Padé approximant approach to singular properties of quantum gases: the ideal cases

Abstract

In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padé approximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.

Energy spectrum of interacting gas: cluster expansion method

“…As can be seen from the experience in Ref. [40], this treatment will extend the method for weak degenerate to a method for strong degenerate.…”

“…In Ref. [40], we suggest a method that converts the virtual expansion method for high-temperature and low-density gases into a method applying to low-temperature and high-density gases. After the Padé approximant treatment, the virtual expansion method can be used to consider the BEC phase transition and calculate the low-temperature properties of Fermi gases.…”

“…Similarly, the Padé approximation method suggested in Ref. [40] can also be used to extend the cluster expansion method to strong degenerate cases. Similar to the virial expansion, the cluster expansion also gives a truncated power series approximation, which is essentially a polynomial approximation.…”

Eliminating oscillation in partial sum approximation of periodic function

“…In this appendix, we will construct a modified Padé approximant to solve this problem. The Padé approximant is to use a rational function instead of the power series to approximate a function [1]. Since the common Padé approximant is of low efficiency in this case, in this appendix, we introduce a generalized Padé approximant to approximate the series (5).…”

High precision solutions to quantized vortices within Gross–Pitaevskii equation

“…The basic idea of Padé approximants is to construct a rational function which can reproduce the Taylor expansion series of the target function within a given order. The rational function can mimic the singularity of the target function since it contains pole structure, so the convergence radius of the original Taylor expansion can be extended (see an example in [34]). If the target function is constrained by two boundary conditions, the rational function should reproduce two Taylor expansion series simultaneously, which is called two-point Padé approximants [35,36].…”