352 17 Tensor Dynamics
defines time-correlation functions. It is assumed that the distribution underlying the
average is stationary, e.g. pertaining to an equilibrium state. Thus the correlation
function just depends on the time difference, i.e. it is independent of the timet 0.
Thent 0 =0 can be chosen. On the other hand, the choicet 0 =−tyields
〈
ψi( 0 )ψj(−t)〉
0 =
〈
ψj(−t)ψi( 0 )〉
0 ,
which implies the symmetry relation
Cij(t)=Cji(−t). (17.2)Consequently the diagonal correlation functions pertaining toi=jare even func-
tions of the time. The correlation functions withi = jarereferredtoasauto-
correlation functions, those withi=jascross-correlation functions. In the follow-
ing, the notationCij(t)is used for the normalized correlation functions which are
defined by
Cij(t)=〈
ψi(t 0 +t)ψj(t 0 )〉
0{
〈ψi(t 0 )ψi(t 0 )〉 0〈
ψj(t 0 )ψj(t 0 )〉
0}− 1 / 2
, (17.3)
wherei= 1 , 2 ,...,j= 1 , 2 ,..., as before. One hasCii( 0 )=1 for the normalized
auto-correlation functions.
Now a non-equilibrium state is considered where〈ψi〉=〈ψi〉(t)=0 applies.
When the distortion which caused the deviation from equilibrium, is switched off,
the quantity〈ψi〉(t)relaxes to 0, in the long time limit. The original derivation of
the Onsager symmetry relation was based on the assumption that fluctuations and
small macroscopic deviations from thermal equilibrium decay alike [108]. With this
argument, time-correlation functions can also be defined via a linear relation between
time dependent averages
〈ψi〉(t)=Cij(t)〈ψj〉( 0 ), i= 1 , 2 ,..., j= 1 , 2 ,... (17.4)This allows the calculation of time-correlation functions from linear macroscopic
equations.
The averages〈ψi〉may depend on the positionrin space〈ψi〉=〈ψi〉(t,r).The
spatial Fourier transform is
〈ψi〉(t|k)=∫
exp[−ik·r]〈ψi〉(t,r)d^3 r,wherekis the relevant wave vector. Applications in spectroscopy and light scattering
involve wave vector dependent time-correlation functions defined by
〈ψi〉(t|k)=Cij(t|k)〈ψj〉( 0 |k), i= 1 , 2 ,..., j= 1 , 2 ,... (17.5)