A decomposition of irreversible diffusion processes without detailed balance

As a generalization of deterministic, nonlinear conservative dynamical systems, a notion of canonical conservative dynamics with respect to a positive, differentiable stationary density rho(x) is introduced: (x)over dot = j(x) in which del.(rho(x)j(x)) = 0. Such systems have a conserved generalized free energy function F[u] = integral u(x, t)ln(u(x, t)/rho(x))dx in phase space with a density flow u(x, t) satisfying partial derivative u(t) = -del.(ju). Any general stochastic diffusion process without detailed balance, in terms of its Fokker-Planck equation, can be decomposed into a reversible diffusion process with detailed balance and a canonical conservative dynamics. This decomposition can be rigorously established in a function space with inner product defined as = integral rho(-1)(x)phi(x)psi(x)dx. Furthermore, a law for balancing F[u] can be obtained: The non-positive dF[u(x, t)]/dt = E-in(t) - e(p)(t) where the source E-in(t) >= 0 and the sink e(p)(t) >= 0 are known as house-keeping heat and entropy production, respectively. A reversible diffusion has E-in(t) = 0. For a linear (Ornstein-Uhlenbeck) diffusion process, our decomposition is equivalent to the previous approaches developed by Graham and Ao, as well as the theory of large deviations. In terms of two different formulations of time reversal for a same stochastic process, the meanings of dissipative and conservative stationary dynamics are discussed. (C) 2013 AIP Publishing LLC.