Exactly how the interactions between particles lead to a speci¬c response

does not concern us in this section. The only principles we assume the

system to obey are:

(i) (causality) the response never precedes its cause;

(ii) (relaxation) the response will, after termination of the disturbance,

in due time return to its equilibrium value.

Without loss of generality we will assume that the equilibrium value of Y is

zero. So, when X = 0, Y (t) will decay to zero.

A crucial role in the description of linear responses is played by the delta-

18.2 Linear response relations 507

X(t) E E Y (t)

system

— —

X(t) = X0 δ(t) Y (t) = X0 ¦(t)

δ-response

0 0

t

X(t) = X0 H(t) Y (t) = X0 ¦(„ ) d„

0

step-response

0 0

∞

X(t) = X0 [1 ’ H(t)] Y (t) = X0 ¦(„ ) d„

t

steady-state decay

0 0

Figure 18.1 Black-box response to a small perturbation. Responses to a delta-

disturbance, to a step disturbance (Heaviside function) and following a terminated

steady state disturbance are sketched.

response ¦(t) (Fig. 18.1):

— —

if X(t) = X0 δ(t), then Y (t) = X0 ¦(t), (18.1)

—

where X0 is the amplitude of the driving disturbance, taken small enough for

the system response to remain linear. Note that we indicate this δ-function

—

amplitude with a star, to remind us that X0 is not just a special value of

X, but that it is a di¬erent kind of quantity with a dimension equal to the

dimension of X, multiplied by time. Our two principles assure that

¦(t) = 0 for t < 0, (18.2)

lim ¦(t) = 0. (18.3)

t’∞

The shape of ¦(t) is determined by the interactions between particles that

govern the time evolution of Y . We note that ¦(t) is the result of a macro-

scopic experiment and therefore is an ensemble average: ¦(t) is the average

result of delta-disturbances applied to many con¬gurations that are repre-

sentative for an equilibrium distribution of the system.

Because of the linearity of the response, once we know ¦(t), we know the

response to an arbitrary disturbance X(t), as the latter can be reconstructed

508 Linear response theory

from a sequence of δ-pulses. So the response to X(t) is given by

∞

X(t ’ „ )¦(„ ) d„.

Y (t) = (18.4)

0

A special case is the response to a step disturbance, which is zero for t < 0

and constant for t ≥ 0 (i.e., the disturbance is proportional to the Heaviside

function H(t) which is de¬ned as 0 for t < 0 and 1 for t ≥ 0). The response

is then proportional to the integral of the δ-response function. Similarly,

the response after suddenly switching-o¬ a constant disturbance (leaving

the system to relax from a steady state), is given by the integral from t to

∞ of the δ-response function. See Fig. 18.1.

Another special case is a periodic disturbance

X(ω)eiωt ,

X(t) = (18.5)

which is a cosine function if X( ω) is real, and a sine function if X( ω) is purely

imaginary. Inserting this into (18.4) we obtain a response in the frequency

domain, equal to the one-sided Fourier transform of the delta-response:

∞

¦(„ )e’iω„ d„ .

iωt

Y (t) = X( ω)e (18.6)

0

Writing simply

X(t) = X(ω)eiωt and Y (t) = Y (ω)eiωt , (18.7)

with the understanding that the observables are the real part of those com-

plex quantities, (18.6) can be rewritten in terms of the complex frequency

response χ(ω):

Y (ω) = χ(ω)X(ω), (18.8)

with

∞

¦(„ )e’iω„ d„.

χ(ω) = (18.9)

0

The frequency response χ(ω) is a generalized susceptibility, indicating how

Y responds to X. It can be split into a real and imaginary part:

χ(ω) = χ (ω) ’ iχ (ω) (18.10)

∞

χ (ω) = ¦(„ ) cos ω„ d„ (18.11)

0

∞

χ (ω) = ¦(„ ) sin ω„ d„ (18.12)

0

18.2 Linear response relations 509

Note that the zero-frequency value of χ (which is real) equals the steady-

state response to a constant disturbance:

∞

χ(0) = ¦(„ ) d„, (18.13)

0

X(t) = X0 H(t) ’ Y (∞) = X0 χ(0). (18.14)

For the case that X is an electric ¬eld and Y a current density, χ is the

speci¬c conductance σ. Its real part determines the current component in

phase with the periodic ¬eld (which is dissipative), while its imaginary part is

the current component 90—¦ out of phase with the ¬eld. The latter component

does not involve energy absorption from the ¬eld and is usually indicated

with the term dispersion. Instead of the current density, we could also

consider the induced dipole density P as the response; P is related to j since

j = dP/dt. With Y = P , χ becomes µ0 times the electrical susceptibility

χe (see Chapter 13) and σ becomes indistinguishable from iωµ0 χe . Thus

the real part of the electrical susceptibility (or the dielectric constant, or

the square root of the refractive index) corresponds to the non-dissipative

dispersion, while the imaginary part is a dissipative absorption.

The Kramers“Kronig relations

There are interesting relations between the real and imaginary parts of a

frequency response function χ(ω), resulting from the causality principle.

These are the famous Kramers“Kronig relations:1

∞

2 ω χ (ω )