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ory

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Phen

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oto

One

of t

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as o

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dere

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arro

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we

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xpla

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ersib

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t onl

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. Kin

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theo

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ives

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mot

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of e

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, ato

ms,

and

mol

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he c

once

pts

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inet

ic th

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hav

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en a

pplie

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innu

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situ

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clud

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the

prod

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par

ticle

s in

the

early

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vers

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e dy

nam

ics o

f astr

ophy

sical

pla

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, gr

anul

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otio

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smal

l mic

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ater

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nt

quan

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kin

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cons

ider

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clas

sical

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row

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oder

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plic

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num

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bal

ance

d be

twee

n ex

plor

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the

fund

amen

tal c

once

pts

of k

inet

ic th

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(irr

ever

sibili

ty, t

rans

port

pro

cess

es, s

epar

atio

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tim

e sc

ales

, co

nser

vatio

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e gr

aini

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istri

butio

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tc.)

and

the

resu

lts a

nd

pred

ictio

ns o

f the

theo

ry, w

here

the

rele

vant

pro

pert

ies o

f diff

eren

t sys

tem

s are

com

pute

d.

Rod

rigo

Sot

o is

Full

Prof

esso

r at t

he P

hysic

s Dep

artm

ent,

FCFM

, Uni

vers

idad

de

Chi

le,

Sant

iago

de

Chi

le.

Boo

ks i

n th

is se

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are

wri

tten

for

the

final

-yea

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radu

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begi

nnin

g gr

adua

te l

evel

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pro

vide

str

aigh

tforw

ard

intro

duct

ions

to

ke

y to

pics

in

phy

sics

toda

y. B

ackg

roun

d m

ater

ial

and

appl

icat

ions

as

w

ell

as

poin

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to

mor

e ad

vanc

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ork

are

incl

uded

, al

ong

with

am

ple

tuto

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m

ater

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exam

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strat

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p-te

r su

mm

arie

s, an

d gr

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pr

oble

m

sets

(with

som

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s and

hin

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978

0198

7160

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ISB

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oxfo

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phys

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‘Sot

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s writ

ten

the m

oder

n, a

ccess

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text

book

that

a fi

eld a

s im

porta

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s kin

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dese

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ten

by a

pra

ctitio

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ts sty

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frequ

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furth

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mak

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par

ticu-

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wan

ts to

get t

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hand

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ickly.

’G

unna

r Pru

essn

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mpe

rial

Col

lege

Lon

don,

UK

‘Pro

fesso

r Sot

o ha

s don

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arve

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job

of p

rodu

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n in

trodu

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text

on

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equi

libriu

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syste

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ansp

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his b

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writ

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in a

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wid

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rcise

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unde

rgra

duat

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arly

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stud

ents.

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org

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egin

ning

with

fund

amen

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pts a

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etho

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nd p

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class

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appl

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inte

rest

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man

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f phy

sics.’

Dim

itri V

vede

nsky

, Im

peri

al C

olle

ge L

ondo

n, U

KC

over

imag

e: (f

ront

) Rec

ollis

ion

ev

ent,

resp

onsib

le fo

r the

dyn

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al

corr

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in c

lass

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gas

es.

(bac

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catte

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pro

cess

.

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Kinetic Theory and Transport Phenomena Soto

Contents

1 Basic concepts 11.1 Velocity distribution function 11.2 The Maxwell–Boltzmann distribution function 21.3 Densities and fluxes 3

1.3.1 Stress tensor and energy flux 51.3.2 Stress tensor and heat flux in equilibrium 61.3.3 Flux distribution 7

1.4 Collision frequency 71.5 Mean free path 81.6 Transport properties in the mean free path approximation 8

1.6.1 Thermal conductivity 91.6.2 Viscosity 101.6.3 Wall slip 111.6.4 Self-di↵usion 11

1.7 Drude model for electric transport 12Exercises 13

2 Distribution functions 152.1 Introduction 152.2 Hamiltonian dynamics 162.3 Statistical description of the phase space 162.4 Equilibrium distribution 192.5 Reduced distributions 212.6 Microscopic and average observables 23

2.6.1 Global observables 232.6.2 Densities 242.6.3 Fluxes 242.6.4 Conservation equations 27

2.7 BBGKY hierarchy 282.7.1 Equation for the one-particle distribution 30

2.8 Generalisation to mixtures 312.9 Reduced distributions in equilibrium and the pair distri-

bution function 322.10 Master equations 332.11 Application: systems with overdamped dynamics 34Further reading 36Exercises 37

xii Contents

3 The Lorentz model for the classical transport of charges 393.1 Hypothesis of the model 393.2 Lorentz kinetic equation 413.3 Ion distribution function 423.4 Equilibrium solution 433.5 Conservation laws and the collisional invariants 433.6 Kinetic collision models 44

3.6.1 Rigid hard spheres 453.6.2 Thermalising ions: the BGK model 46

3.7 Electrical conduction 463.7.1 Conservation equation 463.7.2 Linear response 473.7.3 Ohm’s law 473.7.4 Electrical conductivity 473.7.5 Frequency response 50

3.8 Relaxation dynamics 503.8.1 Properties of the linear operator 513.8.2 Kinetic gap 523.8.3 Spectrum of the linear operator 533.8.4 Di↵usive behaviour 543.8.5 Rigid hard spheres 543.8.6 Time scales 55

3.9 The Chapman–Enskog method 563.10 Application: bacterial suspensions, run-and-tumble motion 58Further reading 60Exercises 61

4 The Boltzmann equation for dilute gases 634.1 Formulation of the Boltzmann model 63

4.1.1 Hypothesis 634.1.2 Kinematics of binary collisions 64

4.2 Boltzmann kinetic equation 664.2.1 General case 664.2.2 Hard sphere model 67

4.3 General properties 684.3.1 Balance equations and collisional invariants 684.3.2 H-theorem 704.3.3 On the irreversibility problem 73

4.4 Dynamics close to equilibrium 744.4.1 Linear Boltzmann operator 744.4.2 Spectrum of the linear Boltzmann equation 754.4.3 Time scales 77

4.5 BGK model 774.6 Boundary conditions 794.7 Hydrodynamic regime 79

4.7.1 The hydrodynamic equations 794.7.2 Linear response 814.7.3 Variational principle 82

Contents xiii

4.7.4 The Chapman–Enskog method 824.8 Dense gases 86

4.8.1 The Enskog model for hard sphere gases 864.8.2 Virial expansion 88

4.9 Application: granular gases 894.10 Application: the expanding universe 91Further reading 92Exercises 92

5 Brownian motion 955.1 The Brownian phenomenon 955.2 Derivation of the Fokker–Planck equation 965.3 Equilibrium solutions 98

5.3.1 Homogeneous equilibrium solution and thefluctuation–dissipation relation 98

5.3.2 Equilibrium solution under external potentials 995.4 Mobility under external fields 1015.5 Long-time dynamics: di↵usion 102

5.5.1 Solution of the di↵usion equation 1025.5.2 Green–Kubo expression 1045.5.3 Coarse-grained master equation 1055.5.4 Eigenvalue analysis 1065.5.5 Chapman–Enskog method 1075.5.6 Boundary conditions 108

5.6 Early relaxation 1085.7 Rotational di↵usion 1095.8 Application: light di↵usion 1105.9 Application: bacterial alignment 111Further reading 112Exercises 113

6 Plasmas and self-gravitating systems 1156.1 Long-range interactions 1156.2 Neutral plasmas 116

6.2.1 Introduction 1166.2.2 Debye screening 1176.2.3 Vlasov equation 1196.2.4 Stationary solutions 1226.2.5 Dynamical response 122

6.3 Waves and instabilities in plasmas 1256.3.1 Plasma waves 1256.3.2 Landau damping 1266.3.3 Instabilities 130

6.4 Electromagnetic e↵ects 1316.4.1 Magnetic fields 1316.4.2 Hydrodynamic equations 131

6.5 Self-gravitating systems 1326.5.1 Kinetic equation 132

xiv Contents

6.5.2 Self-consistent equilibrium solutions 1336.5.3 Jeans instability 135

6.6 Beyond mean field 1356.6.1 Velocity relaxation and dynamical friction 1356.6.2 Slow relaxation 1366.6.3 Kinetic equations 137

6.7 Application: point vortices in two dimensions 138Further reading 140Exercises 140

7 Quantum gases 1437.1 Boson and fermion ideal gases at equilibrium 143

7.1.1 Description of the quantum state 1437.1.2 Equilibrium distributions 145

7.2 Einstein coe�cients 1467.3 Scattering transition rates 1487.4 Master kinetic equation 1497.5 Equilibrium solutions 1517.6 Where is the molecular chaos hypothesis? 1527.7 Phonons 153

7.7.1 Ideal gas of phonons 1537.7.2 Phonon–phonon interactions 1557.7.3 Phonon–electron interactions 160

7.8 Application: lasers 1617.9 Application: quark–gluon plasma 164Further reading 166Exercises 166

8 Quantum electronic transport in solids 1698.1 Electronic structure 1698.2 Fermi–Dirac distribution, conductors, and insulators 1698.3 Boltzmann–Lorentz equation 171

8.3.1 Distribution function 1718.3.2 Scattering processes 1728.3.3 Semiclassical kinetic equation 1738.3.4 Linear collision operator 174

8.4 Time-independent point defects 1758.4.1 Transition rates 1758.4.2 Spherical models 176

8.5 Relaxation time approximation 1778.6 Electrical conductivity 177

8.6.1 Qualitative description: metals and insulators 1778.6.2 Conductivity of metals 1798.6.3 Finite-temperature e↵ects 1818.6.4 Electron–phonon interactions 1828.6.5 Multiple scattering mechanisms and the

Matthiessen rule 1848.7 Thermal conductivity and Onsager relations 185

Contents xv

8.7.1 Wiedemann–Franz law 1888.8 Transport under magnetic fields 189

8.8.1 Equilibrium solution 1908.8.2 Linear response to electric fields 1908.8.3 Hall e↵ect and the magnetoresistance 191

8.9 Thomas–Fermi screening 1928.10 Application: graphene 193Further reading 196Exercises 196

9 Semiconductors and interband transitions 1999.1 Charge carriers: electrons and holes 1999.2 Doped materials and extrinsic semiconductors 2009.3 Kinetic equation 202

9.3.1 Generation–recombination 2039.4 Hydrodynamic approximation 2049.5 Photoconductivity 2049.6 Application: the diode or p–n junction 205Further reading 207Exercises 207

10 Numerical and semianalytical methods 20910.1 Direct approach 20910.2 Method of moments 209

10.2.1 Local equilibrium moment method 21110.2.2 Grad’s method 211

10.3 Particle-based methods 21210.3.1 Sampling 21210.3.2 Random numbers 21310.3.3 Streaming motion 21410.3.4 Brownian motion 21410.3.5 Long-range forces 21610.3.6 Collisions 21910.3.7 Quantum e↵ects 22110.3.8 Boundary conditions 222

Further reading 222Exercises 223

A Mathematical complements 225A.1 Fourier transform 225A.2 Dirac delta distributions 225A.3 Eigenvalues of a perturbed operator 227

A.3.1 Statement of the problem 227A.3.2 Order O(✏0) 227A.3.3 Order O(✏1) 227

Exercises 229

B Tensor analysis 230B.1 Basic definitions 230

xvi Contents

B.2 Isotropic tensors 232B.3 Tensor products, contractions, and Einstein notation 233B.4 Di↵erential operators 234B.5 Physical laws 234Exercises 235

C Scattering processes 236C.1 Classical mechanics 236

C.1.1 Kinematics of binary collisions 236C.1.2 Geometrical parameterisation 237C.1.3 Scattering for hard sphere, Coulomb, and gravita-

tional potentials 237C.2 Quantum mechanics 238

C.2.1 Time-dependent perturbation theory 238C.2.2 Fermi golden rule 239

Exercises 240

D Electronic structure in crystalline solids 242D.1 Crystalline solids 242D.2 Band structure 242

D.2.1 Bloch theorem 243D.2.2 Energy bands 245D.2.3 Bloch velocity and crystal momentum equation 245D.2.4 Self-consistent potential 246

D.3 Density of states 247D.3.1 Free electron gas 247D.3.2 General case in three dimensions 248

Exercises 249

References 250

Index 255