Your problem is highly nontrivial. The theoretical tool to be used is the renormalization group, which extracts the relevant dynamics of the large scales of the system. But if we were able to use it "in a blind way", then we would have a technique to study the macroscopic dynamics of any microscopic system... and this would made a lot of my colleagues unemployed :) The basic idea is to make "blocks" or perform a bit of "coarse-graining" in your original system and see if you can describe the resulting dynamics with the same microscopic laws, but changing a bit the parameters. If you can, then you're lucky. You get a "flow" in your parameter space, and the fixed points give you the macroscopical dynamics: how the system will behave in the thermodynamic limit.

The alternative approach, which is used very often, is to try to write the most general local partial differential equation which is compatible with all your physical requirements and symmetries. These equations will have "open" parameters that you will put later on, in a semi-empirical way. You can see examples in A.L. Barabasi and E.H. Stanley, "Fractal concepts in surface growth", and many other places.

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