Jianfeng Lu - Duke University, Mathematics In this talk, we will discuss our recent works on ring polymer representation for multi-level quantum system for thermal average calculations. The proposed representation keeps the discreteness of the electronic states: besides position and momentum, each bead in the ring polymer is also characterized by a surface index indicating the electronic energy surface. A path integral molecular dynamics with surface hopping (PIMDSH) dynamics is developed to sample the equilibrium distribution of ring polymer configurational space. The sampling can be accelerated by taking the infinite swapping limit. (Joint work with Zhennan Zhou)