Equilibrium and non-equilibrium statistical thermodynamics:
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Cambridge [u.a.]
Cambridge Univ. Press
2004
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| Edition: | Reprinted |
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| Physical Description: | XVI, 616 S. Ill., graph. Darst. |
| ISBN: | 0521821436 9780521821438 9780521528955 |
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| 100 | 1 | |a Le Bellac, Michel |d 1939- |e Verfasser |0 (DE-588)131512188 |4 aut | |
| 245 | 1 | 0 | |a Equilibrium and non-equilibrium statistical thermodynamics |c Michel Le Bellac, Fabrice Mortessagne and G. George Batrouni |
| 250 | |a Reprinted | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2004 | |
| 300 | |a XVI, 616 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 0 | 7 | |a Statistische Thermodynamik |0 (DE-588)4126251-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Statistische Thermodynamik |0 (DE-588)4126251-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Mortessagne, Fabrice |d 1966- |e Verfasser |0 (DE-588)129538868 |4 aut | |
| 700 | 1 | |a Batrouni, Ghassan George |d 1956- |e Verfasser |0 (DE-588)129538841 |4 aut | |
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| _version_ | 1804137591770972161 |
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| adam_text | Contents
Preface
page
xv
Thermostatics
1
1.1
Thermodynamic equilibrium
1
1.1.1
Microscopic and macroscopic descriptions
1
1.1.2
Walls
3
1.1.3
Work, heat, internal energy
5
1.1.4
Definition of thermal equilibrium
8
1.2
Postulate of maximum entropy
9
1.2.1
Internal constraints
9
1.2.2
Principle of maximum entropy
10
1.2.3
Intensive variables: temperature, pressure, chemical potential
12
1.2.4
Quasi-static and reversible processes
17
1.2.5
Maximum work and heat engines
20
1.3
Thermodynamic potentials
22
1.3.1
Thermodynamic potentials and Massieu functions
22
1.3.2
Specific heats
24
1.3.3
Gibbs-Duhem relation
26
1.4
Stability conditions
27
1.4.1
Concavity of entropy and convexity of energy
27
1.4.2
Stability conditions and their consequences
28
1.5
Third law of thermodynamics
31
1.5.1
Statement of the third law
31
1.5.2
Application to metastable states
32
1.5.3
Low temperature behaviour of specific heats
33
1.6
Exercises
35
1.6.1
Massieu functions
35
1.6.2
Internal variable in equilibrium
35
1.6.3
Relations between thermodynamic coefficients
36
vi
Contents
1.6.4
Contact
between two systems
37
1.6.5
Stability conditions
37
1.6.6
Equation of state for a fluid
37
1.7
Problems
38
1.7.1
Reversible and irreversible free expansions of an ideal
gas
38
1.7.2
van
der Waals
equation of state
39
1.7.3
Equation of state for a solid
40
1.7.4
Specific heats of a rod
41
1.7.5
Surface tension of a soap film
42
1.7.6
Joule-Thomson process
43
1.7.7
Adiabatic demagnetization of a paramagnetic salt
43
1.8
Further reading
45
2
Statistical entropy and Boltzmann distribution
47
2.1
Quantum description
47
2.1.1
Time evolution in quantum mechanics
47
2.1.2
The density operators and their time evolution
49
2.1.3
Quantum phase space
51
2.1.4
(P, V, E) relation for a mono-atomic ideal gas
53
2.2
Classical description
55
2.2.1
Liouville s theorem
55
2.2.2
Density in phase space
56
2.3
Statistical entropy
59
2.3.1
Entropy of a probability distribution
59
2.3.2
Statistical entropy of a mixed quantum state
60
2.3.3
Time evolution of the statistical entropy
63
2.4
Boltzmann distribution
64
2.4.1
Postulate of maximum of statistical entropy
64
2.4.2
Equilibrium distribution
65
2.4.3
Legendre transformation
67
2.4.4
Canonical and grand canonical ensembles
68
2.5
Thermodynamics revisited
70
2.5.1
Heat and work: first law
70
2.5.2
Entropy and temperature: second law
72
2.5.3
Entropy of mixing
74
2.5.4
Pressure and chemical potential
77
2.6
Irreversibility and the growth of entropy
79
2.6.1
Microscopic reversibility and macroscopic irreversibility
79
2.6.2
Physical basis of irreversibility
81
2.6.3
Loss of information and the growth of entropy
83
Contents
vii
2.7
Exercises
86
2.7.1
Density operator for spin-
1 /2 86
2.7.2
Density of states and the dimension of space
88
2.7.3
Liouville theorem and continuity equation
88
2.7.4
Loaded dice and statistical entropy
89
2.7.5
Entropy of a composite system
89
2.7.6
Heat exchanges between system and reservoir
89
2.7.7
Galilean transformation
90
2.7.8
Fluctuation-response theorem
90
2.7.9
Phase space volume for
N
free particles
92
2.7.10
Entropy of mixing and osmotic pressure
92
2.8
Further reading
93
3
Canonical and grand canonical ensembles: applications
95
3.1
Simple examples in the canonical ensemble
95
3.1.1
Mean values and fluctuations
95
3.1.2
Partition function and thermodynamics of an ideal gas
98
3.1.3
Paramagnetism
101
3.1.4
Ferromagnetism and the Ising model
105
3.1.5
Thermodynamic limit
112
3.2
Classical statistical mechanics
115
3.2.1
Classical limit
115
3.2.2
Maxwell distribution
116
3.2.3
Equipartition theorem
119
3.2.4
Specific heat of a diatomic ideal gas
121
3.3
Quantum oscillators and rotators
122
3.3.1
Qualitative discussion
122
3.3.2
Partition function of a diatomic molecule
125
3.4
From ideal gases to liquids
127
3.4.1
Pair correlation function
129
3.4.2
Measurement of the pair correlation function
132
3.4.3
Pressure and energy
134
3.5
Chemical potential
136
3.5.1
Basic formulae
136
3.5.2
Coexistence of phases
137
3.5.3
Equilibrium condition at constant pressure
138
3.5.4
Equilibrium and stability conditions at constant
μ
140
3.5.5
Chemical reactions
142
3.6
Grand canonical ensemble
146
3.6.1
Grand partition function
146
3.6.2
Mono-atomic ideal
sas
149
viii Contents
3.6.3
Thermodynamics and
fluctuations
150
3.7
Exercises
152
3.7.1
Density of states
152
3.7.2
Equation of state for the Einstein model of a solid
152
3.7.3
Specific heat of a ferromagnetic crystal
153
3.7.4
Nuclear specific heat of a metal
153
3.7.5
Solid and liquid vapour pressures
154
3.7.6
Electron trapping in a solid
155
3.8
Problems
156
3.8.1
One-dimensional Ising model
156
3.8.2
Negative temperatures
158
3.8.3
Diatomic molecules
160
3.8.4
Models of a boundary surface
161
3.8.5
Debye-Hiickel approximation
165
3.8.6
Thin metallic film
166
3.8.7
Beyond the ideal gas: first term of virial expansion
168
3.8.8
Theory of nucleation
171
3.9
Further reading
173
4
Critical phenomena
175
4.1
Ising model revisited
177
4.1.1
Some exact results for the Ising model
177
4.1.2
Correlation functions
184
4.1.3
Broken symmetry
188
4.1.4
Critical exponents
192
4.2
Mean field theory
194
4.2.1
A convexity inequality
194
4.2.2
Fundamental equation of mean field theory
195
4.2.3
Broken symmetry and critical exponents
198
4.3
Landau s theory
203
4.3.1
Landau functional
203
4.3.2
Broken continuous symmetry
207
4.3.3
Ginzburg-Landau Hamiltonian
210
4.3.4
Beyond Landau s theory
212
4.3.5
Ginzburg criterion
214
4.4
Renormalization group: general theory
217
4.4.1
Spin blocks
217
4.4.2
Critical exponents and scaling transformations
223
4.4.3
Critical manifold and fixed points
227
4.4.4
Limit distributions and correlation functions
233
Contents ix
4.4.5
Magnetization and free energy
236
4.5
Renormalization group: examples
239
4.5.1
Gaussian fixed point
239
4.5.2
Non-Gaussian fixed point
242
4.5.3
Critical exponents to order
ε
248
4.5.4
Scaling operators and anomalous dimensions
251
4.6
Exercises
253
4.6.1
High temperature expansion and Kramers-Wannier duality
253
4.6.2
Energy-energy correlations in the Ising model
255
4.6.3
Mean field critical exponents for
Τ
<
Tc
255
4.6.4
Accuracy of the variational method
255
4.6.5
Shape and energy of an Ising wall
256
4.6.6
The Ginzburg-Landau theory of superconductivity
257
4.6.7
Mean field correlation function in r-space
259
4.6.8
Critical exponents for
η
» 1 259
4.6.9
Renormalization of the Gaussian model
261
4.6.10
Scaling fields at the Gaussian fixed point
262
4.6.11
Critical exponents to order
ε
for
η φ Ι
262
4.6.12
Irrelevant exponents
263
4.6.13
Energy-energy correlations
263
4.6.14
Derivation of the Ginzburg-Landau Hamiltonian from
the Ising model
264
4.7
Further reading
265
5
Quantum statistics
267
5.1
Bose-Einstein and Fermi-Dirac distributions
268
5.1.1
Grand partition function
269
5.1.2
Classical limit: Maxwell-Boltzmann statistics
271
5.1.3
Chemical potential and relativity
272
5.2
Ideal Fermi gas
273
5.2.1
Ideal Fermi gas at zero temperature
273
5.2.2
Ideal Fermi gas at low temperature
276
5.2.3
Corrections to the ideal Fermi gas
281
5.3
Black body radiation
284
5.3.1
Electromagnetic radiation in thermal equilibrium
284
5.3.2
Black body radiation
287
5.4
Debye model
289
5.4.1
Simple model of vibrations in solids
289
5.4.2
Debye approximation
294
5.4.3
Calculation of thermodynamic functions
296
Contents
5.5 Ideal
Bose
gas
with a fixed number of particles
299
5.5.1
Bose-Einstein condensation
299
5.5.2
Thermodynamics of the condensed phase
304
5.5.3
Applications: atomic condensates and helium-4
308
5.6
Exercises
312
5.6.1
The Maxwell-Boltzmann partition function
312
5.6.2
Equilibrium radius of a neutron star
312
5.6.3
Two-dimensional Fermi gas
312
5.6.4
Non-degenerate Fermi gas
313
5.6.5
Two-dimensional
Bose
gas
314
5.6.6
Phonons and
magnons
314
5.6.7
Photon-electron-positron equilibrium in a star
315
5.7
Problems
316
5.7.1 Pauli
paramagnetism
316
5.7.2
Landau diamagnetism
318
5.7.3
White dwarf stars
319
5.7.4
Quark-gluon plasma
321
5.7.5
Bose-Einstein condensates of atomic gases
323
5.7.6
Solid-liquid equilibrium for helium-3
325
5.7.7
Superfluidity for hardcore bosons
329
5.8
Further reading
334
Irreversible processes: macroscopic theory
335
6.1
Flux, affinities, transport coefficients
336
6.1.1
Conservation laws
336
6.1.2
Local equation of state
339
6.1.3
Affinities and transport coefficients
341
6.1.4
Examples
342
6.1.5
Dissipation and entropy production
345
6.2
Examples
349
6.2.1
Coupling between thermal and particle diffusion
349
6.2.2
Electrodynamics
350
6.3
Hydrodynamics of simple fluids
353
6.3.1
Conservation laws in a simple fluid
353
6.3.2
Derivation of current densities
358
6.3.3
Transport coefficients and the Navier-Stokes equation
360
6.4
Exercises
364
6.4.1
Continuity equation for the density of particles
364
6.4.2
Diffusion equation and random walk
364
6.4.3
Relation between viscosity and diffusion
364
6.4.4
Derivation of the energy current
365
6.4.5
Lord Kelvin s model of Earth cooling
365
Contents xi
6.5 Problems 366
6.5.1
Entropy current in hydrodynamics
366
6.5.2
Hydrodynamics of the perfect fluid
368
6.5.3
Thermoelectric effects
369
6.5.4
Lsomerization reactions
371
6.6
Further reading
373
Numerical simulations
375
7.1
Markov chains, convergence and detailed balance
375
7.2
Classical Monte Carlo
379
7.2.1
Implementation
379
7.2.2
Measurements
380
7.2.3
Autocorrelation, thermalization and error bars
382
7.3
Critical slowing down and cluster algorithms
384
7.4
Quantum Monte Carlo: bosons
388
7.4.1
Formulation and implementation
389
7.4.2
Measurements
396
7.4.3
Quantum spin-
1/2
models
398
7.5
Quantum Monte Carlo:
fermions
400
7.6
Finite size scaling
404
7.7
Random number generators
408
7.8
Exercises
410
7.8.1
Determination of the critical exponent
υ
410
7.8.2
Finite size scaling in infinite geometries
410
7.8.3
Bosons on a single site
411
7.9
Problems
411
7.9.1
Two-dimensional Ising model: Metropolis
411
7.9.2
Two-dimensional Ising model: Glauber
413
7.9.3
Two-dimensional clock model
414
7.9.4
Two-dimensional XY model: Kosterlitz-Thouless
transition
419
7.9.5
Two-dimensional XY model: superfluidity and
critical velocity
423
7.9.6
Simple quantum model: single spin in transverse field
431
7.9.7
One-dimensional Ising model in transverse field:
quantum phase transition
433
7.9.8
Quantum anharmonic oscillator: path integrals
435
7.10
Further reading
441
Irreversible processes: kinetic theory
443
8.1
Generalities, elementary theory of transport coefficients
443
8.1.1
Distribution function
443
8.1.2
Cross section, collision time, mean free path
444
xii Contents
8.1.3 Transport
coefficients
in
the mean free path approximation
449
8.2
Boltzmann-Lorentz model
453
8.2.1
Spatio-temporal evolution of the distribution function
453
8.2.2
Basic equations of the Boltzmann-Lorentz model
455
8.2.3
Conservation laws and continuity equations
457
8.2.4
Linearization: Chapman-Enskog approximation
458
8.2.5
Currents and transport coefficients
462
8.3
Boltzmann equation
464
8.3.1
Collision term
464
8.3.2
Conservation laws
469
8.3.3
H-theorem
472
8.4
Transport coefficients from the Boltzmann equation
476
8.4.1
Linearization of the Boltzmann equation
476
8.4.2
Variational method
478
8.4.3
Calculation of the viscosity
481
8.5
Exercises
484
8.5.1
Time distribution of collisions
484
8.5.2
Symmetries of an integral
485
8.5.3
Positivity
conditions
485
8.5.4
Calculation of the collision time
485
8.5.5
Derivation of the energy current
486
8.5.6
Equilibrium distribution from the Boltzmann equation
486
8.6
Problems
487
8.6.1
Thermal diffusion in the Boltzmann-Lorentz model
487
8.6.2
Electron gas in the Boltzmann-Lorentz model
488
8.6.3
Photon diffusion and energy transport in the Sun
492
8.6.4
Momentum transfer in a shear flow
495
8.6.5
Electrical conductivity in a magnetic field and quantum
Hall effect
497
8.6.6
Specific heat and two-fluid model for helium II
502
8.6.7
Landau theory of Fermi liquids
505
8.6.8
Calculation of the coefficient of thermal conductivity
510
8.7
Further reading
512
9
Topics in non-equilibrium statistical mechanics
513
9.1
Linear response: classical theory
514
9.1.1
Dynamical susceptibility
514
9.1.2
Nyquist theorem
518
9.1.3
Analyticky
properties
520
9.1.4
Spin diffusion
522
9.2
Linear response: quantum theory
526
Contents xiii
9.2.1 Quantum
fluctuation
response
theorem
526
9.2.2
Quantum
Kubo
function
528
9.2.3
Fluctuation-dissipation theorem
530
9.2.4
Symmetry properties and dissipation
531
9.2.5
Sum rules
533
9.3
Projection method and memory effects
535
9.3.1
Phenomenological introduction to memory effects
536
9.3.2
Projectors
538
9.3.3
Langevin-Mori equation
540
9.3.4
Brownian motion: qualitative description
543
9.3.5
Brownian motion: the m/M
-> 0
limit
545
9.4
Langevin equation
547
9.4.1
Definitions and first properties
547
9.4.2
Ornstein-Uhlenbeck process
549
9.5
Fokker-Planck equation
552
9.5.1
Derivation of Fokker-Planck from Langevin equation
552
9.5.2
Equilibrium and convergence to equilibrium
554
9.5.3
Space-dependent diffusion coefficient
556
9.6
Numerical integration
558
9.7
Exercises
562
9.7.1
Linear response: forced harmonic oscillator
562
9.7.2
Force on a Brownian particle
563
9.7.3
Green-Kubo formula
564
9.7.4
Mori s scalar product
564
9.7.5
Symmetry properties of
χ- ,
565
9.7.6
Dissipation
566
9.7.7
Proof of the/-sum rale in quantum mechanics
566
9.7.8
Diffusion of a Brownian particle
567
9.7.9
Strong friction limit: harmonic oscillator
568
9.7.10
Green s function method
569
9.7.11
Moments of the Fokker-Planck equation
569
9.7.12
Backward velocity
570
9.7.13
Numerical integration of the Langevin equation
570
9.7.14
Metastable states and escape times
571
9.8
Problems
572
9.8.1
Inelastic light scattering from a suspension of particles
572
9.8.2
Light scattering by a simple fluid
576
9.8.3
Exactly solvable model of a Brownian particle
580
9.8.4
Ito
versus Stratonovitch dilemma
582
9.8.5
Kramers equation
584
xiv Contents
9.9
Further reading
585
Appendix
587
A.
1
Legendre transform
587
A.
1.1
Legendre transform with one variable
587
A.
1.2
Multivariate Legendre transform
588
A.2
Lagrange
multipliers
589
A.3 Traces, tensor products
591
A.3.1 Traces
591
A.3.
2
Tensor products
592
A.4 Symmetries
593
A.4.1 Rotations
593
A.4.2 Tensors
596
A.5 Useful integrals
598
A.5.
1
Gaussian integrals
598
A.5.
2
Integrals of quantum statistics
600
A.6 Functional derivatives
601
A.7 Units and physical constants
604
References
605
Index
611
|
| adam_txt |
Contents
Preface
page
xv
Thermostatics
1
1.1
Thermodynamic equilibrium
1
1.1.1
Microscopic and macroscopic descriptions
1
1.1.2
Walls
3
1.1.3
Work, heat, internal energy
5
1.1.4
Definition of thermal equilibrium
8
1.2
Postulate of maximum entropy
9
1.2.1
Internal constraints
9
1.2.2
Principle of maximum entropy
10
1.2.3
Intensive variables: temperature, pressure, chemical potential
12
1.2.4
Quasi-static and reversible processes
17
1.2.5
Maximum work and heat engines
20
1.3
Thermodynamic potentials
22
1.3.1
Thermodynamic potentials and Massieu functions
22
1.3.2
Specific heats
24
1.3.3
Gibbs-Duhem relation
26
1.4
Stability conditions
27
1.4.1
Concavity of entropy and convexity of energy
27
1.4.2
Stability conditions and their consequences
28
1.5
Third law of thermodynamics
31
1.5.1
Statement of the third law
31
1.5.2
Application to metastable states
32
1.5.3
Low temperature behaviour of specific heats
33
1.6
Exercises
35
1.6.1
Massieu functions
35
1.6.2
Internal variable in equilibrium
35
1.6.3
Relations between thermodynamic coefficients
36
vi
Contents
1.6.4
Contact
between two systems
37
1.6.5
Stability conditions
37
1.6.6
Equation of state for a fluid
37
1.7
Problems
38
1.7.1
Reversible and irreversible free expansions of an ideal
gas
38
1.7.2
van
der Waals
equation of state
39
1.7.3
Equation of state for a solid
40
1.7.4
Specific heats of a rod
41
1.7.5
Surface tension of a soap film
42
1.7.6
Joule-Thomson process
43
1.7.7
Adiabatic demagnetization of a paramagnetic salt
43
1.8
Further reading
45
2
Statistical entropy and Boltzmann distribution
47
2.1
Quantum description
47
2.1.1
Time evolution in quantum mechanics
47
2.1.2
The density operators and their time evolution
49
2.1.3
Quantum phase space
51
2.1.4
(P, V, E) relation for a mono-atomic ideal gas
53
2.2
Classical description
55
2.2.1
Liouville's theorem
55
2.2.2
Density in phase space
56
2.3
Statistical entropy
59
2.3.1
Entropy of a probability distribution
59
2.3.2
Statistical entropy of a mixed quantum state
60
2.3.3
Time evolution of the statistical entropy
63
2.4
Boltzmann distribution
64
2.4.1
Postulate of maximum of statistical entropy
64
2.4.2
Equilibrium distribution
65
2.4.3
Legendre transformation
67
2.4.4
Canonical and grand canonical ensembles
68
2.5
Thermodynamics revisited
70
2.5.1
Heat and work: first law
70
2.5.2
Entropy and temperature: second law
72
2.5.3
Entropy of mixing
74
2.5.4
Pressure and chemical potential
77
2.6
Irreversibility and the growth of entropy
79
2.6.1
Microscopic reversibility and macroscopic irreversibility
79
2.6.2
Physical basis of irreversibility
81
2.6.3
Loss of information and the growth of entropy
83
Contents
vii
2.7
Exercises
86
2.7.1
Density operator for spin-
1 /2 86
2.7.2
Density of states and the dimension of space
88
2.7.3
Liouville theorem and continuity equation
88
2.7.4
Loaded dice and statistical entropy
89
2.7.5
Entropy of a composite system
89
2.7.6
Heat exchanges between system and reservoir
89
2.7.7
Galilean transformation
90
2.7.8
Fluctuation-response theorem
90
2.7.9
Phase space volume for
N
free particles
92
2.7.10
Entropy of mixing and osmotic pressure
92
2.8
Further reading
93
3
Canonical and grand canonical ensembles: applications
95
3.1
Simple examples in the canonical ensemble
95
3.1.1
Mean values and fluctuations
95
3.1.2
Partition function and thermodynamics of an ideal gas
98
3.1.3
Paramagnetism
101
3.1.4
Ferromagnetism and the Ising model
105
3.1.5
Thermodynamic limit
112
3.2
Classical statistical mechanics
115
3.2.1
Classical limit
115
3.2.2
Maxwell distribution
116
3.2.3
Equipartition theorem
119
3.2.4
Specific heat of a diatomic ideal gas
121
3.3
Quantum oscillators and rotators
122
3.3.1
Qualitative discussion
122
3.3.2
Partition function of a diatomic molecule
125
3.4
From ideal gases to liquids
127
3.4.1
Pair correlation function
129
3.4.2
Measurement of the pair correlation function
132
3.4.3
Pressure and energy
134
3.5
Chemical potential
136
3.5.1
Basic formulae
136
3.5.2
Coexistence of phases
137
3.5.3
Equilibrium condition at constant pressure
138
3.5.4
Equilibrium and stability conditions at constant
μ
140
3.5.5
Chemical reactions
142
3.6
Grand canonical ensemble
146
3.6.1
Grand partition function
146
3.6.2
Mono-atomic ideal
sas
149
viii Contents
3.6.3
Thermodynamics and
fluctuations
150
3.7
Exercises
152
3.7.1
Density of states
152
3.7.2
Equation of state for the Einstein model of a solid
152
3.7.3
Specific heat of a ferromagnetic crystal
153
3.7.4
Nuclear specific heat of a metal
153
3.7.5
Solid and liquid vapour pressures
154
3.7.6
Electron trapping in a solid
155
3.8
Problems
156
3.8.1
One-dimensional Ising model
156
3.8.2
Negative temperatures
158
3.8.3
Diatomic molecules
160
3.8.4
Models of a boundary surface
161
3.8.5
Debye-Hiickel approximation
165
3.8.6
Thin metallic film
166
3.8.7
Beyond the ideal gas: first term of virial expansion
168
3.8.8
Theory of nucleation
171
3.9
Further reading
173
4
Critical phenomena
175
4.1
Ising model revisited
177
4.1.1
Some exact results for the Ising model
177
4.1.2
Correlation functions
184
4.1.3
Broken symmetry
188
4.1.4
Critical exponents
192
4.2
Mean field theory
194
4.2.1
A convexity inequality
194
4.2.2
Fundamental equation of mean field theory
195
4.2.3
Broken symmetry and critical exponents
198
4.3
Landau's theory
203
4.3.1
Landau functional
203
4.3.2
Broken continuous symmetry
207
4.3.3
Ginzburg-Landau Hamiltonian
210
4.3.4
Beyond Landau's theory
212
4.3.5
Ginzburg criterion
214
4.4
Renormalization group: general theory
217
4.4.1
Spin blocks
217
4.4.2
Critical exponents and scaling transformations
223
4.4.3
Critical manifold and fixed points
227
4.4.4
Limit distributions and correlation functions
233
Contents ix
4.4.5
Magnetization and free energy
236
4.5
Renormalization group: examples
239
4.5.1
Gaussian fixed point
239
4.5.2
Non-Gaussian fixed point
242
4.5.3
Critical exponents to order
ε
248
4.5.4
Scaling operators and anomalous dimensions
251
4.6
Exercises
253
4.6.1
High temperature expansion and Kramers-Wannier duality
253
4.6.2
Energy-energy correlations in the Ising model
255
4.6.3
Mean field critical exponents for
Τ
<
Tc
255
4.6.4
Accuracy of the variational method
255
4.6.5
Shape and energy of an Ising wall
256
4.6.6
The Ginzburg-Landau theory of superconductivity
257
4.6.7
Mean field correlation function in r-space
259
4.6.8
Critical exponents for
η
» 1 259
4.6.9
Renormalization of the Gaussian model
261
4.6.10
Scaling fields at the Gaussian fixed point
262
4.6.11
Critical exponents to order
ε
for
η φ Ι
262
4.6.12
Irrelevant exponents
263
4.6.13
Energy-energy correlations
263
4.6.14
'Derivation' of the Ginzburg-Landau Hamiltonian from
the Ising model
264
4.7
Further reading
265
5
Quantum statistics
267
5.1
Bose-Einstein and Fermi-Dirac distributions
268
5.1.1
Grand partition function
269
5.1.2
Classical limit: Maxwell-Boltzmann statistics
271
5.1.3
Chemical potential and relativity
272
5.2
Ideal Fermi gas
273
5.2.1
Ideal Fermi gas at zero temperature
273
5.2.2
Ideal Fermi gas at low temperature
276
5.2.3
Corrections to the ideal Fermi gas
281
5.3
Black body radiation
284
5.3.1
Electromagnetic radiation in thermal equilibrium
284
5.3.2
Black body radiation
287
5.4
Debye model
289
5.4.1
Simple model of vibrations in solids
289
5.4.2
Debye approximation
294
5.4.3
Calculation of thermodynamic functions
296
Contents
5.5 Ideal
Bose
gas
with a fixed number of particles
299
5.5.1
Bose-Einstein condensation
299
5.5.2
Thermodynamics of the condensed phase
304
5.5.3
Applications: atomic condensates and helium-4
308
5.6
Exercises
312
5.6.1
The Maxwell-Boltzmann partition function
312
5.6.2
Equilibrium radius of a neutron star
312
5.6.3
Two-dimensional Fermi gas
312
5.6.4
Non-degenerate Fermi gas
313
5.6.5
Two-dimensional
Bose
gas
314
5.6.6
Phonons and
magnons
314
5.6.7
Photon-electron-positron equilibrium in a star
315
5.7
Problems
316
5.7.1 Pauli
paramagnetism
316
5.7.2
Landau diamagnetism
318
5.7.3
White dwarf stars
319
5.7.4
Quark-gluon plasma
321
5.7.5
Bose-Einstein condensates of atomic gases
323
5.7.6
Solid-liquid equilibrium for helium-3
325
5.7.7
Superfluidity for hardcore bosons
329
5.8
Further reading
334
Irreversible processes: macroscopic theory
335
6.1
Flux, affinities, transport coefficients
336
6.1.1
Conservation laws
336
6.1.2
Local equation of state
339
6.1.3
Affinities and transport coefficients
341
6.1.4
Examples
342
6.1.5
Dissipation and entropy production
345
6.2
Examples
349
6.2.1
Coupling between thermal and particle diffusion
349
6.2.2
Electrodynamics
350
6.3
Hydrodynamics of simple fluids
353
6.3.1
Conservation laws in a simple fluid
353
6.3.2
Derivation of current densities
358
6.3.3
Transport coefficients and the Navier-Stokes equation
360
6.4
Exercises
364
6.4.1
Continuity equation for the density of particles
364
6.4.2
Diffusion equation and random walk
364
6.4.3
Relation between viscosity and diffusion
364
6.4.4
Derivation of the energy current
365
6.4.5
Lord Kelvin's model of Earth cooling
365
Contents xi
6.5 Problems 366
6.5.1
Entropy current in hydrodynamics
366
6.5.2
Hydrodynamics of the perfect fluid
368
6.5.3
Thermoelectric effects
369
6.5.4
Lsomerization reactions
371
6.6
Further reading
373
Numerical simulations
375
7.1
Markov chains, convergence and detailed balance
375
7.2
Classical Monte Carlo
379
7.2.1
Implementation
379
7.2.2
Measurements
380
7.2.3
Autocorrelation, thermalization and error bars
382
7.3
Critical slowing down and cluster algorithms
384
7.4
Quantum Monte Carlo: bosons
388
7.4.1
Formulation and implementation
389
7.4.2
Measurements
396
7.4.3
Quantum spin-
1/2
models
398
7.5
Quantum Monte Carlo:
fermions
400
7.6
Finite size scaling
404
7.7
Random number generators
408
7.8
Exercises
410
7.8.1
Determination of the critical exponent
υ
410
7.8.2
Finite size scaling in infinite geometries
410
7.8.3
Bosons on a single site
411
7.9
Problems
411
7.9.1
Two-dimensional Ising model: Metropolis
411
7.9.2
Two-dimensional Ising model: Glauber
413
7.9.3
Two-dimensional clock model
414
7.9.4
Two-dimensional XY model: Kosterlitz-Thouless
transition
419
7.9.5
Two-dimensional XY model: superfluidity and
critical velocity
423
7.9.6
Simple quantum model: single spin in transverse field
431
7.9.7
One-dimensional Ising model in transverse field:
quantum phase transition
433
7.9.8
Quantum anharmonic oscillator: path integrals
435
7.10
Further reading
441
Irreversible processes: kinetic theory
443
8.1
Generalities, elementary theory of transport coefficients
443
8.1.1
Distribution function
443
8.1.2
Cross section, collision time, mean free path
444
xii Contents
8.1.3 Transport
coefficients
in
the mean free path approximation
449
8.2
Boltzmann-Lorentz model
453
8.2.1
Spatio-temporal evolution of the distribution function
453
8.2.2
Basic equations of the Boltzmann-Lorentz model
455
8.2.3
Conservation laws and continuity equations
457
8.2.4
Linearization: Chapman-Enskog approximation
458
8.2.5
Currents and transport coefficients
462
8.3
Boltzmann equation
464
8.3.1
Collision term
464
8.3.2
Conservation laws
469
8.3.3
H-theorem
472
8.4
Transport coefficients from the Boltzmann equation
476
8.4.1
Linearization of the Boltzmann equation
476
8.4.2
Variational method
478
8.4.3
Calculation of the viscosity
481
8.5
Exercises
484
8.5.1
Time distribution of collisions
484
8.5.2
Symmetries of an integral
485
8.5.3
Positivity
conditions
485
8.5.4
Calculation of the collision time
485
8.5.5
Derivation of the energy current
486
8.5.6
Equilibrium distribution from the Boltzmann equation
486
8.6
Problems
487
8.6.1
Thermal diffusion in the Boltzmann-Lorentz model
487
8.6.2
Electron gas in the Boltzmann-Lorentz model
488
8.6.3
Photon diffusion and energy transport in the Sun
492
8.6.4
Momentum transfer in a shear flow
495
8.6.5
Electrical conductivity in a magnetic field and quantum
Hall effect
497
8.6.6
Specific heat and two-fluid model for helium II
502
8.6.7
Landau theory of Fermi liquids
505
8.6.8
Calculation of the coefficient of thermal conductivity
510
8.7
Further reading
512
9
Topics in non-equilibrium statistical mechanics
513
9.1
Linear response: classical theory
514
9.1.1
Dynamical susceptibility
514
9.1.2
Nyquist theorem
518
9.1.3
Analyticky
properties
520
9.1.4
Spin diffusion
522
9.2
Linear response: quantum theory
526
Contents xiii
9.2.1 Quantum
fluctuation
response
theorem
526
9.2.2
Quantum
Kubo
function
528
9.2.3
Fluctuation-dissipation theorem
530
9.2.4
Symmetry properties and dissipation
531
9.2.5
Sum rules
533
9.3
Projection method and memory effects
535
9.3.1
Phenomenological introduction to memory effects
536
9.3.2
Projectors
538
9.3.3
Langevin-Mori equation
540
9.3.4
Brownian motion: qualitative description
543
9.3.5
Brownian motion: the m/M
-> 0
limit
545
9.4
Langevin equation
547
9.4.1
Definitions and first properties
547
9.4.2
Ornstein-Uhlenbeck process
549
9.5
Fokker-Planck equation
552
9.5.1
Derivation of Fokker-Planck from Langevin equation
552
9.5.2
Equilibrium and convergence to equilibrium
554
9.5.3
Space-dependent diffusion coefficient
556
9.6
Numerical integration
558
9.7
Exercises
562
9.7.1
Linear response: forced harmonic oscillator
562
9.7.2
Force on a Brownian particle
563
9.7.3
Green-Kubo formula
564
9.7.4
Mori's scalar product
564
9.7.5
Symmetry properties of
χ-',
565
9.7.6
Dissipation
566
9.7.7
Proof of the/-sum rale in quantum mechanics
566
9.7.8
Diffusion of a Brownian particle
567
9.7.9
Strong friction limit: harmonic oscillator
568
9.7.10
Green's function method
569
9.7.11
Moments of the Fokker-Planck equation
569
9.7.12
Backward velocity
570
9.7.13
Numerical integration of the Langevin equation
570
9.7.14
Metastable states and escape times
571
9.8
Problems
572
9.8.1
Inelastic light scattering from a suspension of particles
572
9.8.2
Light scattering by a simple fluid
576
9.8.3
Exactly solvable model of a Brownian particle
580
9.8.4
Ito
versus Stratonovitch dilemma
582
9.8.5
Kramers equation
584
xiv Contents
9.9
Further reading
585
Appendix
587
A.
1
Legendre transform
587
A.
1.1
Legendre transform with one variable
587
A.
1.2
Multivariate Legendre transform
588
A.2
Lagrange
multipliers
589
A.3 Traces, tensor products
591
A.3.1 Traces
591
A.3.
2
Tensor products
592
A.4 Symmetries
593
A.4.1 Rotations
593
A.4.2 Tensors
596
A.5 Useful integrals
598
A.5.
1
Gaussian integrals
598
A.5.
2
Integrals of quantum statistics
600
A.6 Functional derivatives
601
A.7 Units and physical constants
604
References
605
Index
611 |
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| author | Le Bellac, Michel 1939- Mortessagne, Fabrice 1966- Batrouni, Ghassan George 1956- |
| author_GND | (DE-588)131512188 (DE-588)129538868 (DE-588)129538841 |
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| id | DE-604.BV023275661 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:37:48Z |
| indexdate | 2024-07-09T21:14:44Z |
| institution | BVB |
| isbn | 0521821436 9780521821438 9780521528955 |
| language | English |
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| physical | XVI, 616 S. Ill., graph. Darst. |
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| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Le Bellac, Michel 1939- Verfasser (DE-588)131512188 aut Equilibrium and non-equilibrium statistical thermodynamics Michel Le Bellac, Fabrice Mortessagne and G. George Batrouni Reprinted Cambridge [u.a.] Cambridge Univ. Press 2004 XVI, 616 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Statistische Thermodynamik (DE-588)4126251-7 gnd rswk-swf Statistische Thermodynamik (DE-588)4126251-7 s DE-604 Mortessagne, Fabrice 1966- Verfasser (DE-588)129538868 aut Batrouni, Ghassan George 1956- Verfasser (DE-588)129538841 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016460562&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Le Bellac, Michel 1939- Mortessagne, Fabrice 1966- Batrouni, Ghassan George 1956- Equilibrium and non-equilibrium statistical thermodynamics Statistische Thermodynamik (DE-588)4126251-7 gnd |
| subject_GND | (DE-588)4126251-7 |
| title | Equilibrium and non-equilibrium statistical thermodynamics |
| title_auth | Equilibrium and non-equilibrium statistical thermodynamics |
| title_exact_search | Equilibrium and non-equilibrium statistical thermodynamics |
| title_exact_search_txtP | Equilibrium and non-equilibrium statistical thermodynamics |
| title_full | Equilibrium and non-equilibrium statistical thermodynamics Michel Le Bellac, Fabrice Mortessagne and G. George Batrouni |
| title_fullStr | Equilibrium and non-equilibrium statistical thermodynamics Michel Le Bellac, Fabrice Mortessagne and G. George Batrouni |
| title_full_unstemmed | Equilibrium and non-equilibrium statistical thermodynamics Michel Le Bellac, Fabrice Mortessagne and G. George Batrouni |
| title_short | Equilibrium and non-equilibrium statistical thermodynamics |
| title_sort | equilibrium and non equilibrium statistical thermodynamics |
| topic | Statistische Thermodynamik (DE-588)4126251-7 gnd |
| topic_facet | Statistische Thermodynamik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016460562&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT lebellacmichel equilibriumandnonequilibriumstatisticalthermodynamics AT mortessagnefabrice equilibriumandnonequilibriumstatisticalthermodynamics AT batrounighassangeorge equilibriumandnonequilibriumstatisticalthermodynamics |