Phenomena - DFI UChilersoto/KT/TOC_KineticTheoryand... · Phenomena Soto. Contents ... 1.6...
Transcript of Phenomena - DFI UChilersoto/KT/TOC_KineticTheoryand... · Phenomena Soto. Contents ... 1.6...
Kin
etic
The
ory
an
d Tr
ansp
ort
Phen
omen
aR
odri
go S
oto
One
of t
he q
uesti
ons a
bout
whi
ch h
uman
ity h
as o
ften
won
dere
d is
the
arro
w o
f tim
e. W
hy d
oes t
empo
ral e
volu
tion
seem
irre
vers
ible
? Tha
t is,
we
ofte
n se
e ob
ject
s bre
ak
into
pie
ces,
but w
e ne
ver s
ee th
em re
cons
titut
e sp
onta
neou
sly. T
his o
bser
vatio
n w
as
first
put i
nto
scie
ntifi
c te
rms b
y th
e so
-cal
led
seco
nd la
w o
f the
rmod
ynam
ics:
entro
py
neve
r dec
reas
es. H
owev
er, t
his l
aw d
oes n
ot e
xpla
in th
e or
igin
of i
rrev
ersib
ly; i
t onl
y qu
antifi
es it
. Kin
etic
theo
ry g
ives
a c
onsis
tent
exp
lana
tion
of ir
reve
rsib
ility
bas
ed o
n a
sta
tistic
al d
escr
iptio
n of
the
mot
ion
of e
lect
rons
, ato
ms,
and
mol
ecul
es. T
he c
once
pts
of k
inet
ic th
eory
hav
e be
en a
pplie
d to
innu
mer
able
situ
atio
ns in
clud
ing
elec
troni
cs,
the
prod
uctio
n of
par
ticle
s in
the
early
uni
vers
e, th
e dy
nam
ics o
f astr
ophy
sical
pla
smas
, gr
anul
ar av
alan
ches
, or t
he m
otio
n of
smal
l mic
roor
gani
sms i
n w
ater
, with
exc
elle
nt
quan
titat
ive
agre
emen
t. Thi
s boo
k pr
esen
ts th
e fu
ndam
enta
ls of
kin
etic
theo
ry,
cons
ider
ing
clas
sical
par
adig
mat
ic e
xam
ples
(cla
ssic
al a
nd q
uant
um g
ases
, pla
smas
, B
row
nian
par
ticle
s, an
d el
ectro
nic
tran
spor
t), to
geth
er w
ith m
oder
n ap
plic
atio
ns a
nd
num
eric
al m
etho
ds. T
he te
xt is
bal
ance
d be
twee
n ex
plor
ing
the
fund
amen
tal c
once
pts
of k
inet
ic th
eory
(irr
ever
sibili
ty, t
rans
port
pro
cess
es, s
epar
atio
n of
tim
e sc
ales
, co
nser
vatio
ns, c
oars
e gr
aini
ng, d
istri
butio
n fu
nctio
ns, e
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
ries
are
wri
tten
for
the
final
-yea
r un
derg
radu
ate
and
begi
nnin
g gr
adua
te l
evel
, and
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
ters
to
mor
e ad
vanc
ed w
ork
are
incl
uded
, al
ong
with
am
ple
tuto
rial
m
ater
ial,
exam
ples
, illu
strat
ions
, cha
p-te
r su
mm
arie
s, an
d gr
aded
pr
oble
m
sets
(with
som
e an
swer
s and
hin
ts).
978
0198
7160
51
ISB
N 9
78-0
-19-
8716
05-1
�
�
�
oxfo
rd m
aste
r se
ries
in s
tati
stic
al,
com
puta
tion
al, a
nd t
heo
reti
cal
phys
ics
|oms in sctp| �
oxfo
rd m
aste
r se
ries
in s
tati
stic
al,
com
puta
tion
al, a
nd t
heo
reti
cal
phys
ics
‘Sot
o ha
s writ
ten
the m
oder
n, a
ccess
ible
text
book
that
a fi
eld a
s im
porta
nt a
s kin
etic
theo
ry
dese
rves.
Writ
ten
by a
pra
ctitio
ner, i
ts sty
le, w
ith cl
ear m
otiv
atio
ns, l
ots o
f figu
res,
frequ
ent r
efer-
ence
s to
furth
er re
adin
g and
poi
nter
s to
the m
inefi
elds a
waiti
ng th
e kee
n no
vice,
mak
es it
par
ticu-
larly
use
ful f
or a
nyon
e who
wan
ts to
get t
heir
hand
s dirt
y qu
ickly.
’G
unna
r Pru
essn
er, I
mpe
rial
Col
lege
Lon
don,
UK
‘Pro
fesso
r Sot
o ha
s don
e a m
arve
llous
job
of p
rodu
cing a
n in
trodu
ctory
text
on
non-
equi
libriu
m
syste
ms a
nd tr
ansp
ort p
heno
men
a. T
his b
ook,
writ
ten
in a
n in
vitin
g inf
orm
al st
yle w
ith a
wid
e ra
nge o
f exe
rcise
s, is
suita
ble f
or a
dvan
ced
unde
rgra
duat
e or e
arly
grad
uate
stud
ents.
The
org
anisa
-tio
n, b
egin
ning
with
fund
amen
tal c
once
pts a
nd m
etho
ds, a
nd p
roce
edin
g to
class
ical a
nd m
oder
n
appl
icatio
ns, w
ill sp
ark
inte
rest
from
man
y ar
eas o
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
amic
al
corr
elat
ions
in c
lass
ical
gas
es.
(bac
k) L
ight
-mat
ter s
catte
ring
pro
cess
.
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|i⟩
|f⟩
|f⟩
ε iε f
h̄ω
hω
h̄ω
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