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arXiv:2604.15000

Thermodynamic Geometry of Relaxation

Hao Wang, Li Zhao, Shuai Deng, Yu-Han Ma

cond-mat.stat-mechphysics.class-phquant-phSubmitted Apr 16, 2026• Announced Apr 17, 2026
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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 $λ_s \propto (T-T_c)/T_c$, indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.

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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 TcT_cTc​, the slow-mode relaxation rate vanishes linearly as λs∝(T−Tc)/Tcλ_s \propto (T-T_c)/T_cλs​∝(T−Tc​)/Tc​, indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.
The second law does more than set the direction of thermodynamic evolution: it endows nonequilibrium transformations with an underlying geometry. In this work, we provide a unified geometric description of entropy production in classical and quantum thermodynamics based on Wasserstein-2 structures arising from gradient flows of free energy. We review how relaxation to equilibrium, in overdamped diffusions, discrete detailed-balanced Markov chains, and dissipative Lindblad dynamics, can be formulated as a gradient flow on the space of states. The associated Wasserstein-2 distance bounds entropy production, yielding a finite-time refinement of the second law. We extend this framework beyond purely dissipative dynamics by introducing generalized Wasserstein-2 metrics that incorporate conservative (Hamiltonian) dynamics in both classical inertial systems and open quantum systems, yielding intrinsic distances that exactly characterize minimal entropy production under fixed dissipative mobilities. We establish equivalence bounds between purely dissipative and Hamiltonian-dissipative geometries, explicitly quantifying how inertial or coherent dynamics can reduce dissipation. Finally, when restricted to equilibrium distributions, we recover the thermodynamic length of linear response-including the quantum thermodynamic length-thereby linking optimal transport, thermodynamic length, and counterdiabatic protocols within a single geometric framework. All in all, our results extend the Riemannian program of thermodynamics further from equilibrium and provide a geometric foundation for optimal protocols beyond the overdamped setting.
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Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we reformulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and β--athermality.
Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we reformulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2 -dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and β --athermality.
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Spin relaxation is conventionally described by two independent phenomenological rates - longitudinal (R1R_1R1​) and transverse (R2R_2R2​) - whose separation obscures a deeper structural unity. Here we develop a geometric framework in which dissipation is represented by a single covariant relaxation tensor acting in Liouville space, from which R1R_1R1​ and R2R_2R2​ emerge as complementary projections. This tensor structure is not merely formal but is experimentally accessible through pulse sequences that probe noncommuting directions in spin space. Using hyperpolarized 13C^{13}C13C spins in diamond with nitrogen-vacancy centers, we show that commuting pulse trains yield effective relaxation matrices that are approximately diagonal, while noncommuting sequences produce off-diagonal components that vary with transmitter frequency offset and pulse ordering, providing evidence that relaxation is a directional process governed by a tensor rather than a pair of scalar rates. Complementary measurements of geometric phase demonstrate that noncommuting dynamics introduce ordering-dependent effects that are separable from dissipation, consistent with the interpretation of relaxation as geometric transport on the state manifold. This framework unifies Bloch, Redfield, and Lindblad descriptions within a coordinate-independent formulation and provides a natural language for relaxation in driven, anisotropic, and non-equilibrium spin systems.