The entropy production of diffusion processes on manifolds and its circulation decompositions

In non-equilibrium statistical mechanics, the entropy production is used to describe flowing in or pumping out of the entropy of a time-dependent syst em. Even if a system is in a steady state (invariant in time), Prigogine su ggested that there should be a positive entropy production if it is open. I n 1979, the first author of this paper and Qian Min-Ping discovered that th e entropy production describes the irreversibility of stationary Markov cha ins, and proved the circulation decomposition formula of the entropy produc tion. They also obtained the entropy production formula for drifted Brownia n motions on Euclidean space R-n (see a report without proof in the Proc. I st World Congr. Bernoulli Sec.). By the topological triviality of R-n, ther e is no discrete circulation associated to the diffusion processes on R-n. In this paper, the entropy production formula for stationary drifted Browni an motions on a compact Riemannian manifold M is proved. Furthermore, the e ntropy production is decomposed into two parts - in addition to the first p art analogous to that of a diffusion process on R-n, some discrete circulat ions intrinsic to the topology of M appear! The first part is called the hi dden circulation and is then explained as the circulation of a lifted proce ss on M x S-1 around the circle S-1 The main result of this paper is the ci rculation decomposition formula which states that the entropy production of a stationary drifted Brownian motion on M is a linear sum of its circulati ons around the generators of the fundamental group of M and the hidden circ ulation.