I come from a background in statistical mechanics (not algorithm design or complexity theory), and the following question occurred to me that I could use some expert help in beginning to understand. The question is the following:
Given a statistical mechanics model for which I can find a closed-form for the partition function in the thermodynamic limit, does there exist an efficient algorithm for computing the partition function for a finite size analogue of said problem? By efficient, I mean an algorithm whose run time scales like a polynomial in the system size.
I'm aware that answering this question in generality would be extremely difficult, but my interest in the statement stems from the fact that I can't think of any examples in support of or against it, namely due to my inexperience with complexity theory. For example, we can write down the partition function for the 2D classical Ising model; does there exist an efficient algorithm for obtaining the partition function for the 2D Ising model with $N$ sites? In this case, my intuition says this should be the case, but I'm not sure! For more complicated models, I'm even less sure!
If anyone can offer examples for or counterexamples against this statement to I'd like to hear them, just for my own edification.