无细致平衡的不可逆扩散过程的分解
A Decomposition of Irreversible Diffusion Processes Without Detailed Balance
摘要：
作为确定性非线性保守动力系统的推广，引入了关于正的、可微的平稳密度$ρ(x)$的{ \em正则保守动力学}的概念：$\dot{x}=j(x)$其中$\nabla\cdot\big(ρ(x)j(x)\big)=0$ .这种系统在相空间中具有守恒的“广义自由能函数”$F[u]$ $=$ $\int u(x,t)\ln\big(u(x,t)/ρ(x)\big)dx$，密度流$u(x,t)$满足$\partial u_t =-\nabla\cdot(ju)$。任何一般的无细致平衡的随机扩散过程，根据其Fokker-Planck方程，都可以分解为具有细致平衡的可逆扩散过程和正则保守动力学.这种分解可以在内积定义为$\langleϕ,ψ\rangle=\intρ^{-1}(x)ϕ(x)ψ(x)dx$的函数空间中严格建立。此外，还可以得到平衡$F[u]$的定律：非正$dF[u(x,t)]/dt$ $=$ $E_{in}(t)-e_p(t)$其中“源”$E_{in}(t)\ge 0$和“汇”$e_p(t)\ge 0$分别被称为内务热产生和熵产生。可逆扩散具有$E_{in}(t)=0$。对于线性（Ornstein-Uhlenbeck）扩散过程，我们的分解等价于R. Graham和P. Ao的理论，以及大偏差理论。根据同一随机过程时间反演的两种不同表述，讨论了{ \em耗散}和{ \em保守}平稳动力学的含义。
英文摘要：
As a generalization of deterministic, nonlinear conservative dynamical systems, a notion of {\em canonical conservative dynamics} with respect to a positive, differentiable stationary density $ρ(x)$ is introduced: $\dot{x}=j(x)$ in which $\nabla\cdot\big(ρ(x)j(x)\big)=0$. Such systems have a conserved "generalized free energy function" $F[u]$ $=$ $\int u(x,t)\ln\big(u(x,t)/ρ(x)\big)dx$ 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 $\langleϕ,ψ\rangle=\intρ^{-1}(x)ϕ(x)ψ(x)dx$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
