I am currently rereading J. Krug’s paper “Atom Mobility for the Solid-on-Solid Model.” I think I understand more of it than last time, so I will now attempt to discuss the continuum limit for the evolution of a height profile.

Recall, in the solid on solid model, we move from one height profile to another one via atom hops:

,

where the indexing is done modulo .

In the continuum limit, we describe a height profile as . We have a conservation law:

where the is the *mass transport term*. I’m not entirely sure about the origin of this term but, we may define by

where describes *adatom mobility* and is the chemical potential at . While I don’t understand this completely, I believe this says that atoms hop because of differences in chemical potential.

Now as we have pointed out (see e.g. my thesis proposal), we may write the chemical potential , where the quantity has a functional dependence on , the expected slope of the height at time . Explicitly we have

and upon inverting we have:

where

where and is the bond strength.

The time dependence here is because the underlying probability distribution changes in time according to the master equation). Note this is different from Krug, where he defines instead of the expectation. Either way, we may write and hence

Here corresponds to the effective stiffness of the surface, and we denote it by . The inverse of this, , turns out to be the variance of . Therefore, we may write the chemical potential

where I have assumed (but not proved that) . If instead we use Krug’s definition for , we have

Then the conservation law above may be written as:

or with Krug’s definition

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