1204.6496 (Hong Qian)

A Decomposition of Irreversible Diffusion Processes Without Detailed
Balance    [PDF]

Hong Qian

As an generalization of deterministic, nonlinear conservative 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 "free energy function" $F[u]=\int u(x,t)\ln\big(u(x,t)/\rho(x)\big)dx$ in the phase space with density flow $u(x,t)$ satisfying $u_t =-\nabla\cdot(ju)$. Any general stochastic diffusion process without detailed balance, in terms of its Fokker-Planck equation, can then be decomposed into a reversible diffusion process with detailed balance and a canonical conserved system. This decomposition can be rigorously established in a functional space with inner product defined as $<\phi,\psi>=\int\rho^{-1}(x)\phi(x)\psi(x)dx$. Furthermore, a law for balancing $F[u]$ can be obtained: The non-positive $dF[u(x,t)]/dt=Q_{hk}(t)-e_p(t)$ where the "source" $Q_{hk}(t)\ge 0$ and the "sink" $e_p(t)\ge 0$ are known as energy pumping and entropy production, respectively. A reversible diffusion has $Q_{hk}=0$. 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.

View original: http://arxiv.org/abs/1204.6496