Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions

TL;DR

This paper clarifies the relationships between complex and simplified variational problems in predicting the equilibrium states of 2D Euler flows, making the theory more practically applicable.

Contribution

It establishes the connections between various variational formulations, justifying the use of simpler models for statistical equilibria of 2D flows.

Findings

Simpler variational problems are justified as equivalent to the original complex formulation.

The relations between different variational approaches are explicitly established.

Simplification facilitates practical analysis of 2D Euler equilibrium states.

Abstract

The Robert-Sommeria-Miller equilibrium statistical mechanics predicts the final organization of two dimensional flows. This powerful theory is difficult to handle practically, due to the complexity associated with an infinite number of constraints. Several alternative simpler variational problems, based on Casimir's or stream function functionals, have been considered recently. We establish the relations between all these variational problems, justifying the use of simpler formulations.