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 $λ_s \propto (T-T_c)/T_c$ , indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.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 $λ_s \propto (T-T_c)/T_c$, indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.

cond-mat.stat-mechCross-listedarXiv:2604.15000Apr 17, 2026

Relaxation to Thermodynamical Equilibrium from Simulations of Quantum Dynamics

Barbara Fresch

Feb 26, 20100 citations

Nonlinear Dynamical Equation for Irreversible, Steepest-Entropy-Ascent Relaxation to Stable Equilibrium

Gian Paolo Beretta, Guillaume Adenier, Andrei Yu. Khrennikov +3 more

We discuss the structure and main features of the nonlinear evolution equation proposed by this author as the fundamental dynamical law within the framework of Quantum Thermodynamics. The nonlinear equation generates a dynamical group providing a unique deterministic description of irreversible, conservative relaxation towards equilibrium from any non-equilibrium state, and satisfies a very restrictive stability requirement equivalent to Hatsopoulos-Keenan statement of the second law of thermodynamics. Here, we emphasize its mathematical structure and its applicability also within other contexts, such as Classical and Quantum Statistical Mechanics, and Information Theory.We discuss the structure and main features of the nonlinear evolution equation proposed by this author as the fundamental dynamical law within the framework of Quantum Thermodynamics. The nonlinear equation generates a dynamical group providing a unique deterministic description of irreversible, conservative relaxation towards equilibrium from any non-equilibrium state, and satisfies a very restrictive stability requirement equivalent to Hatsopoulos-Keenan statement of the second law of thermodynamics. Here, we emphasize its mathematical structure and its applicability also within other contexts, such as Classical and Quantum Statistical Mechanics, and Information Theory.

arXiv:0710.2061AIP conference proceedingsJan 1, 20071 citation

Approach to Thermodynamic Equilibrium (and other Stationary States)

Willis E. Lamb

Springer eBooksJan 1, 19730 citations

Approach to Thermodynamic Equilibrium (And Other Stationary States)

Willis E. Lamb

NATO ASI series. Series B : PhysicsJan 1, 19731 citation

T_c

, the slow-mode relaxation rate vanishes linearly as

λ_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.