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E-raamat: Interacting Systems far from Equilibrium: Quantum Kinetic Theory

(Professor of Mathematics and Many-Body Theory, Münster University of Applied Sciences, Germany)
  • Formaat: 576 pages
  • Ilmumisaeg: 08-Dec-2017
  • Kirjastus: Oxford University Press
  • Keel: eng
  • ISBN-13: 9780192517784
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Interacting Systems far from Equilibrium: Quantum Kinetic TheorySuurem pilt
  • Formaat: 576 pages
  • Ilmumisaeg: 08-Dec-2017
  • Kirjastus: Oxford University Press
  • Keel: eng
  • ISBN-13: 9780192517784
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This book presents an up-to-date formalism of non-equilibrium Green's functions covering different applications ranging from solid state physics, plasma physics, cold atoms in optical lattices up to relativistic transport and heavy ion collisions. Within the Green's function formalism, the basic sets of equations for these diverse systems are similar, and approximations developed in one field can be adapted to another field. The central object is the self-energy which includes all non-trivial aspects of the system dynamics. The focus is therefore on microscopic processes starting from elementary principles for classical gases and the complementary picture of a single quantum particle in a random potential. This provides an intuitive picture of the interaction of a particle with the medium formed by other particles, on which the Green's function is built on.

Arvustused

Prof. Morawetz has generated an advanced monograph presenting a coherent theory of quantum manybody kinetics. I expect the intended audience will find his presentation highly relevant. * Andrew Resnick, Contemporary Physics *

Part I Classical Kinetic Concepts
1 Historical Background
3(8)
1.1 Introduction
3(2)
1.2 Virial corrections to the kinetic equation
5(3)
1.2.1 Classical gas of hard spheres
6(1)
1.2.2 Hard-sphere corrections to the quantum Boltzmann equation
7(1)
1.2.3 Quantum nonlocal corrections in gases
7(1)
1.3 Quantum nonlocal corrections in dense Fermi systems
8(1)
1.4 Quantum nonlocal corrections to kinetic theory
9(2)
2 Elementary Principles
11(33)
2.1 Motion versus forces
11(5)
2.1.1 Principle of motion
11(1)
2.1.2 Principle of finite-range forces
12(4)
2.1.3 Summary
16(1)
2.2 Random versus deterministic
16(7)
2.2.1 Chaos
17(1)
2.2.2 Velocity distribution
18(5)
2.3 Information versus chaos
23(3)
2.3.1 Entropy as a measure of negative information or disorder
23(2)
2.3.2 Maximum entropy and equilibrium thermodynamics
25(1)
2.4 Collisions versus drift
26(10)
2.4.1 Collisions
27(2)
2.4.2 Drift
29(2)
2.4.3 Equivalence of low-angle collisions and momentum drift
31(5)
2.4.4 Summary
36(1)
2.5 Explicit versus hidden forces
36(8)
2.5.1 Mean field
38(1)
2.5.2 Virial of forces
39(2)
2.5.3 Virial theorem
41(2)
2.5.4 Summary
43(1)
3 Classical Kinetic Theory
44(39)
3.1 Nonlocal and non-instant binary collisions
45(8)
3.1.1 Hard spheres
45(1)
3.1.2 Sticky point particles
45(1)
3.1.3 Realistic particles
46(5)
3.1.4 Kinetic equation without a mean field
51(2)
3.1.5 Summary
53(1)
3.2 Effect of the mean field on collisions
53(4)
3.2.1 Momentum gains
54(1)
3.2.2 Energy gain
55(2)
3.2.3 Summary
57(1)
3.3 Internal mechanism of energy conversion
57(1)
3.4 Equation of continuity
58(3)
3.4.1 Correlated density
58(3)
3.5 Pressure
61(5)
3.5.1 Partial pressure of effective molecules
63(1)
3.5.2 Inaccessible volume
63(1)
3.5.3 Mean-field contribution to pressure
64(2)
3.6 Latent heat
66(2)
3.7 Entropy production
68(1)
3.8 Nonequilibrium hydrodynamic equations
69(2)
3.9 Two concepts of quasiparticles
71(7)
3.9.1 Virial offerees
72(1)
3.9.2 Landau's quasiparticle concept
72(1)
3.9.3 Quasiparticle pressure
73(1)
3.9.4 Quasiparticle energy
74(4)
3.9.5 Two forms of quasiparticles
78(1)
3.10 Summary
78(5)
Part II Inductive Ways to Quantum Transport
4 Scattering on a Single Impurity
83(13)
4.1 Bound state
83(4)
4.1.1 Secular equation
85(1)
4.1.2 Resolvent of the host crystal
86(1)
4.2 Scattering and Lippmann--Schwinger equation
87(3)
4.2.1 Born approximation of the scattered wave
88(2)
4.2.2 T-matrix
90(1)
4.3 Collision delay
90(2)
4.4 Optical theorem
92(3)
4.4.1 Scattering on two impurities
94(1)
4.5 Dissipativeness
95(1)
5 Multiple Impurity Scattering
96(26)
5.1 Divergence of multiple scattering expansion 9
7(91)
5.2 Averaged wave function
98(5)
5.2.1 Mean field
101(1)
5.2.2 Dyson equation
102(1)
5.2.3 Born approximation of selfenergy
102(1)
5.3 Selfconsistent Born approximation
103(2)
5.4 Averaged T-matrix approximation
105(2)
5.4.1 Double counts in the averaged T-matrix approximation
106(1)
5.5 Effective medium
107(4)
5.5.1 Averaged T-matrix approximation in the virtual crystal
107(3)
5.5.2 Averaged T-matrix approximation in the selfconsistent effective crystal
110(1)
5.6 Coherent potential approximation
111(3)
5.6.1 Lorentz-Lorenz local-field correction
112(1)
5.6.2 Coherent potential approximation as the local-field correction
113(1)
5.7 Energy spectrum
114(8)
5.7.1 Densities of states
115(1)
5.7.2 Bottom of allowed energies in the main band
116(1)
5.7.3 Effective mass
117(1)
5.7.4 Effect of the collision delay on the density of states
118(1)
5.7.5 Test of approximations in the atomic limit
119(3)
6 Selfenergy
122(1)
6.1 Averaged amplitude of wave function
122(1)
6.2 Straightforward perturbative expansion
122(2)
6.2.1 Transport vertex in the non-selfconsistent Born approximation
123(1)
6.3 Generalised Kadanoff and Baym formalism
124(4)
6.3.1 GKB equation
126(1)
6.3.2 Transport vertex in the Born approximation
126(1)
6.3.3 Transport vertex in the selfconsistent averaged T-matrix approximation
127(1)
6.3.4 Coherent potential approximation of the transport vertex
128(1)
6.4 Optical theorem
128(1)
6.5 Application scheme
129(1)
6.6 Elimination of surrounding interaction channels-secular equation
130(3)
6.6.1 Wave function renormalisation
131(1)
6.6.2 Effective mass
131(1)
6.6.3 Two actions of selfenergy
132(1)
6.7 Summary
133(5)
Part III Deductive Way to Quantum Transport
7 Nonequilibrium Green's Functions
137(140)
7.1 Method of equation of motion
137(5)
7.1.1 Enclosure of hierarchy
139(3)
7.2 Quantum transport equation
142(2)
7.3 Information contained in Green's functions
144(8)
7.3.1 Density and current
145(1)
7.3.2 Total energy content
145(1)
7.3.3 Conservation laws
146(1)
7.3.4 Equilibrium information
147(2)
7.3.5 Matsubara technique
149(2)
7.3.6 Equilibrium pressure
151(1)
7.4 Summary
152(1)
8 Spectral Properties
153(11)
8.1 Spectral function
153(2)
8.1.1 Causality and Kramers-Kronig relation
153(1)
8.1.2 Sum rules
154(1)
8.2 Quasiparticle and extended quasiparticle picture
155(3)
8.3 Comparison with equilibrium
158(2)
8.4 The problem of ansatz
160(4)
8.4.1 Further spectral functions
162(1)
8.4.2 Optical theorems
162(2)
9 Quantum Kinetic Equations
164(30)
9.1 Kadanoff and Baym equation in quasiclassical limit
164(8)
9.1.1 Gradient expansion
164(1)
9.1.2 Quasiclassical Kadanoff and Baym equation
165(2)
9.1.3 Collision-less Landau equation
167(1)
9.1.4 Landau equation with collisions
168(1)
9.1.5 Missing satellites
169(1)
9.1.6 Causality and gradient corrections
170(2)
9.1.7 Conclusion from off-shell contributions
172(1)
9.2 Separation of on-shell and off-shell motion
172(2)
9.3 Differential transport equation
174(3)
9.3.1 Quasiclassical limit
176(1)
9.4 Extended quasiparticle picture
177(5)
9.4.1 Precursor of kinetic equation
180(2)
9.5 Numerical examples for equilibrium
182(3)
9.6 Direct gradient expansion of non-Markovian equation
185(6)
9.6.1 Connection between the Wigner and quasiparticle distributions
188(1)
9.6.2 Landau--Silin equation
188(3)
9.7 Alternative approaches to the kinetic equation
191(3)
9.7.1 First quasiclassical approximation
91(101)
9.7.2 Thermo field dynamics approach
192(2)
10 Approximations for the Selfenergy
194(36)
10.1 Hartree--Fock
194(3)
10.2 Random phase approximation
197(9)
10.2.1 Long-range Coulomb interaction
197(1)
10.2.2 Density-density fluctuations
198(5)
10.2.3 Lenard-Balescu collision integral
203(3)
10.3 Selfenergy and effective mass in quasi two-dimensional systems
206(6)
10.3.1 Polarisation function in 2D
206(2)
10.3.2 Integrals over dielectric functions
208(1)
10.3.3 Selfenergy and effective mass
208(4)
10.4 Vertex correction to RPA polarisation
212(1)
10.5 Suucture factor and pair-correlation function
213(5)
10.5.1 Fock approximation
214(1)
10.5.2 Born approximation
215(1)
10.5.3 Collision integral in Born approximation
216(1)
10.5.4 Pair-correlation function in equilibrium
217(1)
10.6 Ladder approximation
218(8)
10.6.1 Analytic pieces
220(1)
10.6.2 Selfenergy
221(1)
10.6.3 Scattering T-matrix
222(1)
10.6.4 Missing particle-hole channels
222(4)
10.7 Bethe--Salpeter equation in quasiparticle approximation
226(2)
10.8 Local and instant quantum Boltzmann kinetic equation
228(2)
11 Variational Techniques of Many-Body Theory
230(37)
11.1 Dyson time ordering and Dirac interaction representation
230(2)
11.2 Representation of Green's functions
232(4)
11.2.1 Screening for Coulomb systems
235(1)
11.3 Hedin equation
236(3)
11.3.1 Under which conditions does RPA become exact in the high-density limit of fermions?
238(1)
11.4 Ward identities
239(1)
11.5 Asymmetric and cummulant expansion
240(12)
11.5.1 Binary collision approximation
244(1)
11.5.2 Three-particle approximations
244(1)
11.5.3 Screened-ladder approximation
245(3)
11.5.4 Maximally crossed diagrams
248(1)
11.5.5 Ladder diagrams
249(1)
11.5.6 Pair-pair correlation
250(2)
11.6 Linear response
252(4)
11.6.1 Basic relations
252(1)
11.6.2 Connection to diagrammatic expansions
252(2)
11.6.3 Schema for constructing higher-order diagrams
254(1)
11.6.4 Linearising kinetic equations
255(1)
11.7 Response in finite systems
256(3)
11.8 Renormalisation techniques
259(8)
11.8.1 Low-energy degrees of freedom
259(3)
11.8.2 Results for separable interaction
262(3)
11.8.3 Functional renormalisation
265(2)
12 Systems with Condensates and Pairing
267(48)
12.1 Condensation phenomena in correlated systems
267(4)
12.1.1 Bose-Einstein condensation in correlated systems
267(1)
12.1.2 Measurement of the quasiparticle spectrum
268(1)
12.1.3 Quasi-classical approach to superfluidity
269(1)
12.1.4 Quasi-classical approach to superconductivity
270(1)
12.2 Cold interacting Bose gas
271(7)
12.2.1 Ideal Bose gas
271(1)
12.2.2 Bogoliubov transformation for a cold interacting Bose gas
272(5)
12.2.3 Popov approximation
277(1)
12.3 Generalised Soven scheme
278(6)
12.3.1 Missing pole structure
278(2)
12.3.2 Link to anomalous propagators
280(3)
12.3.3 Bogoliubov-DeGennes equation
283(1)
12.4 Gap equations
284(6)
12.4.1 Generalised Soven scheme of coherent potential approximation
287(3)
12.5 Interacting Bose gas at finite temperatures
290(7)
12.5.1 Condensed phase
292(5)
12.6 Comparison of approximations
297(3)
12.6.1 Equation of state
298(2)
12.7 Superconductivity
300(1)
12.7.1 Critical temperature and density of states
301(1)
12.8 Stability of the pairing condensate
301(7)
12.8.1 Galitskii T-matrix
306(1)
12.8.2 Kadanoff-Martin theory
307(1)
12.9 Excitation of Cooper pairs from the condensate
308(7)
12.9.1 Galitskii T-matrix and Kadanoff-Martin theory
308(1)
12.9.2 Conclusions
309(1)
12.9.3 Relation of pairing density to correlated density
310(5)
Part IV Nonlocal Kinetic Theory
13 Nonlocal Collision Integral
315(11)
13.1 Gradient expansion
315(4)
13.1.1 Two-particle matrix products
316(1)
13.1.2 Convolution of initial states
317(1)
13.1.3 Convolution of T-matrix and hole Green's function
318(1)
13.2 Binary and ternary collisions
319(2)
13.3 Complete kinetic equation
321(2)
13.4 Particle-hole versus space-time symmetry
323(2)
13.4.1 Kinetic equation for Monte-Carlo simulations
325(1)
13.5 Summary
325(1)
14 Properties of Non-Instant and Nonlocal Corrections
326(16)
14.1 Recovering classical Δ's from quantum formulae
327(2)
14.1.1 Collision delay Δt
327(1)
14.1.2 Hard-sphere displacement
328(1)
14.2 Invariances of the nonlocal scattering integral
329(3)
14.2.1 Gauge invariance
329(2)
14.2.2 Galilean invariance
331(1)
14.3 Quantum Δ's for isolated particles
332(9)
14.3.1 Gauge transformation to a system free of mean field
332(1)
14.3.2 Galilean transformation to the barycentric coordinate framework
333(1)
14.3.3 Rotational symmetry
334(1)
14.3.4 Energy conservation and time-reversal symmetry
335(1)
14.3.5 Classical-like parametrisation
336(1)
14.3.6 Representation in partial waves
337(1)
14.3.7 Numerical results for nuclear matter
338(3)
14.4 Summary
341(1)
15 Nonequilibrium Quantum Hydrodynamics
342(29)
15.1 Local conservation laws
342(1)
15.2 Symmetries of collisions
343(4)
15.2.1 Transformation A
343(1)
15.2.2 Transformation B
344(1)
15.2.3 Symmetrisation of collision term
345(2)
15.3 Drift contributions to balance equations
347(4)
15.3.1 Density balance from drift
347(1)
15.3.2 Energy balance from the drift
348(1)
15.3.3 Balance of forces from the drift
349(1)
15.3.4 Entropy balance
350(1)
15.4 Molecular contributions to observables from collision integral
351(8)
15.4.1 Expansion properties
351(1)
15.4.2 Correlated observables
352(3)
15.4.3 Molecular current contributions from collision integral
355(3)
15.4.4 Remaining gains
358(1)
15.5 Balance equations and proof of H-theorem
359(5)
15.5.1 Equation of continuity
359(1)
15.5.2 Energy balance
360(1)
15.5.3 Navier-Stokes equation
361(1)
15.5.4 Entropy balance
361(1)
15.5.5 Proof of H-theorem
362(2)
15.6 Equivalence to extended quasiparticle picture
364(2)
15.6.1 Generalised Beth--Uhlenbeck formula
364(1)
15.6.2 Correlated density in terms of the collision delay
365(1)
15.7 Limit of Landau theory
366(1)
15.8 Summary
367(4)
Part V Selected Applications
16 Diffraction on a Barrier
371(21)
16.1 One-dimensional barrier
371(2)
16.1.1 Convenient formulation of the boundary problem
371(2)
16.2 Matrix inversion with surface Green's functions
373(5)
16.2.1 Summary of the method
377(1)
16.3 Reflection and transmission
378(2)
16.3.1 Current and current fluctuations
379(1)
16.4 Transport coefficients: parallel stacked organic molecules
380(3)
16.5 Disordered barrier
383(5)
16.5.1 Configurational averaging
383(1)
16.5.2 Open system
384(2)
16.5.3 Dissipative diffraction: example GaAs/A1As layer
386(2)
16.6 Three-dimensional barrier in a multi-band crystal
388(4)
16.6.1 Diffraction in multi-band crystals
390(2)
17 Deep Impurities with Collision Delay
392(10)
17.1 Transport properties
392(3)
17.2 Dielectric function
395(2)
17.3 Compensation of virial and quasiparticle corrections
397(5)
17.3.1 Long wave-length limit
397(1)
17.3.2 Virial correction to Fermi momentum
398(1)
17.3.3 DC Conductivity
399(3)
18 Relaxation-Time Approximation
402(13)
18.1 Local equilibrium
402(2)
18.2 Transport coefficients
404(1)
18.3 Transport coefficients for metals
405(1)
18.4 Transport coefficients in high-temperature gases
406(3)
18.4.1 Stress or momentum current-density tensor
408(1)
18.5 Exact solution of a linearised Boltzmann equation
409(6)
19 Transient Time Period
415(26)
19.1 Formation of correlations
417(5)
19.1.1 Levinson equation
417(2)
19.1.2 Formation of correlations in plasma
419(2)
19.1.3 Formation of correlations in nuclear matter
421(1)
19.2 Quantum quenches and sudden switching
422(7)
19.2.1 Atoms in a lattice after sudden quench
424(2)
19.2.2 Femtosecond laser response
426(3)
19.3 Failure of memory-kinetic equations
429(5)
19.3.1 Double counts
429(2)
19.3.2 Double count of correlation energy explained by extended quasiparticle picture
431(3)
19.4 Initial correlations
434(6)
19.4.1 Formation of correlations with initial correlations
438(2)
19.5 Summary
440(1)
20 Field-Dependent Transport
441(17)
20.1 Gauge invariance
441(2)
20.1.1 Equation for Wigner distribution
442(1)
20.1.2 Spectral function and ansatz
442(1)
20.2 Kinetic equation in dynamically screened approximation
443(3)
20.3 Feedback and relaxation effects
446(2)
20.4 Conductivity with electron-electron interaction
448(2)
20.5 Isothermal conductivity
450(3)
20.5.1 Quasi two-dimensional example
450(3)
20.6 Adiabatic conductivity
453(1)
20.6.1 Quasi two-dimensional example
453(1)
20.7 Debye--Onsager relaxation effect
454(4)
20.7.1 Thermally averaged dynamically screened result
455(1)
20.7.2 Asymmetric dynamically screened result
456(2)
21 Kinetic Theory of Systems with SU(2) Structure
458(31)
21.1 Transport in electric and magnetic fields
458(4)
21.1.1 Basic notation
458(1)
21.1.2 Spin
459(1)
21.1.3 Conductivity for crossed electric and magnetic fields: orbital motion
460(2)
21.2 Systems with electro-magnetic fields and spin-orbit coupling
462(1)
21.2.1 Various spin-orbit coupling as SU(2) structure
462(1)
21.3 Meanfield kinetic equations
463(8)
21.3.1 Gradient expansion
463(1)
21.3.2 Mean field
464(2)
21.3.3 Commutators
466(2)
21.3.4 Coupled kinetic equations
468(1)
21.3.5 Quasi stationary solution
469(2)
21.4 Normal and anomal currents
471(1)
21.5 Linear response
472(4)
21.5.1 Linearisation to external electric field
472(1)
21.5.2 Conductivities without magnetic fields
473(1)
21.5.3 Collective modes
474(2)
21.6 Response with magnetic fields
476(7)
21.6.1 Retardation subtleties by magnetic field
479(1)
21.6.2 Classical-Hall effect
480(1)
21.6.3 Quantum-Hall effect
481(2)
21.7 Spin-Hall effect, conductivity of graphene
483(6)
21.7.1 Dirac (Weyl) dispersion
483(1)
21.7.2 Conductivity
483(3)
21.7.3 Hall conductivity
486(1)
21.7.4 Pseudo-spin conductivity
487(2)
22 Relativistic Transport
489(15)
22.1 Model and basic equations
489(2)
22.2 Coupled Green's functions for meson and baryons
491(5)
22.3 Equilibrium and saturation thermodynamic properties
496(3)
22.4 Kinetic equations
499(5)
23 Simulations of Heavy-Ion Reactions with Nonlocal Collisions
504(15)
23.1 Scenario of low-energy heavy-ion reactions
504(1)
23.1.1 Numerical simulations of heavy-ion reactions
505(1)
23.2 Instant nonlocal approximation
505(5)
23.2.1 Displacements from the on-shell shifts
507(2)
23.2.2 Realistic displacements
509(1)
23.3 Numerical simulation results
510(5)
23.4 Quasiparticle renormalisation
515(4)
23.4.1 Experimental charge density distribution
517(2)
Appendix A Density-Operator Technique
519(8)
A.1 BBGKY hierarchy
519(1)
A.2 Derivation of the Levinson equation and non-Markovian energy conservation
520(2)
A.3 Debye-Onsager relaxation effect from classical hierarchy
522(5)
A.3.1 With background friction
524(2)
A.3.2 Without background
526(1)
Appendix B Complex Time Path
527(4)
Appendix C Derived Optical Theorem
531(6)
Appendix D Proof of Drift and Gain Compensation into the Rate of Quasiparticles
537(8)
Appendix E Separable Interactions
545(6)
E.1 Yamaguchi form factor
546(5)
References 551(16)
Index 567
Klaus Morawetz graduated in 1992 with a PhD in theoretical physics. He held various postdoc positions at Arizona University, Tennessee TEC, Michigan University , LNS.INFN Catania, and at the Nils Bohr Institute in Copenhagen. He completed a one year research stay at France (LPC Caen), two years at the Max Planck Institute MPIPKS Dresden, one year HZDR Rossendorf, and four years as assistant at Chemnitz University of Technology. In 2008 he was guest professor at ICCMP in Brasilia. Since 2009 he has held a professorship for mathematics and theoretical many-body physics at University of Applied Sciences Münster, researching interacting many-body systems out of equilibrium.
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