Springer Lecture Notes in Physics 716

300 U. C. Täuber
simultaneously by setting D → D (i∇) a , where a =0anda = 2 respectively
represent the nonconserved and conserved cases. Explicitly, we thus obtain
∂S α (x,t)
∂t
= −D (i∇) a δH[S]
δS α (x,t) + ζα (x,t)
= −D (i∇) a[ r − ∇ 2 + u ∑
[S β (x)] 2] S α (x,t)
6
β
+D (i∇) a h α (x,t)+ζ α (x,t) , (7.18)
with
〈
ζ α (x,t) ζ β (x ′ ,t ′ ) 〉 =2k B TD(i∇) a δ(x − x ′ ) δ(t − t ′ ) δ αβ . (7.19)
Notice already that the presence or absence of a conservation law for the order
parameter implies different dynamics for systems described by identical static
behaviour. Before proceeding with the analysis of the relaxational models,
we remark that in general there may exist additional reversible contributions
to the systematic forces F α [S], see Subsect. 7.1.6, and/or dynamical modecouplings
to additional conserved, slow fields, which effect further splitting
into several distinct dynamic universality classes [6, 7, 13].
Let us now evaluate the dynamic response and correlation functions in the
Gaussian (mean-field) approximation in the high-temperature phase. To this
end, we set u = 0 and thus discard the nonlinear terms in the Hamiltonian
(7.7) as well as in Eq. (7.18). The ensuing Langevin equation becomes linear in
the fields S α , and is therefore readily solved by means of Fourier transforms.
Straightforward algebra and regrouping some terms yields
[
−iω + Dq
a ( r + q 2)] S α (q,ω)=Dq a h α (q,ω)+ζ α (q,ω) . (7.20)
With 〈ζ α (q,ω)〉 = 0, this gives immediately
χ αβ
(q,ω)〉
0 (q,ω)=∂〈Sα ∂h β (q,ω) ∣ = Dq a G 0 (q,ω) δ αβ , (7.21)
h=0
with the response propagator
G 0 (q,ω)= [ −iω + Dq a (r + q 2 ) ] −1
. (7.22)
As is readily established by means of the residue theorem, its Fourier backtransform
in time obeys causality,
G 0 (q,t)=Θ(t)e −Dqa (r+q 2 ) t . (7.23)
Setting h α = 0, and with the noise correlator (7.19) in Fourier space
〈
ζ α (q,ω) ζ β (q ′ ,ω ′ ) 〉 =2k B TDq a (2π) d+1 δ(q + q ′ ) δ(ω + ω ′ ) δ αβ , (7.24)