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

TL;DR

This paper offers a thermodynamic interpretation of diffusion processes with stationary circulation, distinguishing between conservative supercurrents at equilibrium and driven nonequilibrium cycles, and introduces a new energy conservation framework.

Contribution

It introduces a novel interpretation of stationary circulation in diffusion processes as a Maxwell-Boltzmann equilibrium with non-dissipative supercurrents, linking conservative dynamics to thermodynamic principles.

Findings

Stationary circulation can be orthogonal to the probability density gradient, indicating non-dissipative supercurrents.

A decomposition of the drift term characterizes equilibrium diffusion processes with conservative dynamics.

A nonequilibrium cycle involves external driving and spontaneous movements, extending thermodynamic concepts to complex mesoscopic systems.

Abstract

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.