122
Linear response theory
VI.2.2 Dissipation
Consider a system subject to a harmonic perturbation Ŵ cos ωt with constant Ŵ. A straightforward application of time-dependent perturbation theory in quantum mechanics yields for the
transition rate from an arbitrary initial state |φi i to the set of corresponding final states |φf i
2 π X hφf | Ŵ |φi i δ(ωfi − ω) + δ(ωif − ω) .
Γi→f = 2
(VI.30)
2~
f
If the system absorbs energy from the perturbation, then Ef > Ei , so that only the first δ-term
contributes. On the other hand, energy release from the system, corresponding to “induced emission”,
requires Ef < Ei , i.e. involves the second δ-term.
We apply this result to the system of Sec. VI.1.1, initially at thermodynamic equilibrium. Let
f (t) = fω cos ωt, with constant real fω , be a “force” coupling to an operator  acting on the system,
leading to perturbation (VI.7) in the Hamiltonian.
The probability per unit time that the system absorb a quantum ~ω of energy starting from
an initial state |φn i is given by Eq. (VI.30) with |φi i = |φn i and keeping only the first δ-term.
Multiplying this probability by the initial state population πn and by the absorbed energy ~ω, and
summing over all possible initial states, one finds the total energy received per unit time by the
system
2
πf 2 X
Pgain = ω2
πn ~ω Ann0 δ(ωn0 n − ω).
2~
0
n,n
An analogous reasoning gives for the total energy emitted by the system per unit time
−Ploss =
2 X
2
πfω2 X
Ann0 2 δ(ωnn0 − ω) = πfω
π
~ω
πn0 ~ω Ann0 δ(ωn0 n − ω),
n
2
2
2~
2~
0
0
n,n
n,n
where in the second identity we have exchanged the roles of the two dummy indices. Note that
since the system is in thermodynamic equilibrium, the lower energy levels are more populated, so
that the absorption term is larger than the emission term.
Adding Pgain and Ploss yields the net total energy exchanged—and actually absorbed by the
system—per unit time, dEtot /dt, when it is submitted to a perturbation −fω Â cos ωt, namely
2
dEtot
πf 2 ω X
= ω
(πn − πn0 )Ann0 δ(ωn0 n − ω).
dt
2~
0
n,n
Under consideration of the definition (VI.20) of the spectral density, this also reads
dEtot
f2
= ω ω ξ˜A†A (ω).
dt
2
(VI.31)
The spectral function ξ˜A†A (ω) thus characterizes the dissipation of energy in the system when it is
submitted to a small perturbation proportional to  cos ωt.
VI.3 Properties and interrelations of the correlation functions
127
VI.3.2 Properties and relations of the time-correlation functions
VI.3.2
b Interrelations between time-correlation functions
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
The explicit expression (VI.26) of the generalized susceptibility shows that it is simply related
to the inverse Fourier transform (VI.19) of the spectral density according to
χBA (τ ) = 2iΘ(τ ) ξBA (τ ).
(VI.48)
Since χBA (τ ), which was defined for Hermitian operators only, is real-valued (see last remark of
Sec. VI.1.2 a), one recovers property VI.47b.
Let us define an operator Ȧˆ by the relation
1
Ȧˆ ≡
Â, Ĥ0 ,
i~
(VI.49)
i.e. such that its matrix elements are given by (Ȧ)nn0 = (En0 − En )Ann0 /i~ = iωnn0 Ann0 . If  is an
(81)
The properties involving CBA , SBA or ξBA can be read at once from their definitions (VI.12) resp. (VI.16), or
invoking their respective spectral representations. Those pertaining to KBA can be shown with the help of the
decomposition (VI.23), in particular using the invariance of the ratio under the exchange n ↔ n0 .
128
Linear response theory
observable, then Ȧˆ coincides with the value taken at t = 0 by the derivative dÂI (t)/dt for a system
evolving with Ĥ0 only, i.e. in the absence of external perturbation.
Replacing  by Ȧˆ in the spectral form (VI.23) of Kubo’s correlation function, one finds
X πn − πn0
KB Ȧ (τ ) = i
Bnn0 An0 n e−iωn0 n τ ,
β~
0
n,n
i.e.
KB Ȧ (τ ) =
2i
ξBA (τ ).
β
(VI.50)
In turn, relation (VI.48), becomes
χBA (τ ) = βΘ(t)KB Ȧ (τ ).
(VI.51)
This relation is sometimes referred to as Kubo formula, since in his original article [50] Kubo
expressed the linear response to an perturbation with the help of βKB Ȧ (τ ) instead of the retarded
propagator χBA (τ ) used in Sec. VI.1.2.
Identifying the right-hand sides of Eqs. (VI.38) and (VI.37) and differentiating the resulting
relation with respect to time, one finds
dKBA (t)
2i
= − ξBA (t).
dt
β
Equation (VI.50) then yields
dKBA (t)
= −KB Ȧ (t).
(VI.52)
dt
138
Linear response theory
VI.4 Examples and applications
The presentation of this Chapter is missing.
VI.4.1 Green–Kubo relation
An important application of the formalism of linear response is the calculation of the transport coefficients which appear in the phenomenological constitutive relations of non-equilibrium
thermodynamics.
VI.4.1
a Linear response revisited
::::::::::::::::::::::::::::::::::
Under consideration of relation (VI.48) or (VI.51) and dropping the nonlinear terms, the Kubo
formula (VI.8) can be recast as either
Z ∞
ξBA (τ ) f (t − τ ) dτ
(VI.83)
B̂I (t) n.eq. = B̂ eq. + 2i
0
or equivalently
B̂I (t) n.eq. = B̂ eq. +
∞
Z
0
βKB Ȧ (τ ) f (t − τ ) dτ,
(VI.84)
where the latter is in fact the form originally given by Kubo.
Let us assume that the expectation value of B̂ at equilibrium vanishes: for instance, B̂ is a flux
as introduced in Chapter I. To emphasize this interpretation, we denote the responding observable
by Ĵ b , instead of B̂. Accordingly, to increase the similarity with Sec. I.2, we call the generalized
force Fa (t) instead of f (t), and we rename Âa the observable coupling to Fa (t).
Fourier transforming relations (VI.83) or (VI.84) leads to
˜
Ĵ b (ω) n.eq. = Lba (ω) F˜a (ω),
(VI.85a)
where we also adopted a new notation for the susceptibility:
Z ∞
Z ∞
iωτ
ξJb Aa(τ ) eiωτ dτ.
Lba (ω) ≡ β
KJb Ȧa(τ ) e dτ = 2i
(VI.85b)
0
0
Summing over different generalized forces, whose effects simply add up at the linear approximation,
the Fourier-transformed Kubo formula (VI.85a) is a straightforward generalization of Eq. (I.31)
accounting for frequency dependent fluxes and affinities. The real novelty is that the “generalized
kinetic coefficient” Lba (ω) is no longer a phenomenological factor as in Sec. I.2: instead, there is
now an explicit formula to compute it using time-correlation functions at equilibrium of the system,
Eq. (VI.85b).
In Chapter I, the considered affinities and fluxes were implicitly quasi-static—the gradients
of intensive thermodynamic quantities were time-independent—, which is the regime for which
transport coefficients are defined. For instance, the electrical conductivity σel. is defined as the
proportionality factor between a constant electric field and the ensuing direct current. Accordingly,
the kinetic coefficients Lba in the constitutive relation I.31 are actually the zero-frequency limits of
the corresponding susceptibilities:
Lba
1
= lim
ω→0 kB T
Z
0
∞
iωτ
KJb Ȧa(τ ) e
1
dτ =
kB T
Z
0
∞
KJb Ȧa(τ ) dτ.
(VI.86)
This constitutes the general form of the Green–Kubo-relation [57, 58, 50].
VI.4.1
b Example: electrical conductivity
:::::::::::::::::::::::::::::::::::::::::
As example of application of the Green–Kubo-relation, consider a system made of electrically
charged particles, with charges qi , submitted to a classical external electric field E~ (t), which plays
139
VI.4 Examples and applications
the role of the generalized force. In the dipolar approximation, this field perturbs the Hamiltonian
by coupling to the electric dipole moment of the system
X
~ˆ where D
~ˆ ≡
Ŵ (t) = −E~ (t) · D
qi~rˆi
with ~rˆi the position operator of the i-th particle.
i
Quite obviously, we are interested in the electric current due to this perturbation, namely
X d~rˆi (t)
ˆ
J~ el. (t) ≡
qi
.
dt
i
~ˆ along the direction of E~ —let us
In the notations of the present Chapter, Â is the component of D
for simplicity denote this component by D̂z —, while the role of B̂ is played by any component Jˆel.,j
ˆ
of J~ el. . Additionally, one sees that B̂ = Ȧˆ in the case B̂ = Jˆel.,z .
Focussing on the latter case, Kubo’s linear response formula reads in Fourier space
Jˆel.,z (ω) = χ̃Jz Dz (ω)Ez (ω) ≡ σzz (ω)Ez (ω),
where for obvious reasons we have introduced for the relevant generalized susceptibility the alternative notation σzz (ω). The zero-frequency limit of σzz is the electrical conductivity, which according
to the Green–Kubo-relation (VI.86) is given by
Z ∞
1
KJz Jz (τ ) dτ,
σel. =
kB T 0
i.e. by the integral of the time-autocorrelation function of the component of the electric current
along the direction of the electric field.
Let us again emphasize that given a microscopic model of the system, the canonical correlation
function can be calculated, which then allows one to compute the electrical conductivity, which in
paragraph I.2.3 c was a mere phenomenological coefficient.