elements of nonequilibrium statistical physics: boltzmann ... · phys813: quantum statistical...

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


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


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


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


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′′=


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


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


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


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


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=


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 = −


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


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


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


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≡

= + + =

elements of nonequilibrium statistical physics: boltzmann ... · phys813: quantum statistical...

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