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kinetic equation, Eq. (7.55), with respect to the momentum variable.

For a discussion of the classical Boltzmann transport coe¬cients for a degenerate
Fermi system, electrical and thermal conductivities, we refer the reader to chapter 5 of
reference [1]. Here we just note that, for the case of a time-independent electric ¬eld,
the solution to the Boltzmann equation, Eq. (7.56), to linear order is immediately
obtained giving for the conductivity, σ0 , the Boltzmann result
ne2 „tr
σ0 = (7.64)
where „tr ≡ „tr ( F ) is the transport relaxation time in the Born approximation
dˆ F
|Vimp (pF ’ pF )|2 (1 ’ pF · pF ) .
= 2πni N0 (7.65)
„tr ( F ) 4π
The appearance of the transport time expresses the simple fact that small angle
scattering is not e¬ective in degrading the current. For isotropic scattering the mo-
mentum and transport relaxation times are identical, as each scattering direction is
14 This so-called Landau criterion is not su¬cient for the applicability of the Boltzmann equation
in low-dimensional systems, d ¤ 2. This is a subject we shall discuss in detail in Chapter 11.
192 7. Quantum kinetic equations

weighted equally. The transport relaxation time is the characteristic time a particle
can travel before the direction of its velocity is randomized.

Exercise 7.4. Show that the retarded impurity self-energy, Eq. (7.51), in equilibrium
and for |E ’ F | F and |p ’ pF | pF just becomes the constant

ΣR (E, p) = ’i (7.66)
dˆ F
|Vimp (pF ’ pF )|2 .
= 2πni N0 (7.67)
„ 4π

For later use, we end this section on dynamics due to impurity scattering by
considering Boltzmannian motion and its large scale features, Brownian motion.

7.4.1 Boltzmannian motion in a random potential
In later chapters we shall discuss quantum corrections to classical transport. How-
ever, in many cases we often still need to know only the classical kinetics of the
particle motion. We therefore take this opportunity to discuss the Boltzmannian
motion of a particle scattered by impurities, although we shall not need these results
before we discuss destruction of phase coherence due to electron“phonon interaction
in Chapter 11. The Boltzmann theory is a stochastic description of the classical mo-
tion of a particle in a weakly disordered potential. At each instant the particle has
attributed a probability for a certain position and velocity (or momentum). In the
absence of external ¬elds the Boltzmann equation for a particle in a random potential
has the form
‚f (x, p, t) ‚f (x, p, t) dp
+ v· W (p, p ) [f (x, p, t)’f (x, p , t)] , (7.68)
(2π )3
‚t ‚x

where we have introduced the notation v = vp = p/m for the particle velocity.
The Boltzmann equation is ¬rst order in time (the state of a particle is completely
determined in classical mechanics by specifying its position and momentum), and
the solution for such a Markov process can be expressed in terms of the conditional
probability F for the particle to have position x and momentum p at time t given it
had position x and momentum p at time t

f (x, p, t) = dx F (x, p, t; x , p , t ) f (x , p , t ) . (7.69)

For elastic scattering only the direction of momentum can change, and consequently
we need only integrate over the direction of the momentum. In the absence of ex-
ternal ¬elds the motion in between scattering events is along straight lines, and the
conditional probability describes how the particle by impurity scattering, is thrown
between di¬erent straight-line segments, i.e. a Boltzmannian path.
7.4. Impurity scattering 193

We de¬ne the Boltzmann propagator as the conditional probability for the initial
condition that it vanishes for times t < t , the retarded Green™s function for the
Boltzmann equation. The equation obeyed by the Boltzmann propagator is thus,
assuming isotropic scattering,

‚ ‚ 1 1 dˆ
+ vp · F (p, x, t; p , x , t ) ’
+ F (p, x, t; p , x , t )
‚t ‚x „ „ 4π

ˆp ˆ
δ(ˆ ’ p ) δ(x ’ x ) δ(t ’ t ) ,
= (7.70)

where δ is the spherical delta function

dˆ ˆ
δ(ˆ ’ p ) f (p ) = f (p) .
pˆ (7.71)

The equation for the Boltzmann propagator is solved by Fourier transformation, and
we obtain
dq dω iq·(x’x )’iω(t’t )
F (p, x, t; p , x , t ) = e F (p, p ; q, ω) , (7.72)

1 1/„ ˆp ˆ
I(q, ω) + δ(ˆ ’ p )
F (p, p ; q, ω) =
’iω + p · q/m + 1/„ ’iω + p · q/m + 1/„
I(q, ω) = , (7.74)
ql ’ arctan ql/(1 ’ iω„ )
where l = v„ is the mean free path.
We note, by direct integration, the property

F (x, p, t; x , p , t ) = dx F (x, p, t; x , p , t ) F (x , p , t ; x , p , t )

the signature of a Markov process. This property will be utilized in Section 11.3.1
in the calculation of the dephasing rate in weak localization due to electron“phonon

7.4.2 Brownian motion
If we are interested only in the long-time and large-distance behavior of the particle
motion, |x ’ x | l, t ’ t „ , the wave vectors and frequencies of importance in
15 For a Markov process, the future is independent of the past when the present is known, i.e.
the causality principle of classical physics in the context of a stochastic dynamic system, here the
process in question is Boltzmannian motion.
194 7. Quantum kinetic equations

the Boltzmann propagator, Eq. (7.73), satisfy ql, ω„ 1, and we obtain the di¬usion
I(q, ω) , (7.76)
’iω + D0 q 2
where D0 = vl/3 is the di¬usion constant in the considered case of three dimensions
(and isotropic scattering). By Fourier transforming we ¬nd that, in the di¬usion
approximation, the dependence on the magnitude of the momentum (velocity) in the
momentum directional averaged Boltzmann propagator appears only through the
di¬usion constant, t > t ,

dqdω eiq·(x’x )’iω(t’t )
D(x, t; x, t ) ≡ F (p, x, t; p , x , t ) =
’iω + D0 q 2
(4π)2 (2π)4

e’(x’x ) /4D0 (t’t )

= . (7.77)
(4πD0 (t ’ t ))d/2
This di¬usion propagator describes the di¬usive or Brownian motion of the particle,
the conditional probability for the particle to di¬use from point x to x in time span
t ’ t , described by the one parameter, the di¬usion constant. The absence of the
explicit appearance of the magnitude of the velocity re¬‚ects the fact that the local
velocity is a meaningless quantity in Brownian motion.

Exercise 7.5. Show that
≡ dx x2 D(x, t; x , t ) = x + 2dD0 (t ’ t ) ,
x2 (7.78)
t,x ,t

where the d on the right-hand side is the spatial dimension.

If we are interested only in the long-time and large-distance behavior of the Boltz-
mannian motion we can, as noted above, get a simpli¬ed description of the classical
motion of a particle in a random potential. We are thus not interested in the zigzag
Boltzmannian trajectories, but only in the smooth large-scale behavior. It is instruc-
tive to relate the large-scale behavior to the velocity (or momentum) moments of the
distribution function, and the corresponding physical quantities, density and current
density. Expanding the distribution function on spherical harmonics

f (x, p, t) = f0 ( p , x, t) + p · f ( p , x, t) + · · · (7.79)

we have the particle current density given in terms of the ¬rst moment
1 dp 1 dp
p p · f ( p , x, t) = p2 f ( p , x, t)
j(x, t) = (7.80)
(2π )3 3
m 3m (2π )
and the density given in terms of the zeroth moment
n(x, t) = f0 ( p , x, t) . (7.81)
(2π )3
7.4. Impurity scattering 195

Taking the spherical average


... ... (7.82)

of the force-free Boltzmann equation, Eq. (7.68), we obtain the zeroth moment equa-
‚f0 ( p , x, t)
∇x · f ( p , x, t) = 0 .
+ (7.83)
‚t 3m
Integrating this equation with respect to momentum gives the continuity equation

‚n(x, t)
+ ∇x · j(x, t) = 0 . (7.84)
This result is of course independent of whether external ¬elds are present or not.
This is seen directly from the Boltzmann equation by integrating with respect to
momentum as we have the identity

Ix,p,t [f ] = 0 (7.85)

simply re¬‚ecting that the collision integral respects particle conservation.
Taking the ¬rst moment of the Boltzmann equation, p . . . ,

dˆ ‚f (x, p, t) ‚f (x, p, t)
+ vp · ’ Ix,p,t [f ] =0 (7.86)
4π ‚t ‚x

we obtain the ¬rst moment equation

p2 p2 ‚f0 (x, p, t)
‚ 1
+ f (x, p, t) + = 0, (7.87)
3 ‚t „ ( p ) 3m ‚x

where we have repeatedly used the angular average formulas

dˆ dˆ
p p
p± pβ = δ±β , p± pβ pγ = 0 . (7.88)
4π 3 4π

We have thus reduced the kinetic equation to a closed set of equations relating the
two lowest moments of the distribution function, f0 and f , and we get the equation
satis¬ed by the zeroth moment f0 :

‚ 1 ‚f0 (x, p, t)

+ x f0 (x, p, t) = 0. (7.89)
‚t „ ( p ) ‚t

In a metal the derivatives of the zeroth harmonic of the distribution function for
the conduction electrons, ‚t f0 ( p , x, t) and ”x f0 ( p , x, t), are peaked at the Fermi
energy, and we can use the approximations
dp dp
p2 ”x f0 ( p , x, t) p2 ”x f0 ( p , x, t) (7.90)
3 (2π )3
(2π )
196 7. Quantum kinetic equations

dp ‚ 1 ‚f0 ( p , x, t) ‚ 1 ‚n(x, t)
+ + , (7.91)
(2π )3 ‚t „ ( p ) ‚t ‚t „ ‚t
where as usual „ ≡ „ ( pF ). Assuming only low-frequency oscillations in the density,
ω„ 1,
‚2n 1 ‚n
‚t2 „ ‚t
and we obtain from Eq. (7.89) the continuity equation on di¬usive form

’ D0 n(x, t) = 0 . (7.93)
Since ∇x f0 ( p , x, t) is peaked at the Fermi energy, we can use the approximation
dp dp
p2 ∇x f0 ( p , x, t) ∇x f0 ( p , x, t)
p2 (7.94)
3 (2π )3
(2π )
and assuming only low-frequency current oscillations
‚j(x, t) 1
|j(x, t)| (7.95)
‚t „
we obtain from the ¬rst moment equation, Eq. (7.87), the di¬usion expression for
the current density
‚n(x, t)
j(x, t) = ’D0 . (7.96)
If we assume that the particle is absent prior to time t , at which time the particle
is created at point x , the di¬usion equation, Eq. (7.93), gets a source term, and we
obtain for the conditional probability or di¬usion propagator D(x, t; x , t )

n(x, t) = dx D(x, t; x , t ) n(x , t ) (7.97)

the equation

’ D0 D(x, t; x , t ) = δ(x ’ x ) δ(t ’ t ) (7.98)
with the initial condition
D(x, t; x , t ) = 0 , for t<t . (7.99)
We can solve the equation for the di¬usion propagator, the retarded Green™s function
for the di¬usion equation, by referring to the solution of the free particle Schr¨dinger
Green™s function equation, Eq. (C.24), and letting it ’ t, and /2m ’ D0 , and we
obtain 2 (x’x )

e 4D 0 (t ’t )
D(x, t; x , t ) = θ(t ’ t ) . (7.100)
(4πD0 (t ’ t ))d/2
7.4. Impurity scattering 197

Exercise 7.6. Show that the Di¬uson or di¬usion propagator has the path integral
xt =x xt =x
Dxt e’SE [xt¯] = Dxt e’ d t L E (x t )
¯ ™¯
D(x, t; x , t ) = (7.101)
¯ ¯

xt =x xt =x

where the Euclidean action SE [xt ] is speci¬ed by the Euclidean Lagrangian


LE (xt ) = . (7.102)
The probability density of di¬usive paths is therefore given by
’ t ¯
’SE [xt ] t
PD [xt ] ≡ ¯
e =e . (7.103)
4D 0

Note that the velocity entering the above Wiener measure is not the local velocity
but the velocity averaged over Boltzmannian paths.16
Exercise 7.7. Show that, for a di¬using particle, we have the Gaussian property for
the characteristic function

Dxt PD [xt ] eiq·(x(t)’x(t ))
¯ ¯
= e’D0 q |t’t |
iq·(x(t)’x(t ))
<e >D = . (7.104)
Dxt PD [xt ]
¯ ¯

Exercise 7.8. Consider the Di¬uson or di¬usion propagator speci¬ed by the ladder

p+ p+ p+ p+
E+ p+ E+ p+ E+ p+

+ ···
DE (q, ω) = + +
p’ p’ p’ p’
Ep’ Ep’ Ep’


⎜ E+ p+

u⎜ 1

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