Sampling microcanonical measures of the 2D Euler equations through Creutz's algorithm: a phase transition from disorder to order when energy is increased

TL;DR

This paper demonstrates how Creutz's algorithm can sample microcanonical measures of the 2D Euler equations, revealing a novel phase transition from disorder to order as energy increases, and confirming theoretical predictions.

Contribution

It introduces the use of Creutz's algorithm for sampling 2D Euler microcanonical measures and uncovers a new phase transition from disorder to order with increasing energy.

Findings

Creutz's algorithm successfully samples 2D Euler microcanonical measures.

A first-order phase transition from disordered to ordered states is observed.

The results confirm the mean-field statistical mechanics theory predictions.

Abstract

The 2D Euler equations is the basic example of fluid models for which a microcanical measure can be constructed from first principles. This measure is defined through finite-dimensional approximations and a limiting procedure. Creutz's algorithm is a microcanonical generalization of the Metropolis-Hasting algorithm (to sample Gibbs measures, in the canonical ensemble). We prove that Creutz's algorithm can sample finite-dimensional approximations of the 2D Euler microcanonical measures (incorporating fixed energy and other invariants). This is essential as microcanonical and canonical measures are known to be inequivalent at some values of energy and vorticity distribution. Creutz's algorithm is used to check predictions from the mean-field statistical mechanics theory of the 2D Euler equations (the Robert-Sommeria-Miller theory). We found full agreement with theory. Three different ways to compute the temperature give consistent results. Using Creutz's algorithm, a first-order phase transition never observed previously, and a situation of statistical ensemble inequivalence are found and studied. Strikingly, and contrasting usual statistical mechanics interpretations, this phase transition appears from a disordered phase to an ordered phase (with less symmetries) when energy is increased. We explain this paradox.