Boundary time crystals through the lens of nonequilibrium thermodynamics: A thermodynamic framework for emergent temporal order

Boundary Time Crystals (BTCs) represent a distinctive form of nonequilibrium temporal order in which persistent oscillations emerge in open dissipative quantum systems without external periodic driving. Although their dynamical characterization is well established through the spectral properties of the Liouvillian, the thermodynamic meaning of persistent temporal order remains less systematically articulated. This work develops a nonequilibrium-thermodynamic framework for interpreting BTCs as emergent dissipative structures sustained by irreversible energy and entropy exchange. The framework connects Liouvillian temporal dynamics with thermodynamic forces and fluxes, entropy production, dissipative organization, and self-organization, thereby providing a common language for describing temporal persistence and irreversible activity within open quantum systems. In addition, phenomenological measures are introduced to characterize the balance between coherent temporal dynamics and dissipative relaxation, together with a normalized entropy-production measure for comparing dynamical regimes. A concrete numerical demonstration based on the established collective-spin BTC model illustrates the framework by contrasting non-BTC and BTC regimes, finite-size temporal decay, and the associated dissipative activity. The results support a thermodynamic interpretation in which persistent temporal order is maintained as a nonequilibrium organized state rather than as a consequence of equilibrium thermodynamics. The proposed framework is complementary to microscopic quantum-thermodynamic treatments of BTCs and does not replace the established Liouvillian criterion for identifying temporal-crystalline order. Instead, it provides a broader conceptual bridge between temporal symmetry breaking, irreversible thermodynamics, and dissipative self-organization, offering a basis for future quantitative studies of the thermodynamic structure and stability of time-crystalline phases.