Thermodynamics of the General Diffusion Process: Equilibrium Supercurrent and Nonequilibrium Driven Circulation with Dissipation

Unbalanced probability circulation, which yields cyclic motions in phase space, is the defining characteristics of a stationary diffusion process without detailed balance. In over-damped soft matter systems, such behavior is a hallmark of the presence of a sustained external driving force accompanied with dissipations. In an under-damped and strongly correlated system, however, cyclic motions are often the consequences of a conservative dynamics. In the present paper, we give a novel interpretation of a class of diffusion processes with stationary circulation in terms of a Maxwell-Boltzmann equilibrium in which cyclic motions are on the level set of stationary probability density function thus non-dissipative, e.g., a supercurrent. This implies an orthogonality between stationary circulation $J^{ss}(x)$ and the gradient of stationary probability density $f^{ss}(x)>0$. A sufficient and necessary condition for the orthogonality is a decomposition of the drift $b(x)=j(x)+ D(x)\nabla\varphi(x)$ where $\nabla\cdot j(x)=0$ and $j(x)$ $\cdot\nabla\varphi(x)=0$. Stationary processes with Maxwell-Boltzmann equilibrium has an underlying conservative dynamics $\dot{x}= j(x)\equiv$ $\big(f^{ss}(x)\big)^{-1}J^{ss}(x)$, and a first integral $\varphi(x)\equiv-\ln f^{ss}(x)=$ const, akin to a Hamiltonian system. At all time, an instantaneous free energy balance equation exists for a given diffusion system; and an extended energy conservation law among a family of diffusion processes with different parameter $\alpha$ can be established via a Helmholtz theorem. For the general diffusion process without the orthogonality, a nonequilibrium cycle emerges, which consists of external driven $\varphi$-ascending steps and spontaneous $\varphi$-descending movements, alternated with iso-$\varphi$ motions. The theory presented here provides a rich mathematical narrative for complex mesoscopic dynamics.