Off-diagonal long-range order, cycle probabilities, and condensate fraction in the ideal Bose gas

TL;DR

This paper explores the deep connection between cycle probabilities, off-diagonal long-range order, and Bose--Einstein condensation in the ideal Bose gas, providing exact relations and analyzing cycle distributions at various temperatures.

Contribution

It establishes a precise link between cycle probabilities and off-diagonal long-range order, deriving explicit formulas for cycle distributions in the thermodynamic limit.

Findings

Sum of long cycle probabilities equals off-diagonal long-range order

Derived explicit form of cycle probabilities in the thermodynamic limit

Analyzed maximum cycle length distribution at zero and finite temperature

Abstract

We discuss the relationship between the cycle probabilities in the path-integral representation of the ideal Bose gas, off-diagonal long-range order, and Bose--Einstein condensation. Starting from the Landsberg recursion relation for the canonic partition function, we use elementary considerations to show that in a box of size L^3 the sum of the cycle probabilities of length k >> L^2 equals the off-diagonal long-range order parameter in the thermodynamic limit. For arbitrary systems of ideal bosons, the integer derivative of the cycle probabilities is related to the probability of condensing k bosons. We use this relation to derive the precise form of the \pi_k in the thermodynamic limit. We also determine the function \pi_k for arbitrary systems. Furthermore we use the cycle probabilities to compute the probability distribution of the maximum-length cycles both at T=0, where the ideal Bose gas reduces to the study of random permutations, and at finite temperature. We close with comments on the cycle probabilities in interacting Bose gases.