Onsager's Inequality, the Landau-Feynman Ansatz and Superfluidity

TL;DR

This paper rigorously proves Onsager's inequality for a soluble Bose fluid model, reinforcing the theoretical foundation of superfluidity criteria and highlighting the importance of the thermodynamic limit in quantum fluids.

Contribution

It provides a rigorous proof of Onsager's inequality for Girardeau's model, emphasizing the role of the thermodynamic limit and extending the understanding of superfluidity criteria.

Findings

Proves Onsager's inequality for Girardeau's model

Highlights the importance of the thermodynamic limit at nonzero wave vectors

Supports heuristic density functional arguments in superfluidity theory

Abstract

We revisit an inequality due to Onsager, which states that the (quantum) liquid structure factor has an upper bound of the form (const.) x |k|, for not too large modulus of the wave vector k. This inequality implies the validity of the Landau criterion in the theory of superfluidity with a definite, nonzero critical velocity. We prove an auxiliary proposition for general Bose systems, together with which we arrive at a rigorous proof of the inequality for one of the very few soluble examples of an interacting Bose fluid, Girardeau's model. The latter proof demonstrates the importance of the thermodynamic limit of the structure factor, which must be taken initially at k different from 0. It also substantiates very well the heuristic density functional arguments, which are also shown to hold exactly in the limit of large wave-lengths. We also briefly discuss which features of the proof may be present in higher dimensions, as well as some open problems related to superfluidity of trapped gases.