We are now in a position to come back to coarsening in the random magnet. Each domain wall will experience now a combination of curvature and pinning forces. For simplicity, let us first treat the curvature effect as a uniform external force f per unit area. Then we can come back and make this self-consistent later.
If this external force is small, the domain will typically get stuck in a local potential well, the local restoring forces acting to counterbalance the external force. At strictly zero temperature, this would be the whole story. The configuration of the wall would be frozen and history dependent, and equilibrium would never be reached.
At finite temperature, however, thermal activation can eventually overcome the local barriers. It will definitely do so, since the energy in the presence of the external force is unbounded, i.e.
For small forces, the domain must, however, move a large region (in
) over a large distance (in 
) in order to lower the
energy enough to compensate for the change in energy due to disorder.
We therefore expect the motion of the interface to proceed in rather
large and infrequent jumps (see Fig. 5). To determine
the rate at which these jumps occur, we need to estimate the energy
barrier that must be activated over. We suppose a section of linear
size L moves a distance 
, since this is the distance
needed to get a roughly independent sample. Roughly speaking, the
total energy change has two terms,
where the first term is the energy variation due to disorder, and the
second is the energy decrease, 
, due to motion in the direction
of the field. This has a maximum at a scale
with a maximum energy, i.e. a barrier of
where
We have certainly been somewhat cavalier in estimating this barrier,
so the final result for 
should be taken with a grain of salt.
Generally, we would expect a power-law relation such as
Eq. 60, but where may be more non-trivial. In fact,
recent work by several authors suggests that this relation between the
barrier heights and typical sample-to-sample energy variations is
probably correct. Using Eq. 60, we can estimate an
activation time
Figure 5: Creep and coarsening in a random-bond magnet. Domains are
compact on large scales, but motion occurs on much smaller scales,
due to curvature forces. This motion proceeds via activated jumps of
length L and width .
This extremely long time-scale (and consequent slow motion of the
domain) is known as ``creep''. We can use this result to obtain an
estimate for the coarsening scale at time t. Let us now use the
fact that the local force 
, the scale of the curvature of
the domain. Note that the typical size of the jumping regions satisfies
since 
. This means that many jumps must occur before the
curvature changes significantly, and we may therefore indeed treat f
as static on the time scale of a single jump. Therefore we can
imagine the motion of a domain is governed by a scale-dependent
differential equation,
Separating variables, we have
The integral over R is dominated by the largest value of R, so we have
and therefore the coarsening law
This a dramatically slower coarsening than in the pure case! It explains why dirty magnets can fail to reach true equilibrium even on the longest laboratory time scales.
Such logarithmic domain growth has indeed been observed experimentally[7].
You will hear or have heard much more detailed investigations of these issues in the context of aging. Here I have assumed that the coarsening is adequately described by a single time-scale. If the energy barriers involved have even an exponential tail to their distribution, the amplification effect of the activated time-scale can invalidate that assumption and these arguments need to be more carefully reconsidered. The logarithmic growth of domains should, however, be a fairly generic feature.
