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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Further aspects of an interpolative quantum statistics

R. Ramanathan

Published Aug 1, 1993• 6 citations
Publisher

Abstract

Further aspects of an interpolative quantum statistics are presented. It is shown, by imposing a positivity condition on the distribution function, that this statistics subsumes Gentile's intermediate statistics and is far richer than Gentile's statistics, apart from being a genuine quantum statistics.
DOI
10.1103/physreve.48.843

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A review of the interpolative quantum statistics is presented. It is shown, by imposing a positivity condition on the distribution function, that this statistics subsumes Gentile's intermediate statistics and is far richer than Gentile's statistics, apart from being a genuine quantum statistics. It is also capable of accounting for the possible very small violations of statistics in physical processes. It is finally pointed out that this statistics offers an elegant and useful parametrization of Haldane's 'statistical interaction', while retaining all the interesting features of the latter. Some of the statistical mechanics that these distribution functions lead to are also examined.
Comparison of statistical models (experiments) is an important branch of mathematical statistics, which gives deep insights in many aspects of foundation of statistics. So far, there are two quantum versions of the concept: Comparison with respect to classical tasks and full quantum tasks. In the latter version, a quantum statistical model is more informative than another if and only if a trace preserving positive map sends the former to the latter. On the other hand, in the former version, the existence of a trace preserving positive map sending the former to the latter is a useful sufficient condition. A natural question is whether this is necessary or not. We resolve this question negatively by giving a counter example.
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.