Springer Lecture Notes in Physics 716

320 U. C. Täuber
7.1.7 Critical Relaxation, Initial Slip, and Ageing
We begin with a brief discussion of the coarsening dynamics of systems described
by model A/B kinetics that are rapidly quenched from a disordered
state at T ≫ T c to the critical point T ≈ T c [27, 28]. The situation may be
modeled as a relaxation from Gaussian random initial conditions, i.e., the
probability distribution for the order parameter at t = 0 can be taken as
(
P[S, t =0]∝ e −H0[S] =exp − ∆ ∫
d d x ∑ )
[S α (x, 0) − a α (x)] 2 , (7.108)
2
α
where the functions a α (x) specify the most likely initial configurations. Power
counting for the parameter ∆ gives [∆] =µ 2 , whence it is a relevant perturbation
that will flow to ∆ →∞under the RG. Asymptotically, therefore, the
system will be governed by sharp Dirichlet boundary conditions. Whereas the
response propagators remains a causal function of the time difference between
applied perturbation and effect, G 0 (q,t−t ′ )=Θ(t−t ′ )e −Dqa (r+q 2 )(t−t ′) ,see
Eq. (7.23), time translation invariance is broken by the initial state in the
Dirichlet correlator of the Gaussian model,
C D (q; t, t ′ )= 1 (
r + q 2 e −Dqa (r+q 2 ) |t−t ′| ) − e −Dqa (r+q 2 )(t+t ′ )
. (7.109)
Away from criticality, i.e., for r>0andq ≠ 0, temporal correlations decay
exponentially fast, and the system quickly approaches the stationary equilibrium
state. However, as T → T c , the equilibration time diverges according to
t c ∼|τ| −zν →∞, and the system never reaches thermal equilibrium. Twotime
correlation functions will then depend on both times separately, in a
specific manner to be addressed below, a phenomenon termed critical “ageing”
(for more details, see Refs. [29, 30]).
The field-theoretic treatment of the model A/B dynamical action (7.35),
(7.36) with the initial term (7.108) follows the theory of boundary critical
phenomena [31]. However, it turns out that additional singularities on the
temporal “surface” at t + t ′ = 0 appear only for model A, and can be incorporated
into a single new renormalisation factor; to one-loop order, one
finds [27, 28]
a =0 : ˜Sα R (x, 0) = (Z 0 Z ˜S) 1/2 ˜Sα (x, 0) , Z 0 =1− n +2
6
u R
ɛ
. (7.110)
This in turn leads to a single independent critical exponent associated with
the initial time relaxation, the initial slip exponent, which becomes for the
purely relaxational models A and B with nonconserved and conserved order
parameter:
a =0 : θ = γ∗ 0
2 z = n +2
4(n +8) ɛ + O(ɛ2 ) , a =2 : θ =0. (7.111)