Microscopic inclusion statistics in a discrete one-body spectrum

Area of Science:

• Quantum mechanics
• Statistical mechanics

Background:

• Exclusion statistics governs particle behavior, preventing simultaneous occupation of quantum states.
• Bosonic statistics allows multiple particles in the same state, but with standard statistical weights.

Purpose of the Study:

• To formulate a microscopic theory of inclusion statistics.
• To explore the statistical properties of quantum states under inclusion statistics.
• To investigate the implications for a quantum gas in a harmonic potential.

Main Methods:

• Derivation of microscopic occupation multiplicities for one-body quantum states.
• Factorization of multiplicities into clusters of neighboring occupied states.
• Application to a one-dimensional quantum gas in a harmonic trap.

Main Results:

• Inclusion statistics demonstrates a tendency for particles to coalesce.
• Occupation multiplicities exhibit enhanced statistical weights for clustered states.
• A Calogero-like n-body inclusion spectrum emerges for the quantum gas.

Conclusions:

• Inclusion statistics provides a novel framework contrasting with exclusion principles.
• The derived properties offer new insights into quantum particle interactions and correlations.
• The Calogero-like spectrum suggests potential for new physical phenomena in confined quantum systems.