Considerable progress has been made in understanding elastic depinning, in which we neglect the possibility of phase slip. A classic model is simply to use a ``Langevin'' dynamics, e.g., for the CDW phase,
where I have dropped the thermal noise present at . is just the continuum Hamiltonian in Eq. 172. This is known as the Fukuyama-Lee-Rice equation.
A priori, at T=0, there is no guarantee of reaching a steady state at long times. In purely elastic models of random manifolds and random media, an exact result, due to Alan Middleton, tells us that these models have a unique long-time steady state in the sliding phase (of course, the pinned state is hysteretic even in elastic models). This ``theorem'' breaks down once phase slips are allowed in the model, and more complex behavior including multiple steady states and chaotic dynamics is allowed.
One extremely simple model of CDW depinning can be easily solved, which is in effect the case of d=0, or a single degree of freedom. The equation of motion in this case is
where and . This is in fact the equation describing the dynamics of an overdamped Josephson junction in an external current. Then corresponds to the phase difference between two superconductors across the junction. The force F is proportional to the current, and the voltage .
It is clear that, for , the phase asymptotically reaches a constant value, where
The threshold force, , occurs at the point at which this equation can no longer be satisfied, i.e. . For , the voltage is always non-zero. But we can solve Eq. 221 by separation of variables
taking for simplicity as the initial condition. For , the integral is dominated by the maxima of Y. For long times (large ), there are approximately such maxima, each contributing equally to the integral (since the integrand is periodic). Expanding Y near a maxima, e.g. , we have
The implicit solution is then
with . The exponent is one of several interesting quantities to study more generally at a depinning transition.
Space does not permit me to discuss the full details of the theory of depinning in finite dimensions. The interested reader is encouraged to look at the references at the end of the notes. Instead, I will present some of the scaling arguments that provide a physical picture of depinning. For variety, I do this here for the case of a driven domain wall. Very similar considerations apply to CDWs, and are discussed in depth in Refs.. We can imagine field-cooling a dirty ferromagnet to arrive at a well-ordered state. After turning off the field, we can force the introduction of a domain wall by imposing opposite magnetizations on two ends of the sample. Applying a field at this point imposes a force on the domain wall in the direction that increases the magnetization along the field.
At zero temperature, where no thermal activation is possible, the domain may be stuck in a local minima of the random potential. For small applied forces, only transient motion will result, ending in an stationary configuration for the domain with asymptotically zero velocity. If enough force is applied to overcome all the local pinning forces, the domain will slide with a non-zero mean velocity. Somewhere in between there must be a depinning point, at which the domain's mean velocity goes to zero. We would like to understand as much as possible the approach to this point from above and below, as well as the general behavior of the system in both the pinned and moving ``states''.
Consider for concreteness a dynamical model for a driven interface (an N=1 random manifold) with overdamped dynamics,
where is a kinetic (drag) coefficient and F is an external force. Using the form of the Hamiltonian, we have
where , , and is the quenched random local force. In fact, a systematic analytical treatment can be made for this model, and I encourage you to look at the relevant references. The approach is technically similar to the functional RG approach just discussed for equilibrium systems, though the physics is in many ways quite different. Here I will just discuss the behavior in a phenomenological way and describe the results we can obtain just from scaling.
Let us first consider the behavior at low forces. As the force is increased from zero, the interface will slide up against the potential barriers around the initial state, with only small smooth rearrangements of the configuration. Eventually, however, the applied force will be sufficient to overcome some local pinning force and that region will jump forward into a new metastable configuration. At low forces, this events will be widely separated and typically small. There will, of course, be such events at every force in an infinite sample, but they will be quite far apart. Likewise, certain rare regions of the sample may have anomalously low pinning forces, and these regions will enable very rare large jumps even at low forces. We expect that the ``rare'' events will actually be exponentially unlikely, since, for instance, the probability of finding an area of linear size L with anomalously low pinning forces is roughly
where a is some correlation length for the random potential, and .
Both large and small events may be thought of as avalanches. As we increase the force f, the typical size of avalanches will increase, as will the distance between avalanches. Another effect reinforcing this trend is that smaller avalanches will essentially trigger neighboring areas to jump as well. As we increase f close to , the critical or threshold force, even the typical avalanche size becomes much larger than the correlation length for the disorder. Then we can expect a scaling form for the distribution of avalanche size induced upon increasing the force an infinitesimal amount. Let's denote the probability (per unit volume) of finding an avalanche of size larger than as
where this form applies only for , and is the typical avalanche size. We expect that the scaling function
for large x.
We should also note that these avalanches will typically begin and end in rough configurations of the interface. They can thus be characterized by self-affine scaling, just as in equilibrium. So we should define an ``avalanche roughness'' exponent,
for . Because this roughness is only really defined in the scaling limit where and diverge, it is really a property of the threshold system. The divergence of should also be characterized by a scaling law,
Large avalanches likewise require a long time to move. As we approach threshold from below, any small change of the force thus causes a rather long-lived disturbance. The lifetime of these jumps may be denoted
defining a conventional dynamical critical exponent z.
We seem to have developed a profusion of unknown exponents! Let us take stock for a moment and attempt to derive some relations between them. To do so, consider the response of the interface to an infinitesimal perturbation. Let us imagine adding a small external force to the equation of motion
We may then define a sort of susceptibility for the interface as
We can use the rotational invariance of the system to constrain . To do so, let us specialize to the case where is independent of time. Then we can make the change of variables
This removes from the right-hand-side of Eq. 228, but it re-appears inside the random force
where FT indicates a Fourier transform. As in the equilibrium problem, however, the distribution of is unaffected by this shift, so the average properties of the shifted equation are as if . Thus the only contribution to comes from the constant shift of u, and we have
We thus expect to have the general scaling form
If we now first take , then , we expect to get a finite result, and we have
However, in physical terms, we know that
Equating the two requires
We can also determine the avalanche exponent . Again, this mean susceptibility is the mean change in u as f is increased. For an avalanche of size , this change is if the avalanche contains the point of interest. For a volume of size L, the probability density of such avalanches is
The mean change in u at a point in this volume is then
Integrating this over gives
Doing this integral and comparing gives .
This gives rather a complete picture below threshold. What happens above threshold? Well, as , the avalanches become larger and larger and eventually one giant avalanche brings the system into motion at . At that point, the system begins to slide. We may estimate the velocity above threshold by
Near threshold, the motion is ``jerky'', with large regions sticking for long times. We may reinterpret the correlation lengths and times at these sticking scales and times on this side of the phase transition.
It only remains to determine the two unknown exponents, say z and . These require an analytic field-theoretical RG treatment to obtain. The results are, however,
where . As in equilibrium, the interface becomes flat for d>4.