Scaling approach to existence of long cycles in Casimir boxes

TL;DR

This paper explores the relationship between generalized Bose-Einstein condensation, long cycles, and off-diagonal-long-range-order in Casimir boxes, establishing a hierarchy based on scaling and coherence length.

Contribution

It introduces a scaling approach linking g-BEC types to long cycles and ODLRO hierarchies in anisotropic Casimir boxes, extending understanding beyond cubic geometries.

Findings

Equivalence of g-BEC, long cycles, and ODLRO in cubic boxes.

Hierarchy of long cycles depending on size scale.

Hierarchy of ODLRO based on condensate coherence length.

Abstract

We analyse the concept of generalized Bose-Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in the Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the Off-Diagonal-Long-Range-Order (ODLRO), which are usually considered as some adequate way to describe the standard BEC on the ground state for the cubic boxes. First we show that these three criterions are equivalent in this latter case. Then, basing on a scaling approach, we revise formu- lation of these concepts to prove that the classification of the g-BEC into three types I,II,III, implies a hierarchy of long cycles (depending on their size scale) as well as a hierarchy of ODLRO which depends on the coherence length of the condensate.