeskojdinamike AT vasilʹevaleksandrn thefieldtheoreticrenormalizationgroupincriticalbehaviortheoryandstochasticdynamics |