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During the course of my first statistical mechanics course we generally concerned ourselves with a bulk amount of our system and considered it in terms of a set of lattice sites that had a state. How do you make the jump from systems like this to ones where you now have geometry concerns. In particular I'm curious about three things in particular:

  1. How would you account for a complex structure in your material? For example, if you have a nano-lattice of a material vs the bulk equivalent.

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  1. How do you account for the geometry of a surface? A simple example would be dissolving a salt cube vs a salt sphere, another the compressing the lattice above, one case being the structure is welded at each point on the surface vs not being welded.

  2. Are there uses of group theory in exploiting geometric symmetries within a material?

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    $\begingroup$ I think there are some good questions in here, but this may be too broad as it stands. Could you refine your question, or make it more specific? $\endgroup$
    – Danu
    Jan 11, 2015 at 21:31
  • $\begingroup$ I don't know very much condensed matter physics, but I can answer part 3 of your question. Check out "Symmetry" by Roy McWeeny. It's a book on finite group theory and its applications. I haven't read the whole thing, but some of the applications involve molecular lattices and crystal optics. It's sophomore math but requires some knowledge of QM. $\endgroup$
    – Ryan Unger
    Jan 12, 2015 at 1:13

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This is only a small part of the answer to your question, but group theory is widely used - point groups for individual molecules and space groups for crystals - used to help describe/make caclulations of electronic structure and vibrations

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  • $\begingroup$ Do you know anything about group theory being used to describe larger scale structures of a material though? $\endgroup$
    – Skyler
    Jan 12, 2015 at 17:36
  • $\begingroup$ I am not familiar with applications of group theory to large scale structures, but I expect it could be used. I am not sure if it is a useful part of large scale calculations or not - I expect that it may be for very regulary structures, but when molecules get to the size of proteins then group theory is not much help because there is not symmetry. A good book about molecular symmetry is by Vincent (first name Alan I think) which is first step to understanding how group theory is applied to small molecules. $\endgroup$
    – tom
    Jan 12, 2015 at 21:24

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