The Ising and Hubbard models are really clever physical approximations from statistical physics. The Wikipedia page on the Ising model is extensive and every physicist should preform the 1D solution at some point in their life, and for others just taking a look at these models and considering their application to the social sciences is worth considering. Anyways onto Hip-Hop Physics from the

*American Scientist*(by Brian Hayes who also runs the bit-player) which discusses the challenges inherent in solving even these simple mathematical models (I first saw this link on 3QuarksDaily and have shamelessly reproduced it here, but good science writing must be propogated):

Mathematical models and computer simulations usually begin as aids to understanding, introduced when some aspect of natural science proves too knotty for direct analysis. Facing an intractable problem, we strip away all the messy details of the real world and build a toy universe, one simple enough that we can hope to master it. Often, though, even the dumbed-down model defies exact solution or accurate computation. Then the model itself becomes an object of scientific inquiry—a puzzle to be solved.Read on...

A good example is the Ising model in solid-state physics, which attempts to explain the nature of magnetism in materials such as iron. (I wrote about the Ising model in an earlierComputing Sciencecolumn; see “The World in a Spin,” September–October 2000.) The Ising model glosses over all the intricacies of atomic structure, representing a magnet as a simple array of electron “spins” on a plain, gridlike lattice. Even in this abstract form, however, the model presents serious challenges. Only a two-dimensional version has been solved exactly; for the three- dimensional model, getting accurate results requires both algorithmic sophistication and major computer power.

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