Furthermore, any dynamics problem is also sensitive to conservation
laws, which play no role in equilibrium. We must therefore study the
dynamics separately in the presence or absence of a conserved density.
Consider first the case in which no conservation law applies. This is
appropriate for certain CDW phases. A well studied example is the
higher-temperature CDW in NBSe 
. This material has
three chains in its unit cell. In a temperature range of 59K to 144K,
only one of the three chains forms a CDW, and the others provide
itinerant metallic carriers which coexist with it. Charge
can thus be exchanged back and forth between the CDW and the Fermi
sea.
Provided thermal effects can be neglected (this is far from obvious in real materials, but seems to be a good approximation in some instances), an appropriate model would appear to be simply a driven random-field XY (RFXY) model, i.e.
where F is a uniform force, and H is the RFXY Hamiltonian,
Here the angular brackets denote a sum over nearest neighbors, K is
a stiffness constant, and 
and 
are the magnitude and
direction of the random field at site i. Because this lattice form
involves only periodic functions, local phase slips are allowed. We
will also need the expression for the current, which is
An argument due to Coppersmith and Millis tells us that once phase
slips are allowed, this depinning transition is actually washed out
even at zero temperature[16]. Let us suppose
that the system is below threshold, 
. Then at long times, the
system is static, and the net force on any region of the sample must
be zero. This net force is the sum of the external force, the
interaction forces, and the random forces acting on the region, i.e.
where 
represents a compact region of the sample.
Because the interaction forces act pairwise, the elastic forces in the
interior of the region cancel. We thus have
where 
denotes the set of boundary spins. The
external elastic force on boundary spin i is
where the prime on the sum indicates that it is taken only over
nearest neighbor sites not contained in . Clearly,
is bounded,
One can now easily argue that this force balance cannot be satisfied
everywhere in an infinite sample. Suppose we divide space up into
regions (say cubes) of linear size L. There is some small, but
finite probability that mean (spatially averaged) magnitude of the
random fields in each region is less than F, no matter how small F
is. If this is the case, the local random fields are not strong
enough to balance the external force alone. The only hope is that the
boundary interactions can complete the balance. But since this
contribution only grows with the area 
of the regions,
it cannot do so. Thus in some finite fraction of these regions, force
balance is impossible, and some phase slip must occur.
Because it occurs in a finite volume fraction of the system, the
spatially averaged velocity must be non-zero for any F. In other
words, there is no threshold field below which the CDW is completely
pinned. The functional form of 
is, however,
expected to be highly non-analytic, because it depends upons these
extremely rare weakly pinned regions of the sample.