A Decomposition of Irreversible Diffusion Processes Without Detailed Balance

As a generalization of deterministic, nonlinear conservative dynamical systems, a notion of {\em canonical conservative dynamics} with respect to a positive, differentiable stationary density $\rho(x)$ is introduced: $\dot{x}=j(x)$ in which $\nabla\cdot\big(\rho(x)j(x)\big)=0$. Such systems have a conserved "generalized free energy function" $F[u]$ $=$ $\int u(x,t)\ln\big(u(x,t)/\rho(x)\big)dx$ in phase space with a density flow $u(x,t)$ satisfying $\partial u_t =-\nabla\cdot(ju)$. 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 $\langle\phi,\psi\rangle=\int\rho^{-1}(x)\phi(x)\psi(x)dx$. 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" $E_{in}(t)\ge 0$ and the "sink" $e_p(t)\ge 0$ 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 R. Graham and P. 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 {\em dissipative} and {\em conservative} stationary dynamics are discussed.