The Mesoscopic Partition Function:A Combined Spatial and Phase-Space Cell Structure

We develop a mesoscopic formulation of equilibrium statistical mechanics based on coarse-grained occupation-number sectors of one-particle phase space. A mesoscopic partition function is constructed by averaging the microscopic Hamiltonian over configurations compatible with a given occupation profile. The construction converges to the canonical Gibbs partition function in the fine-graining limit and remains compatible with interacting many-body systems. Within this framework, thermodynamic extensivity is shown to be equivalent to asymptotic factorisation of the mesoscopic partition function, while residual inter-cell correlations generate subextensive corrections. The resulting formalism provides a mathematically consistent bridge between microscopic Gibbs theory and mesoscopic thermodynamics.