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arXiv:2511.10125 · Journal of Physics A Mathematical and Theoretical

Geometric foundations of thermodynamics in the quantum regime

Álvaro Tejero, Martín de la Rosa

math-phmath.DGquant-phSubmitted Nov 13, 2025• Announced Nov 14, 2025• 0 citations
v1 · Nov 13, 2025v2 · Apr 16, 2026
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Abstract

In this work, we present a comprehensive geometrical formulation of quantum thermodynamics based on contact geometry and principal fiber bundles. The quantum thermodynamic state space is modeled as a contact manifold, with equilibrium Gibbs states forming a Legendrian submanifold that encodes the fundamental thermodynamic relations. A principal fiber bundle over the manifold of density operators distinguishes the quantum state structure from thermodynamic labels: its fibers represent non-equilibrium configurations, and their unique intersections with the equilibrium submanifold enforce thermodynamic consistency. Quasistatic processes correspond to minimizing geodesics under the Bures-Wasserstein metric, leading to minimal dissipation, while the divergence of geodesic length toward rank-deficient states geometrically derives the third law. Non-equilibrium extensions, formulated through pseudo-Riemannian metrics and connections on the principal bundle, introduce curvature-induced holonomy that quantifies irreversibility in cyclic processes. In this framework, the thermodynamic laws in the quantum regime emerge naturally as geometric consequences.

Journal reference
J. Phys. A: Math. Theor. 59 155201 (2026)
DOI
10.1088/1751-8121/ae5c48
Comments
36 pages

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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.