Springer Lecture Notes in Physics 716

322 U. C. Täuber
√
Upon simple rescaling ˜S α → ˜S ′α = ˜S α ˜D/D, S α → S ′α = S
√D/ α ˜D, the
response functional (7.118) recovers its equilibrium form, albeit with modified
nonlinear coupling u → ũ = u ˜D/D. However, the universal asymptotic
properties of these models are governed by the Heisenberg fixed point (7.84),
and the specific value of the (renormalised) coupling, which only serves as
the initial condition for the RG flow, does not matter. In fact, the relaxational
dynamics of the kinetic Ising model with Glauber dynamics (model
A with n = 1) is known to be quite stable against nonequilibrium perturbations
[34, 35], even if these break the Ising Z 2 symmetry [36]. For model J
the above rescaling modifies in a similar manner √ merely the mode-coupling
strength in Eq. (7.99), namely g → ˜g = g ˜D/D [37]. Again, since the dynamic
critical behaviour is governed by the universal fixed point (7.107), thermal
equilibrium becomes effectively restored at criticality. More generally, it
has been established that isotropic detailed balance violations do not affect
the universal properties in other models for critical dynamics that contain
additional conserved variables either: the equilibrium RG fixed points tend to
be asymptotically stable [33].
In systems with conserved order parameter, however, we may in addition
introduce spatially anisotropic violations of Einstein’s relation; for example, in
model B one can allow for anisotropic relaxation −D ∇ 2 →−D ⊥ ∇ 2 ⊥ −D ‖ ∇ 2 ‖ ,
with different rates in two spatial subsectors and concomitantly anisotropic
noise correlations − ˜D ∇ 2 →−˜D ⊥ ∇ 2 ⊥ − ˜D ‖ ∇ 2 ‖. We have thus produced a
truly nonequilibrium situation provided ˜D ⊥ /D ⊥ ≠ ˜D ‖ /D ‖ , which we may
interpret as having effectively coupled the longitudinal and transverse spatial
sectors to heat baths with different temperatures T ⊥