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 system. Even if a system is in a steady state (invariant in time), Prigogine suggested that there should be a positive entropy production if it is open. In 1979, the first author of this paper and Qian Min-Ping discovered that the entropy production describes the irreversibility of stationary Markov chains, and proved the circulation decomposition formula of the entropy production. They also obtained the entropy production formula for drifted Brownian motions on Euclidean space R ⁿ (see a report without proof in the Proc. 1st World Congr. Bernoulli Soc.). By the topological triviality of R ⁿ , there is no discrete circulation associated to the diffusion processes on $R^n$. In this paper, the entropy production formula for stationary drifted Brownian motions on a compact Riemannian manifold M is proved. Furthermore, the entropy production is decomposed into two parts – in addition to the first part analogous to that of a diffusion process on R ⁿ , some discrete circulations intrinsic to the topology of M appear! The first part is called the hidden circulation and is then explained as the circulation of a lifted process on M×S ¹ around the circle S ¹. The main result of this paper is the circulation decomposition formula which states that the entropy production of a stationary drifted Brownian motion on M is a linear sum of its circulations around the generators of the fundamental group of M and the hidden circulation. Received: 4 November 1998 / Accepted: 7 April 1999