‚zi

i=1

By partial integration it can been shown (see Exercise 18.4) that

Y (0) = β Y ”H , (18.23)

where

2n

‚H(z)

”H = ”zi . (18.24)

‚zi

i=1

Combining (18.19), (18.20) and (18.24), we ¬nd a relation between the delta-

response ¦(t) and the autocorrelation function of Y :

β Y ”H

¦(t) = Y (0)Y (t) . (18.25)

—

X0 Y 2

This relation is only simple if Y ”H is proportional to Y 2 . This is the

case if

—

”H ∝ X0 Y, (18.26)

imposing certain conditions on X and Y .

An example will clarify these conditions. Consider a system of (partially)

charged particles with volume V, subjected to a homogeneous electric ¬eld

—

E(t) = E0 δ(t). For simplicity we consider one dimension here; the extension

to a 3D vector is trivial. Each particle with charge qi will experience a

18.3 Relation to time correlation functions 513

force qi E(t) and will be accelerated during the delta disturbance. After

the disturbance, the ith particle will have changed its velocity with ”vi =

—

(qi /mi )E0 . The total Hamiltonian will change as a result of the change in

kinetic energy:

1 —

2

”H = ” mi vi = mi vi ”vi = E0 q i vi . (18.27)

2

i i i

Thus ”H is proportional to the current density j:

1

j= q i vi . (18.28)

V

i

—

So, if we take Y = j, then ”H = E0 V j and (18.25) becomes

V

¦(t) = j(0)j(t) . (18.29)

kB T

Note that we have considered one dimension and thus j is the current density

in one direction. In general j is a vector and ¦ is a tensor; the relation then

is

V

¦±β (t) = j± (0)jβ (t) . (18.30)

kB T

1

In isotropic materials ¦ will be a diagonal tensor and ¦ = tr ¦ is given

3

by

V

j(0) · j(t) .

¦(t) = (18.31)

3kB T

Equation (18.31) relates the correlation function of the equilibrium current

density ¬‚uctuation with the response function of the speci¬c conductance σ,

which is the ratio between current density and electric ¬eld:

j = σE. (18.32)

Using (18.9), we can express the frequency-dependent speci¬c conductance

in terms of current density ¬‚uctuations:

∞

V

j(0) · j(„ ) e’iω„ d„,

σ(ω) = (18.33)

3kB T 0

with the special case for ω = 0:

∞

V

j(0) · j(„ ) d„.

σ0 = (18.34)

3kB T 0

514 Linear response theory

Note that in these equations the average product of two currentdensities,

multiplied by the volume, occurs:

1

V j(o)j(t) = qi vi (0) qi vi (t) , (18.35)

V

i i

which is indeed a statistically stationary quantity when the total volume of

the system is much larger than the local volume over which the velocities

are correlated.

Equation (18.31) is an example of a Kubo formula (Kubo et al., 1985, p.

155), relating a time correlation function to a response function. Equation

(18.34) is an example of a Green“Kubo formula (Green, 1954; Kubo, 1957),

relating the integral of a time correlation function to a transport coe¬cient.

There are many such equations for di¬erent transport properties.

In the conjugate disturbance E and current density j, which obey the

simple relation (18.26), we recognize the generalized force and ¬‚ux of the

thermodynamics of irreversible processes (see (16.98) in Section 16.10 on

page 446). The product of force and ¬‚ux is an energy dissipation that leads

to an irreversible entropy production. Kubo relations also exist for other

force“¬‚ux pairs that are similarly conjugated.

In this section we have considered the speci¬c conductance as example of

the Kubo and Green“Kubo formula. In the following sections other trans-

port properties will be considered.

18.3.1 Dielectric properties

When the material is non-conducting and does not contain free charge carri-

™

ers, the current density j = (1/V ) qi vi is caused by the time derivative P

of the dipole density P = (1/V ) qi xi . In this case there is no steady-state

current and the zero-frequency conductivity vanishes. But we can connect to

the conductivity case by realizing that the following relations exist between

™

time correlation functions of j = P and P :

d ™ ™

P (0)P (t) = P (0)P (t) = ’ P (0)P (t) , (18.36)

dt

d2 ¨ ™ ™ ¨

P (0)P (t) = P (0)P (t) = ’ P (0)P (t) = P (0)P (t) . (18.37)

dt2

These relations are easily derived when we realize that in an equilibrium

system the time axis in correlation functions may be shifted: A(0)B(t) =

18.3 Relation to time correlation functions 515

A(’t)B(0) . One of the relations in (18.37) is particularly useful:

d2

P (0)P (t) = ’ j(0)j(t) , (18.38)

dt2

as it allows us to translate the current ¬‚uctuations into polarization ¬‚uc-

tuations. The equivalence of the Kubo formula (18.31) for the polarization

response ¦P , which is the integral of the current density response ¦(t), is:

Vd

¦P (t) = ’ P (0) · P (t) . (18.39)

3kB T dt

Realizing that P (ω) = µ0 [µr (ω) ’ 1]E(ω) (see Section 13.2 on page 336), we

¬nd for the frequency-dependent dielectric constant:

∞