Tensors for Physics

17.1 Time-Correlation Functions and Spectral Functions 357

In a spatial Fourier transform of this equation, the LaplacianΔis replaced by−k^2.
The resulting time-correlation function is

C(t|k)=exp

[

−(ν+Dak^2 )t

]

,

and the corresponding spectral function is a Lorentzian with the line width deter-
mined by

ν+Dak^2 ,ν∼n, Da∼n−^1. (17.18)

The density dependence of the line width (17.18) shows a minimum at an interme-
diate density. Such a minimum, referred to asDicke narrowing, is actually observed
provided that the collisions change the direction of the velocity of a particle more
effectively than its rotational angular momentum. Relation (17.18) does not apply to
lower densities where the Doppler broadening takes over [178, 179].
In general, the diffusional broadening is anisotropic in the sense that the
k-dependent contribution to the line width is different for the VH and HH scat-
tering geometries. The replacement ofDak^2 aμνin the spatial Fourier transformer
equation (17.17)by

Da

(

k^2 aμν+βkμkκaκν

)

,

leads to such an effect [180, 181]. The parameterβcharacterizes the anisotropy of
the effective diffusion coefficient.

17.2 Nonlinear Relaxation, Component Notation

In the absence of a flow and of any orienting torque, (16.149) describes a nonlinear
relaxation process which can be significantly different from the exponential relax-
ation following from a linear equation. The symmetric traceless second rank tensor
has 5 independent components. A convenient choice of components is introduced
next, based on appropriately defined basis tensors.

17.2.1 Second-Rank Basis Tensors

The tensoraμνis decomposed as

aμν=

∑^4

i= 0

aiTμνi , ai=Tλκi aλκ. (17.19)