The reversibility paradox in matrix hydrodynamics

Abstract:
The “reversibiliy paradox” refers to the seeming contradiction between two major discoveries made during the end of the 19th century: the law of entropy, due to Boltzmann, and the recurrence theorem, due to Poincaré. The discourse is captured in the following question: how can reversible dynamics give rise to irreversible behaviour?

In this talk, I shall discuss how the same enigma shows up in a finite-dimensional model of the 2-D incompressible Euler equations. The model, suggested by V. Zeitlin in 1991, uses quantization theory to replace the vorticity formulation of the 2-D Euler equations by an isospectral flow of matrices. This way, a spatial discretization preserving all the geometric structure is obtained. Recent advances on Zeitlin’s model propose a connection between the long-time behaviour of (generic) 2-D Euler flows and integrability conditions for “blob dynamics”. A major point at issue, however, is whether simulations based on Zeitlin’s model truly reflect the dynamics of 2-D Euler. It is within this context the reversibility paradox enters.

The work is joint with Milo Viviani.