[Paper Review] Universal behaviour of ideal and interacting quantum gases in two dimensions

Dragoş-Victor Anghel|arXiv (Cornell University)|May 4, 2001

Cold Atom Physics and Bose-Einstein Condensates42 references18 citations

TL;DR

This paper establishes the universal thermodynamic equivalence of ideal and interacting quantum gases in two dimensions, showing that systems with constant density of states exhibit identical entropy regardless of exclusion statistics or interactions due to a one-to-one correspondence between excited states. The key contribution is a novel transformation method between different statistics systems, valid beyond bosonisation, with implications for Bose-Einstein condensation and mean-field approximations in 2D quantum gases.

ABSTRACT

I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called {\em thermodynamically equivalent} and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonisation technique, to transform between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In Appendix A I calculate the fluctuation of the ground state population of a condensed Bose gas in grandcanonical ensemble and mean field approximation, while in Appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.

Motivation & Objective

• To establish the thermodynamic equivalence of ideal and interacting quantum gases in two dimensions with constant density of states.
• To explain the universal behavior of entropy in such systems, independent of microscopic statistics or interactions.
• To develop a transformation method between systems of different fractional exclusion statistics based on excited state correspondence.
• To analyze the breakdown of thermodynamic equivalence at Bose-Einstein condensation in mean-field approximations.
• To clarify the limitations of standard statistical distribution techniques in systems with fractional exclusion statistics.

Proposed method

• Derives unified thermodynamic expressions for 2D quantum gases using polylogarithmic functions, enabling compact formulation across statistics.
• Identifies a one-to-one correspondence between excited states of systems with different fractional exclusion statistics (FES), explaining their thermodynamic equivalence.
• Applies the mean field approximation (MFA) and its modified version (MFA’) to map interacting systems to ideal FES gases with effective statistics parameter α.
• Uses grand canonical ensemble and statistical mechanics to calculate ground state population fluctuations in Bose gases under MFA.
• Analyzes the critical temperature for condensation by examining the behavior of the partition function and particle number distribution.
• Demonstrates that condensation breaks thermodynamic equivalence between MFA and MFA’ models, even when the systems appear similar at high temperatures.

Experimental results

Research questions

• RQ1 Why do ideal and interacting 2D quantum gases with constant density of states exhibit identical thermodynamic behavior regardless of statistics or interaction?
• RQ2 What is the microscopic origin of thermodynamic equivalence in systems with different fractional exclusion statistics?
• RQ3 How does the one-to-one correspondence between excited states enable a transformation between systems of different statistics?
• RQ4 Under what conditions does Bose-Einstein condensation break the thermodynamic equivalence between MFA and MFA’ models?
• RQ5 Why do standard statistical distribution techniques fail in systems with fractional exclusion statistics, as revealed by rigorous single-particle state population analysis?

Key findings

• Systems with constant density of states exhibit identical entropy as a function of temperature, volume, and particle number, regardless of exclusion statistics or interaction, establishing thermodynamic equivalence.
• The microscopic reason for this equivalence is a one-to-one correspondence between the excited states of different systems, providing a new transformation method between FES systems distinct from bosonisation.
• The mean field approximation (MFA) for interacting 2D gases can be mapped to an ideal FES gas with statistics parameter α determined by interaction strength, preserving thermodynamic properties.
• In the canonical ensemble, the relative fluctuation of the ground state population vanishes in the thermodynamic limit, validating the use of grand canonical averages even for finite systems.
• Condensation occurs at a finite critical temperature $ T_{ m c} eq 0 $, below which the thermodynamic equivalence between MFA and MFA’ models breaks down.
• The standard method for calculating the most probable distribution fails in systems with fractional exclusion statistics, as rigorous analysis reveals condensation even when such methods suggest otherwise.