Canonical and Grand-Canonical Singular Ensembles within a Thermodynamicized Gravity Framework

Abstract

This paper develops a gravitational-thermodynamic interpretation of two ensemble structures with singular behavior, denoted as canonical ensemble A and grand canonical ensemble B. Ensemble A is modeled as a stellar-type system in which energy plays the dominant thermodynamic role under an effectively fixed particle-number condition, whereas ensemble B is modeled as a galactic-type open system in which both energy and particle number participate nontrivially and relativistic effects become essential. Within this framework, the singular structures of the two ensembles are treated in a unified manner by contour integration and residue analysis. For ensemble A, the dominant contribution is associated with the mass-energy relation and the corresponding energy-driven singular sector. For ensemble B, the coupled influence of mass-energy equivalence and the invariance of the speed of light leads to a more intricate singular configuration in which particle exchange and relativistic kinematics must be incorporated simultaneously. We show that the gravitational-thermodynamic formulation provides a natural bridge between local singular behavior and global thermodynamic observables, allowing the two ensembles to be compared within a common variational and geometric language. The resulting description clarifies why the canonical sector is more suitable for closed or quasi-closed astrophysical systems, while the grand canonical sector is more appropriate for open large-scale structures with appreciable matter and energy exchange. This approach offers a mathematically coherent route for extending ensemble theory toward self-gravitating systems and suggests a broader singularity-based methodology for relativistic thermodynamics.