Quantum statistics with bounded occupancies

Abstract We propose a generalized quantum statistical framework, termed Max–Min statistics, which imposes simultaneous lower ( N min ) and upper ( N max ) bounds on the occupation numbers of single-particle quantum levels. The formalism is developed using two complementary but equivalent approaches: a combinatorial microstate-counting scheme based on modified statistical weights, and an operator-algebraic construction employing a generalized number operator with constrained ladder operations. The resulting distribution function provides a unified description that continuously interpolates between Bose–Einstein, Fermi–Dirac, Gentile, and Ewkon statistics as limiting cases. Thermodynamic consequences of the occupation constraints are examined within the grand canonical ensemble, and the geometric properties of the corresponding thermodynamic state space are analyzed using Ruppeiner geometry. The scalar curvature is shown to vary systematically with the allowable occupation range Δ N = N max − N min , reflecting changes in statistical correlations induced by the imposed bounds. In particular, non-zero lower occupancy constraints lead to distinct thermodynamic behaviors that require regularization via an energy cutoff. These results establish the Max–Min framework as a flexible and mathematically consistent tool for exploring bounded and intermediate quantum statistics from a formal and phenomenological perspective.