In this paper, we extend Energy Structure Theory (EST) to a time-periodic isolated thermodynamic system consisting of two interacting subsystems and derive the necessary and sufficient thermodynamic conditions for the emergence of macroscopic time crystals. The total energy of the system is conserved, Uₜotal=α, where one activated component is taken as the independent variable and the other depends on it according to uB (t) =α-uA (t). The independent component is assumed to vary periodically in time, uA (t) =u₀+Asinωt. Based on the energy-structure formulation, a quasi-statistical entropy is defined as δS_ (q-s) (t) =KMS δln (U ̇A (t) ), where KMS is a constant and U ̇A (t) is the rate of energy exchange between the subsystems. When this rate remains constant, U ̇A (t) const. , the quasi-statistical entropy is invariant, indicating a completely reversible thermodynamic oscillation-a time-crystalline state in macroscopic form. Analytical results demonstrate that such periodic dynamics are fully consistent with both energy conservation and the second law of thermodynamics, thus establishing a unified, self-consistent macroscopic framework for time crystals. To evaluate stability, a small irreversibility is introduced by adding a rate-dependent energy term to the EST equations. Under this perturbation, the time-crystal structure does not collapse; instead, it transitions into a quasi-periodic regime, revealing inherent robustness under weak non-equilibrium conditions. Finally, potential experimental realizations are discussed-particularly micro-electromechanical systems (MEMS) with tunable coupling and thermally insulated cyclic heat engines capable of simulating quasi-steady oscillatory behavior. These findings provide a unified thermodynamic foundation for macroscopic time crystals and offer clear pathways for experimental validation and further theoretical generalization.
Shahsavari et al. (Sun,) studied this question.