Suppose, for instance that we want to model the polymer as a self-avoiding walk on a regular lattice. In fact, although this model has no energy terms (except for the exclusion of a lattice site to occupation by other vertices of the walk), it is a good model of the ``universal'' behaviour of a long linear polymer in dilute solution in a good solvent. For this model, one might fix the length of the walk and define the states of the Markov chain as being all self-avoiding walks of this length. Since all such states are equally likely in the model, the Markov chain should have a unique limit distribution which is uniform.

This model can be decorated in various ways. By adding a short range attractive potential one can model the collapse transition in dilute solutions of polymers. By adding a surface with which the walk can interact, one can model polymer adsorption. By considering self-avoiding polygons (walks which return to their first vertex at their last step) one can investigate knotting in ring polymers.

To learn more about Monte Carlo methods, an excellent reference is Hammersley and Handscomb: Monte Carlo methods, published by Methuen in their series of monographs on probability and statistics.

The University of Bristol operates a useful Markov chain Monte Carlo preprint service.