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Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Elements of Nonequilibrium Statistical Physics:
Boltzmann Equation and Kubo Formula
Branislav K. NikolićDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A.
http://wiki.physics.udel.edu/phys813
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Fundamental Quantities in Statistical Physics: Phase Space Density and Density Operator
( , ) 1d d ρ =∫ p q p q
( , ) ( , )O d d O ρ= ∫ p q p q p q
( , ) ( , )0 ( , ), ( , )d Hdt t
ρ ρ ρ∂= ⇒ =
∂p q p q p q p q
ˆTr 1ρ =
ˆˆTr[ ]O Oρ=
ˆ ˆ[ , ]di Hdtρ ρ=
phase space density
Liouville equation
ensemble average
0 ( , ), ( , ) 0Hρ =p q p qequilibrium
density matrix
von Neumann equation
0ˆˆ , 0Hρ =
equilibrium
ensemble average
equilibrium vs. nonequilibrium statistical physics
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Formal Derivation of Boltzmann Equation (BE) for Plasma Physics
The full phase space density contains much more information than necessary → define s-particle density:
1 1 1 2 22
2 1 1 2, 2 1 1 2 23
1 11
( , , ) ( , , , , , , , )
( , , , ) ( 1) ( , , , , , , , )
! !( , , , ) ( , , ) ( , , , )( )! ( )!
N
i N Ni
N
i N NiN
s s i s si s
f p q t N dV p p q q p q p q t
f p q p q t N N dV p q p q p q t
N Nf p q t dV t p q tN s N s
ρ
ρ
ρ ρ
=
=
= +
= = =
= −
= = =− −
∏∫
∏∫
∏∫ p q
3 3i i idV d p d q=
one-particle density
two-particle density
s-particle density
2
1 ( , ) 1 1
1 11
1
1( , ) ( ) ( ) ( , , ),2 2
( ),
NN Ni s
i i j ii i j i s
ss n s s
s sn n n
pH p q U q V q q dV t Hm t
f V q q fH f dVt q p
ρ ρ= = = +
+ ++
=
∂= + + − ⇒ = − ∂
∂ ∂ − ∂− =
∂ ∂ ∂
∑ ∑ ∏∫
∑∫
p q
Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy
3 211 2 1 2 1 1 1 1 2 1 1 1 1 1 2 1
1 1 1
( , , ) ( , , ) ( , , ) ( , , )pU df d p d v v f p q t f p q t f p q t f p q tt q p m q d
σ ∂ ∂ ∂ ∂ ′ ′− + = − Ω − − ∂ ∂ ∂ ∂ Ω ∫
Assumption of molecular chaos made by Boltzmann replaces 2-particle density with a product of 1-particle densities: 3
int 1nd
3int 1nd Boltzmann equation Vlasov equation
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
BE in Condensed Matter Describes Transport of Dilute Gas of Quasiparticles
For neutral or ionized gases, the BE and its range of validity, can be directly derived from the Hamiltonian for the classical gas of molecules.
In condensed matter, the BE describes the distribution function for the excitation modes (quasiparticles) and not for the constituents (electrons, ion cores, ...).
The definition of quasiparticles is absolutely vital for setting up BE - it effectively maps, as far as the kinetics is concerned, the quantum-mechanical many-particle system of the constituents to a semiclassical gas of excitation modes.
phonon as quasiparticlequasielectron spin-wave or magnon as quasiparticle
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Heuristic Derivation of BEfor Quasielectrons in Condensed Matter
equilibrium
no scattering
0 ( )1( ) ( , , )
1E Fkeqf k f r k t
eβ ε ε= −= =+
( , , ) ( , , )eEf r k t f r vdt k dt t dt= − + −
( , , ) ( , , )scattering
eE ff r k t f r vdt k dt t dt dtt
∂ = − + − + ∂
33
33
( ) ( , , )8
( ) ( ) ( , , )8
en r d k f r k t
ej r d k v k f r k t
π
π
−=
−=
∫
∫
scattering from
disorder of phonons
charge density
current density
BE in condensed matter describes semi-classical transport: Effective mass approximation (which incorporates the quantum effects due to periodicity of the crystal); Born approximation for the collisions in the limit of small perturbation for the electron-phonon or electron-impurity interaction and instantaneous collisions; no memory effects (i.e., no dependence on initial conditions).
scattering
eE ff v f ft r tk∂ ∂ ∂ ∂ + + = ∂ ∂ ∂∂
33 (1 ( )) ( ) (1 ( )) ( )
(2 ) kk k kscattering
f V d k f k w f k f k w f kt π ′ ′
∂ ′ ′ ′= − − − ∂ ∫
assuming that phonon or impurity (or defect) perturbations are small and time-independent, use Born approximation for the scattering rate from occupied to unoccupied Bloch state
2ˆ 2k kw k H kπ′′=
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Relaxation Time Approximation (RTA) for the Scattering Term in BE
Ansatz: The rate at which a system returns to equilibrium is proportional to its deviation from equilibrium (i.e., we make the assumption that scattering merely acts to drive a non-equilibrium system back to equilibrium):
If E≠0 at t<0 and at t≥0, E=0 external electric field is switched off, then for a homogeneous system we find:
In the steady state transport regime induced by a time-independent homogeneous external electric field:
( )( ) ( ) ( )eqe f kf k f k k E
kτ ∂
= + ⋅∂
( ) ( )( )
eq
scattering
f k f kft kτ
−∂ = − ∂
/( 0)eq teq eq
scattering
f ff f f f f t f et t
τ
τ−−∂ ∂ = = − ⇒ − = = − ∂ ∂
( ) ( )0, 0
( )eq
scattering
f k f kf f e f fEt r tk kτ
−∂ ∂ ∂ ∂ = = ⇒ − ⋅ = = − ∂ ∂ ∂∂
momentum dependence of relaxation time is determined phenomenologically in such
way that the dependence of the conductivity upon the electronic density
agrees with experimental data
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
RTA in the Linear-Response Transport Regime
PRB 60, 3963 (1999): Upon linearization BE and Poisson equation decouple
according to the linear Boltzmann equation, the effect of the electric field is to shift the Fermi surface by
elastic scattering cannot restore equilibrium, rather it would cause Fermi surface to expand →inelastic scattering (e.g., from phonons) is needed
to explain relaxation
For small electric field (Ohmic regime), the relaxation time approximation can be linearized:
/x xk e Eδ τ= −
( ) ( ) ( ) ( )
ˆ ( ) ( )
eq eq
x eq x
ef k f k k E f kkeE E f k f k k E
τ
τ
∂+ ⋅
∂ = ⇒ +
x
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Example: Drude Conductivity of Electrons in Disordered Metal from BE in Linearized RTA
3 3
( )( ) ( ) ( ) ( )8 8
eqeq x
x
fe e e kj dk v k f k dk v k f k Ek
τπ π
∂ −= + ∂
∫ ∫
0y zj j= =
isotropic material
0( )( )
Teq eq
x F xx x
f k ff E v E E vk E k E
δ→∂ ∂∂ ∂
= = − −∂ ∂ ∂ ∂
( ) 0eqk k kv v v f k dk−= − ⇒ =∫
23
2 2
3 3
( )8
( ) ( )( ) ( ) ( )8 8( ) ( )
F
eqx x x x
x x xE F E
x E E
fej E dkv k EE
j v k v ke edS dE k E E dS kE v k v k
τ σπ
σ τ δ τπ π =
∂−=
∂
= = − =
∫
∫ ∫
23 3
*
( ) 4 4( ) ( ) ( ) ( )3 3( )
F
x FE F F F F F
E E
v k kdS k k E v E k Emv k
π πτ τ τ=
= =∫
3 2
223
3 * *3
( )4 ( )8 3
B F
F
k T EF F
F Fk n
k e Ee k E nm mπ
τπσ τπ =
= =
assuming spherical
Fermi surface
Drude formula
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Example: Drude Conductivity of Electrons in Doped Graphene from BE in Linearized RTA
RMP 81
, 109
(200
9)
Relaxation time + Born approximation
eigenstates of clean graphene
scattering potential
0xx nσ ∝
Experiment and BE with unscreened Coulomb potential
BE with delta function potential
Boltzmann limit
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Kubo Linear-Response Theory for Time-Dependent Density Matrix
0ˆ ( )f αα
ρ ε α α=∑
0 1 0 1
10 1 1 0
ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )ˆ ˆ ˆˆ ˆ( ), ( ) ( ),
H H H t t
i H t t H tt
ρ ρ ρρ ρ ρ
= + ⇒ = +∂ ≈ + ∂
equilibrium density matrix for noninteracting
fermions in GCE
in linear response system is driven slightly away from
equilibrium by a small perturbation
2ˆˆ ( )ˆ ˆ ˆ ˆ ˆˆ( ) ( ) ( ) ( ) ( )ii p d
i i
e t e net e t t t tm m m
−= ⇒ = − = +∑ ∑p Aj j p A j j current density operator is the sum of
paramagnetic and diamagnetic term
( )2
ext
0 0 1
1 ˆˆ ˆˆ ( )2
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )
ii
p
H e t Vm
H H t t H H t
= − +
≈ − ⋅ = +
∑ p A
j A assume external field is small and keep only terms linear in vector potential
0 0ˆ ˆ( ')/ ( ')/
1 1 1
11 0 0 1 1 0
ˆˆ ˆ ˆ( ) ( '), ( ') ' ( ) 0
ˆ ( ) ˆ ˆ ˆˆ ˆ ˆ( ), ( ) ( ) ( )
tiH t t iH t tit e H t t e dt
t i i iH t H t t t Ht
ρ ρ ρ
ρ ρ ρ ρ
− − −
−∞
= − −∞ =
∂ = − − + ∂
∫
solution
check by direct differentiation
0H αα ε α=
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Von Neumann Equation in Interaction PictureSplit Hamiltonian of quantum system into free (unperturbed) term and external perturbation assumed to be small:
0ˆ ˆ ˆH H V= +
Expectation value is the same in Schrödinger and Dirac (or interaction) picture:
Von Neumann equation in Schrödinger picture:
( )( )0 0 0 0
ˆ ˆ/ /
ˆ ˆ ˆ ˆˆ ˆ/ / / // /
ˆ ˆ ˆˆ ˆ( ) Tr ( ) ( ) Tr ( ) (0)
ˆ ˆˆ ˆTr ( ) (0) Tr ( ) ( )
iHt iHt
iH t iH t iH t iH tiHt iHtI I
A t A t t A t e e
e A t e e e e e A t t
ρ ρ
ρ ρ
−
− −−
= = = =
ˆˆ ˆ( ) , ( )d it H tdtρ ρ = −
Von Neumann equation in Dirac (or interaction) picture:0 0 0 0
0 0
ˆ ˆ ˆ ˆ/ / / /0 0
ˆ ˆ/ /0 0
0 0
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ( ) , ( ) ( ) , ( ) , ( )
ˆ ˆ ˆˆ ˆ, ( ) , ( )
ˆ ˆ ˆˆ ˆ, ( ) , ( )
iH t iH t iH t iH tI I I
iH t iH tI
I I I
d i d i it H t e t e H t e H t edt dt
i iH t e H V t e
i i iH t H t V
ρ ρ ρ ρ ρ
ρ ρ
ρ ρ
− −
−
= + = −
= − +
= − −
ˆˆ ˆ, ( ) , ( )I I Iit V tρ ρ = −
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Kubo Formula for AC Conductivity as Current-Current Correlation Function
( )
0 0
2ˆ ˆ( )/ ( )/
1 1 0 0
2
00
ˆ ˆ ˆ ˆˆˆ ˆ ˆ( ) Tr ( ) ( ) ( ), Tr ( )
ˆ ˆ ˆ ˆ ˆˆ( ) Tr ( ) ( ) ( ) ( ) ( ), ( )
tiH t t iH t t
p
p p
i net t t dt e H t e tm
ne it dt t t t t t t t t t tm
ρ ρ ρ
ρ θ
′ ′− − −
−∞
∞
−∞
′ ′= = − − ′ ′ ′ ′ ′ ′= − Π − + Π − = − −
∫
∫
j j j A
j A A j j
retarded current-current correlation function
2
00
2( ) / ( ) /
0 ,
2
,
ˆ ˆ( ) ( )ˆ ˆˆ ˆ( ) ( ) ( ) ( ) ( )
ˆ ˆ( ) ( ) ( ) ( ), (0)
ˆ ˆ ˆ ˆ( ), (0) ( )
ˆ( ) 2
i tp p
i t i tp p x x
x
ttt i
i ne i t em
et f p p e em
e pm
α β α β
ω
ε ε ε εα
α β
α β
ωω ω σ ω ωω
σ ω ω ωω
ε α β β α
ω α
∞
− − −
∂= − ⇒ = ⇒ =
∂ = Π + Π = −
= −
Π =
∫
∑
∑
A EE A j E
j j
j j
22
,
( ) ( )ˆ
1 1 ˆ( ) ( ) 2 ( ) ( ) ( )
x
x
f fp
i
eP i x i p f fx i x m
α β
α β
α β α βα β
ε εβ β α
ε ε ω η
πδ ω π α β ε ε δ ε ε ωη
−− + +
= − ⇒Π = − − − + + ∑
Ohm law connecting Fourier transformed
current and electric field
AC conductivity
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Kubo Formula and Kubo-Greewood Formula for DC Conductivity
22
0 ,
22
,
1 ˆ(0) 2 lim ( ) ( ) ( )
ˆ ( ) ( )
x
x
e p f fm
e fh d pm
α β α βω α β
β βα β
σ π α β ε ε δ ε ε ωω
ε α β δ ε ε δ ε εε
→
= − − +
∂ = − − − ∂
∑
∑∫
Kubo formula for DC conductivity in exact eigenstate representation
, , ,1ˆ ˆ( , , )
1 1 1( ) ( , , ) ( , , ) 2 ( )2
r a r a r a
a r
k kG G k k k G k
H i i
x G k k G k k i k ki x i x i
α α
αα
α αε
ε η ε ε η
δ ε ε π α α δ ε επ η η
′′ ′= ⇒ = =
− ± − ±
′ ′ ′= − ⇒ − = − − +
∑
∑
23 2
2
2 1 ( , , ) ( , , ) ( , , ) ( , , )2
r a r axx x x
e fd k G k k G k k k G k k G k km iπσ ε ε ε ε ε
π ε∂ ′ ′ ′ ′ ′ = − − − ∂ ∫
Kubo-Greenwood formula for DC conductivity in terms of Green functions
disorder disorder disorder
,
disorder
2 222 2 2 2
disorder disorder
( , , ) ( , , ) ( , , ) ( , , )
1( , , )2
1 1 ( )2 2 2
r a r a
r a
k
x x kk kk k
G k k G k k G k k G k k
G k ki
e k e k e Dm i i i m
ε ε ε ε
εε ε τ
τσ τ δ ε ε νπ ε ε τ ε ε τ
′ ′ ′ ′=
′ =− ±
= − = − = − − − + ∑ ∑
Einstein formula
isotropic scattering off short-ranged impurities
neglect quantum interference effects
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Kubo Formula on the Computer
,2 2
W Wε ∈ − m
PRL 60
, 848
(198
8)
conductance quantization
reproduced only if all terms in
the original Kubo-Greenwood formula
are evaluated
Boltzmann equation and Kubo formulaPHYS813: Quantum Statistical Mechanics
Numerical Exact Calculation of Conductivity of Graphene using Kubo Formula
ησ
iEEnvnnvn
EEEfEf
Lei
nn
xx
nn
nn
nn +−⟩⟩⟨⟨
−−
−= ∑''
'
',2
2 ||''||)()(
Numerically exact evaluation of the Kubo formula in exact state
representation reproduces Drude conductivity
together with quantum interference corrections
known as weak localization
Short-range scattereres Coulomb scattereres
2min e
hσ
π≈
2min e
hσ ≈
PRL 98
, 076
602
(200
7)
cos 02 2 2 22 cos 4I II I II IA A A A A A
θ
θ≡
= + + =