Geometric foundations of thermodynamics in the quantum regime

TL;DR

This paper develops a geometric framework for quantum thermodynamics using contact geometry and fiber bundles, linking thermodynamic states, processes, and laws to geometric structures.

Contribution

It introduces a novel geometric formulation of quantum thermodynamics, connecting state space, equilibrium, and irreversibility through advanced differential geometry.

Findings

Quantum state space modeled as a contact manifold.

Geodesics under Bures-Wasserstein metric relate to quasistatic processes.

Geometric structures explain the unattainability principle and irreversibility.

Abstract

In this work, we present a 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 unattainability aspect of the third law. Non-equilibrium extensions, formulated through pseudo-Riemannian metrics and connections on the principal bundle, introduce curvature-induced holonomy that quantifies a geometric source of irreversibility in cyclic processes. In this framework, the thermodynamic laws in the quantum regime emerge naturally as geometric consequences.