Thermodynamic Geometry of Relaxation

TL;DR

This paper introduces a thermodynamic geometric framework to analyze relaxation dynamics in complex systems, demonstrating critical slowing down near phase transitions with a van der Waals gas example.

Contribution

It develops a novel geometric measure based on the Rayleigh quotient to describe relaxation, completing the thermodynamic geometry framework.

Findings

Relaxation rate vanishes linearly near critical temperature.

The framework applies to systems with multiple dissipation channels.

Provides a general tool for relaxation analysis in complex systems.

Abstract

While the geometry of equilibrium states and driven non-equilibrium processes is clearly understood, a geometric description for relaxation towards equilibrium is still lacking. Here, we propose a thermo-geometric measure based on the Rayleigh quotient, reformulating relaxation as a fundamental competition between entropic stiffness and frictional dissipation. Taking a van der Waals gas with two dissipation channels as an example, we explicitly demonstrate its relaxation landscape. Particularly, we find that upon approaching the critical temperature $T_c$, the slow-mode relaxation rate vanishes linearly as $\lambda_s \propto (T-T_c)/T_c$, indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.