17.1 Time-Correlation Functions and Spectral Functions 353
The pertainingspectral functions Sij are the Fourier-Laplace transform of the
time-correlation functions, viz.
Sij(ω,k)=π−^1 Re∫∞
0exp[iωt]Cij(t|k)dt. (17.6)When the averages〈ψi〉and〈ψj〉are components of irreducible tensors of ranks
andn, the corresponding time-correlation and spectral functions are tensors of rank
+n. With〈ψi〉and〈ψj〉replaced byAμ 1 ···μandBν 1 ···νn, equation (17.5) becomes
Aμ 1 ···μ(t|k)=CμAB 1 ···μ,ν 1 ···νn(t|k)Bν 1 ···νn( 0 |k), (17.7)The pertaining spectral function, evaluated according to (17.6), is denoted by
SABμ 1 ···μ,ν 1 ···νn(ω,k).The symmetry, parity and time reversal consideration discussed in connection with
linear constitutive relations, cf. Sect.16.1, apply to these functions as well. The
depolarized Rayleigh scattering, to be discussed in the next section, corresponds to
a case, where one has=n=2.
Examples for the computation of auto- and cross-correlation functions of the
friction pressure tensor and the tensor polarization of a gas of linear molecules, are
found in [175]. These correlation functions are linked with the viscosity and the flow
birefringence. The influence of a magnetic field also studied there is associated with
the Senftleben-Beenakker effect of the viscosity, cf. Sect.16.3.4.
As originally pointed out by Green and Kubo [101, 176], transport coefficients
can be computed as time-integrals of correlation functions. The relevant equations
are referred to asGreen-Kubo-orasKubo-relations. For details of the method e.g. see
[48–50]. The Green-Kubo relations imply that material coefficients characterizing
non-equilibrium processes can be inferred from fluctuations in an equilibrium state.
Instead of performing a time integral, the material coefficients can also be obtained
from the dependence of the magnitude of the fluctuations on the length of the time
interval, over which the fluctuations are pre-averaged. This has been demonstrated
in [177] for the viscosity and the viscoelasticity of a simple fluid.
17.1.2 Depolarized Rayleigh Scattering
Light scattering is caused by fluctuations of the dielectric tensorεμν.TheRayleigh
scatteringand theBrillouin scatteringare associated with the fluctuations of the
isotropic part which, in turn, are mainly caused by density fluctuations. In this case,
the electric field of the scattered light is parallel to that of the incident light, this
ispolarized scattering. Fluctuations of the anisotropic part εμν lead to a scattered