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Path integrals on classical statistical mechanics

Physics Asked on August 10, 2021

I’m learning a little bit about path integrals by myself lately and notice something quick curious. So far, I’ve learned that path integrals have many applications in physics, including quantum mechanics, quantum field theory, condensed matter physics and so on. Also, I commonly see discussions on how path integrals also arise in statistical mechanics and how statistical mechanics and quantum field theory have so many things in common because of such integrals. However, statistical mechanics is a very wide area and I’ve realized that every reference I know which discusses the applications of path integrals to statistical mechanics actually does it by using to quantum statistical mechanics rather than classic. To me, this is somewhat deceptive, since when I think about ‘statistical mechanics’, the first thing that pops to my mind is classical statistical mechanics, nor the quantum one.

Question: Does path integral applications to statistical mechanics are actually applicable only to quantum statistical mechanics or can we use it to study classical models too? If we can use it to studey classical models, how does it arise in the theory?

NOTE: I know path integrals are also applicable to statistical field theory, which I think can be considered like a classical theory. However, I think this theory studies fields rather than particles, so there are infinite degrees of freedom and so on. My question is mainly concerned in classical statistical mechanics of particles.

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