Eur. Phys. J. Special Topics 224, 781–799 (2015)
https://doi.org/10.1140/epjst/e2015-02427-6

Regular Article

Thermodynamics of the general diffusion process: Equilibrium supercurrent and nonequilibrium driven circulation with dissipation

Department of Applied Mathematics, University of Washington Seattle, WA 98195-3925, USA

Received: 5 October 2014
Revised: 7 May 2015
Published online: 17 July 2015

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)∇φ(x) where ∇⋅ j(x) = 0 and j(x) ⋅∇φ(x) = 0. Stationary processes with such Maxwell-Boltzmann equilibrium has an underlying conservative dynamics ˙x = j(x) ≡ (f^ss(x))⁻¹J^ss(x), and a first integral ϕ(x) ≡ −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 an entire family of diffusion processes with different parameter α can be established via a Helmholtz theorem. For the general diffusion process without the orthogonality, a nonequilibrium cycle emerges, which consists of external driven φ-ascending steps and spontaneous φ-descending movements, alternated with iso-φ motions. The theory presented here provides a rich mathematical narrative for complex mesoscopic dynamics, with contradistinction to an earlier one [H. Qian et al., J. Stat. Phys. 107, 1129 (2002)].

This article is supplemented with comments by H. Ouerdane and a final reply by the author.