Vorticity and level-set variations of invariant current bound steady-state dissipation

TL;DR

This paper investigates how the geometry of invariant currents influences entropy production in non-equilibrium steady states, identifying key descriptors that bound dissipation and analyzing their behavior in different regimes.

Contribution

It introduces two minimal descriptors, vorticity and current variation, that bound entropy production, advancing understanding of geometric effects on dissipation in non-equilibrium systems.

Findings

Vorticity and current variation jointly bound entropy production.

In near-Gaussian regions, vorticity dominates the bound.

Low-noise limit highlights non-potential drift contributions.

Abstract

A non-vanishing entropy production rate is a hallmark of non-equilibrium stationary states and is therefore at the heart of non-equilibrium thermodynamics. It is a manifestation of a steady circulation $J_{\rm inv}$ along the level sets of the invariant density $\rho_{\rm inv}$, and is thus generically used to quantify how far a steady system is driven out of equilibrium. While it is well known that there exists a continuum of distinct steady states with the same invariant measure, the question how the geometry and topology of the invariant current a priori affect dissipation remained elusive. For confined irreversible diffusions we identify two minimal descriptors, the $\rho_{\rm inv}$-weighted vorticity and the variation of $J_{\rm inv}$ along level sets of $\rho_{\rm inv}$, and prove that these jointly bound from above the steady-state entropy production rate. In regions where $\rho_{\rm inv}$ is close to Gaussian the bound is dominated solely by the vorticity of the drift field and in the low-noise (Freidlin-Wentzel) limit by any non-potential contribution to the drift, rendering $J_{\rm inv}$ virtually constant along the level sets of $\rho_{\rm inv}$.