Abstract: Consider any finite state Markov chain. With the help of graph theory and algebraic topology, we construct its minimal lifting with global potential. Or equivalently, we can embed a finite Markov chain into n-torus, such that a closed path is homotopy trivial if and only if its potential change is zero. For the lifted Markov chain, we prove that its instantaneous entropy production rate will converge to that of the original Markov chain in Cesaro's sense. Most of the above results are also valid for diffusion processes on n-torus.