Introduction to relativistic statistical mechanics: classical and quantum
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| Language: | English |
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New Jersey [u.a.]
World Scientific
2011
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVII, 538 S. graph. Darst. |
| ISBN: | 9789814322430 |
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| 245 | 1 | 0 | |a Introduction to relativistic statistical mechanics |b classical and quantum |c Rémi Hakim |
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| adam_text | Titel: Introduction to relativistic statistical mechanics
Autor: Hakim, Rémi
Jahr: 2011
Contents
Preface xvii
Notations and Conventions xix
Introduction xxi
1. The One-Particle Relativistic Distribution Function 1
1.1 The One-Particle Relativistic Distribution Function ... 1
1.1.1 The phase space volume element ........ 5
1.2 The Jiittner-Synge Equilibrium Distribution....... 6
1.2.1 Thermodynamics of the Jiittner-Synge gas ... 9
1.2.2 Thermal velocity.................. 10
1.2.3 Moments of the Jiittner-Synge function..... 12
1.2.4 Orthogonal polynomials.............. 13
1.2.5 Zero mass particles................. 15
1.3 From the Microcanonical Distribution
to the Jiittner-Synge One................. 16
1.4 Equilibrium Fluctuations.................. 19
1.5 One-Particle Liouville Theorem.............. 21
1.5.1 Relativistic Liouville equation from the
Hamiltonian equations of motion......... 22
1.5.2 Conditions for the Jiittner-Synge functions
to be an equilibrium................ 24
1.6 The Relativistic Rotating Gas............... 24
2. Relativistic Kinetic Theory and the BGK Equation 27
2.1 Relativistic Hydrodynamics................ 29
2.1.1 Sound velocity................... 31
2.1.2 The Eckart approach................ 32
2.1.3 The Landau-Lifschitz approach.......... 34
viii Introduction to Relativistic Statistical Mechanics: Classical and Quantum
2.2 The Relaxation Time Approximation........... 35
2.3 The Relativistic Kinetic Theory Approach
to Hydrodynamics ..................... 36
2.4 The Static Conductivity Tensor.............. 40
2.5 Approximation Methods for the Relativistic Boltzmann
Equation and Other Kinetic Equations.......... 41
2.5.1 A simple Chapman-Enskog approximation ... 42
2.6 Transport Coefficients for a System Embedded
in a Magnetic Field..................... 43
3. Relativistic Plasmas 47
3.1 Electromagnetic Quantities in Covariant Form...... 47
3.2 The Static Conductivity Tensor.............. 50
3.3 Debye-Huckel Law..................... 51
3.4 Derivation of the Plasma Modes.............. 52
3.4.1 Evaluation of the various integrals........ 55
3.4.2 Collective modes in extreme cases........ 56
3.5 Brief Discussion of the Plasma Modes........... 57
3.6 The Conductivity Tensor.................. 62
3.7 Plasma-Beam Instability.................. 63
3.7.1 Perturbed dispersion relations
for the plasma-beam system ........... 63
3.7.2 Stability of the beam-plasma system....... 64
4. Curved Space-Time and Cosmology 67
4.1 Basic Modifications..................... 68
4.2 Thermal Equilibrium in a Gravitational Field...... 70
4.2.1 Thermal equilibrium in a static
isotropic metric................... 71
4.3 Einstein-Vlasov Equation................. 71
4.3.1 Linearization of Einstein s equation........ 72
4.3.2 The formal solution to the linearized
Einstein equation.................. 74
4.3.3 The self-consistent kinetic equation
for the gravitating gas............... 76
Contents ix
4.4 An Illustration in Cosmology............... 76
4.4.1 The two-timescale approximation......... 78
4.4.2 Derivation of the dispersion relations
(a rough outline).................. 80
4.5 Cosmology and Relativistic Kinetic Theory ....... 81
4.5.1 Cosmology: a very brief overview......... 82
4.5.2 Kinetic theory and cosmology........... 85
4.5.3 Kinetic theory of the observed universe..... 87
4.5.4 Statistical mechanics in the primeval
universe....................... 88
4.5.5 Particle survival .................. 90
5. Relativistic Statistical Mechanics 94
5.1 The Dynamical Problem.................. 94
5.2 Statement of the Main Statistical Problems....... 96
5.2.1 The initial value problem: observations
and measures.................... 97
5.2.2 Phase space and the Gibbs ensemble....... 100
5.3 Many-Particle Distribution Functions........... 102
5.3.1 Statistics of the particles manifolds....... 103
5.4 The Relativistic BBGKY Hierarchy............ 105
5.4.1 Cluster decomposition of the relativistic
distribution functions............... 107
5.5 Self-interaction and Radiation............... 109
5.5.1 An alternative treatment of radiation
reaction....................... Ill
5.5.2 Remarks on irreversibility ............ 113
5.5.3 Remarks on thermal equilibrium......... 114
5.6 Radiation Quantities.................... 116
5.7 A Few Relativistic Kinetic Equations........... 118
5.7.1 Derivation of the covariant Landau
equation....................... 118
5.7.2 The relativistic Vlasov equation
with radiation effects................ 121
5.7.3 Radiation effects for a relativistic plasma
in a magnetic field................. 124
5.8 Statistics of Fields and Particles.............. 125
x Introduction to Relativistic Statistical Mechanics: Classical and Quantum
6. Relativistic Stochastic Processes and Related Questions 128
6.1 Stochastic Processes in Minkowski Space-Time..... 129
6.1.1 Basic definitions.................. 130
6.1.2 Conditional currents................ 131
6.1.3 Markovian processes in space-time........ 131
6.2 Stochastic Processes in /x Space.............. 133
6.2.1 An overview..................... 134
6.2.2 Markovian processes................ 135
6.2.3 An alternative approach.............. 137
6.2.4 Markovian processes................ 139
6.2.5 A simple illustration................ 140
6.3 Relativistic Brownian Motion............... 142
6.4 Random Gravitational Fields: An Open Problem .... 144
6.4.1 A simple example ................. 148
6.4.2 The case of thermal equilibrium ......... 149
6.4.3 Matter-induced fluctuations............ 150
6.4.4 Random Einstein equations............ 151
7. The Density Operator 152
7.1 The Density Operator for Thermal Equilibrium..... 153
7.1.1 Thermodynamic properties............ 154
7.1.2 The partition function of the relativistic
ideal gas....................... 156
7.1.3 The average occupation number ......... 158
7.2 Relativistic Bosons in Thermal Equilibrium....... 159
7.2.1 The complex scalar field.............. 161
7.2.2 Charge fluctuations ................ 164
7.2.3 A few remarks on the calculation
of various integrals................. 164
7.2.4 Bose-Einstein condensation............ 165
7.2.5 Interactions..................... 167
7.3 Free Fermions in Thermal Equilibrium.......... 171
7.4 Thermodynamic Properties of the Relativistic
Ideal Fermi-Dirac Gas................... 174
7.4.1 Remarks on the numerical calculations
of various physical quantities........... 175
7.4.2 The degenerate Fermi gas............. 175
7.4.3 Thermal corrections: Sommerfeld expansion . . . 177
Contents xi
7.4.4 Corrections for various thermodynamic
quantities...................... 179
7.4.5 High temperature expansion (nondegenerate) . . 180
7.5 White Dwarfs: The Degenerate Electron Gas....... 181
7.5.1 Cooling of white dwarfs.............. 185
7.5.2 Pycnonuclear reactions............... 187
7.6 Functional Representation of the Partition Function . . 187
7.6.1 The partition function for gauge
particles (photons)................. 188
7.6.2 The photons partition function.......... 189
7.6.3 Illustration in the case of the Lorentz gauge . . . 191
8. The Covariant Wigner Function 194
8.1 The Covariant Wigner Function for Spin 1/2
Particles........................... 195
8.1.1 Basic equations................... 197
8.1.2 The equilibrium Wigner function
for free fermions.................. 200
8.1.3 Polarized media................... 201
8.2 Equilibrium Fluctuations of Fermions........... 204
8.3 A Simple Example..................... 207
8.4 The BBGKY Relativistic Quantum Hierarchy...... 208
8.5 Perturbation Expansion of the Wigner Function..... 211
8.6 The Wigner Function for Bosons ............. 213
8.6.1 The example of the p4 theory.......... 216
8.6.2 Four-current fluctuations of the complex
scalar field ..................... 217
8.7 Gauge Properties of the Wigner Function ........ 218
8.7.1 Gauge-invariant Wigner functions ........ 218
8.7.2 A few remarks................... 222
8.7.3 Gauge-invariant Wigner functions
for the photon field................. 223
8.7.4 Another gauge-invariant Wigner function .... 225
8.7.5 Gauge invariance and approximations...... 226
9. Fermions Interacting via a Scalar Field: A Simple Example 228
9.1 Thermal Equilibrium.................... 229
9.2 Collective Modes...................... 233
xii Introduction to Relativistic Statistical Mechanics: Classical and Quantum
9.3 Two-Body Correlations................... 234
9.3.1 A brief discussion.................. 237
9.3.2 Exchange correlations............... 238
9.4 Renormalization - An Illustration of the Procedure . . 240
9.4.1 Regularization of the gap equation........ 241
9.4.2 Regularization of the energy-momentum
tensor........................ 244
9.4.3 Determination of the constants
(AF,BF,CF,DF) .................. 245
9.5 Qualitative Discussion of the Effects
of Renormalization..................... 246
9.6 Thermodynamics of the System.............. 249
9.6.1 The gap equation as a minimum
of the free energy.................. 250
9.6.2 Thermodynamics.................. 251
9.7 Renormalization of the Excitation Spectrum....... 253
9.7.1 Comparison with the semiclassical case..... 257
9.8 A Short Digression on Bosons............... 258
10. Covariant Kinetic Equations in the Quantum Domain 262
10.1 General Form of the Kinetic Equation.......... 264
10.2 An Introductory Example................. 265
10.3 A General Relaxation Time Approximation....... 269
10.3.1 Properties of the kinetic system.......... 270
10.3.2 The collision term................. 272
10.3.3 General form of F(1)................ 274
11. Application to Nuclear Matter 277
11.1 Thermodynamic Properties at Finite Temperature . . . 279
11.1.1 Thermodynamics in some important cases .... 282
11.2 Remarks on the Oscillation Spectra of Mesons...... 285
11.3 Transport Coefficients of Nuclear Matter......... 286
11.3.1 Chapman-Enskog expansion ........... 288
11.3.2 Transport coefficients: Eckart versus
Landau-Lifschitz representations......... 290
11.3.3 Entropy production ................ 293
11.3.4 A brief comparison: BGK versus BUTJ...... 297
11.4 Discussion.......................... 299
Contents xiii
11.5 Dense Nuclear Matter: Neutron Stars........... 302
11.5.1 The static equilibrium of a neutron star..... 303
11.5.2 The composition of matter in a neutron star . . 304
11.5.3 Beyond the drip point............... 307
12. Strong Magnetic Fields 309
12.1 Relations Obeyed by the Magnetic Field......... 312
12.2 The Partition Function................... 314
12.2.1 Magnetization of an electron gas......... 317
12.3 Relativistic Quantum Liouville Equation......... 319
12.3.1 Solution of the inhomogeneous equation..... 321
12.3.2 The initial value problem............. 323
12.4 The Equilibrium Wigner Function for Noninteracting
Electrons........................... 324
12.4.1 Thermodynamic quantities............ 325
12.5 The Wigner Function of the Ideal Magnetized
Electron Gas ........................ 326
12.5.1 The nonmagnetic field limit............ 328
12.5.2 Equations of state................. 329
12.5.3 Is the pressure isotropic? ............. 330
12.5.4 The completely degenerate case.......... 331
12.5.5 Magnetization ................... 333
12.5.6 Landau orbital ferromagnetism:
LOFER states ................... 335
12.6 The Magnetized Vacuum.................. 336
12.6.1 The general structure of the vacuum
Wigner function .................. 336
12.6.2 The Wigner function of the magnetized
vacuum....................... 338
12.6.3 Renormalization of the vacuum
Wigner function .................. 339
12.7 Fluctuations......................... 340
12.7.1 Fluctuations of the four-current.......... 341
12.8 Polarization Tensors of the Magnetized Electron Gas
and of the Magnetized Vacuum.............. 348
12.8.1 The vacuum polarization tensor.......... 349
12.9 Remarks on the Transport Coefficients
of the Magnetized Electron Gas.............. 350
12.10 Astrophysical Aspects ................... 353
xiv Introduction to Relativistic Statistical Mechanics: Classical and Quantum
13. Statistical Mechanics of Relativistic Quasiparticles 356
13.1 Classical Fields....................... 359
13.1.1 Internal symmetries and conserved currents . . . 360
13.1.2 Space-time symmetries .............. 363
13.1.3 A general remark.................. 367
13.2 Quantum Quasiparticles.................. 370
13.2.1 Formal quantization................ 371
13.3 Problems with the Quantization of Quasiparticles .... 374
13.3.1 A first example................... 374
13.3.2 Another example the QED plasma........ 376
13.3.3 Migdal s approach................. 377
13.4 The Covariant Wigner Function.............. 379
13.5 Equilibrium Properties................... 382
13.6 A Simple Example: The / 4 Model............ 385
13.7 Remarks on the Thermodynamics of Quasiparticles . . . 388
13.8 Equilibrium Fluctuations.................. 391
13.9 Remarks on the Negative Energy Modes......... 394
13.10 Interacting Quasibosons.................. 395
13.10.1 The long wavelength and low frequency limit . . 398
14. The Relativistic Fermi Liquid 400
14.1 Independent Quasifermions ................ 400
14.1.1 Quantization and observables........... 402
14.1.2 Statistical expressions............... 405
14.1.3 Thermal equilibrium................ 406
14.2 Interacting Quasifermions................. 407
14.2.1 The long wavelength and low frequency limit . . 409
14.3 Kinetic Equation for Quasiparticles............ 410
14.4 Remarks on the Relativistic Landau Theory....... 412
15. The QED Plasma 422
15.1 Basic Equations....................... 422
15.2 Plasma Collective Modes.................. 423
15.3 The Fluctuation-Dissipation Theorem and Its Inverse . 428
15.4 Four-Current Fluctuations and the Polarization
Tensor............................ 429
15.5 The Polarization Tensor at Order e2 ........... 433
15.6 Quasiparticles in the Relativistic Plasma......... 436
Contents xv
15.6.1 Quasiphotons in thermal equilibrium....... 436
15.6.2 Gauge properties.................. 440
15.6.3 Quasielectron modes in thermal equilibrium . . . 442
Appendix A: A Few Useful Properties of Some Special Functions 446
A.l Kelvin s Functions ..................... 446
A.2 Associated Laguerre Polynomials............. 447
Appendix B: 7 Matrices 448
Appendix C: Outline of Functional Methods 451
C.l Functional Differentiation................. 452
C.2 Functional Integration................... 453
Appendix D: Units 457
D.l Ordinary Units....................... 457
D.2 Other Units of Interest................... 458
Appendix E: Some Useful Formulae for Wigner Functions 460
E.l Useful Formulae for Bosons................ 460
E.2 Useful Formulae for Fermions............... 462
Bibliography 465
Index 529
|
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| isbn | 9789814322430 |
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| physical | XXVII, 538 S. graph. Darst. |
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| spelling | Hakim, Rémi 1936- Verfasser (DE-588)134151232 aut Introduction to relativistic statistical mechanics classical and quantum Rémi Hakim New Jersey [u.a.] World Scientific 2011 XXVII, 538 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Relativistische Mechanik (DE-588)4177685-9 gnd rswk-swf Statistische Mechanik (DE-588)4056999-8 gnd rswk-swf Relativistische Quantenmechanik (DE-588)4177687-2 gnd rswk-swf Statistische Mechanik (DE-588)4056999-8 s Relativistische Mechanik (DE-588)4177685-9 s DE-604 Relativistische Quantenmechanik (DE-588)4177687-2 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022633094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hakim, Rémi 1936- Introduction to relativistic statistical mechanics classical and quantum Relativistische Mechanik (DE-588)4177685-9 gnd Statistische Mechanik (DE-588)4056999-8 gnd Relativistische Quantenmechanik (DE-588)4177687-2 gnd |
| subject_GND | (DE-588)4177685-9 (DE-588)4056999-8 (DE-588)4177687-2 |
| title | Introduction to relativistic statistical mechanics classical and quantum |
| title_auth | Introduction to relativistic statistical mechanics classical and quantum |
| title_exact_search | Introduction to relativistic statistical mechanics classical and quantum |
| title_full | Introduction to relativistic statistical mechanics classical and quantum Rémi Hakim |
| title_fullStr | Introduction to relativistic statistical mechanics classical and quantum Rémi Hakim |
| title_full_unstemmed | Introduction to relativistic statistical mechanics classical and quantum Rémi Hakim |
| title_short | Introduction to relativistic statistical mechanics |
| title_sort | introduction to relativistic statistical mechanics classical and quantum |
| title_sub | classical and quantum |
| topic | Relativistische Mechanik (DE-588)4177685-9 gnd Statistische Mechanik (DE-588)4056999-8 gnd Relativistische Quantenmechanik (DE-588)4177687-2 gnd |
| topic_facet | Relativistische Mechanik Statistische Mechanik Relativistische Quantenmechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022633094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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